testboard: src/FOL/simpdata.ML@540da2415751

     1 (*  Title:      FOL/simpdata.ML

     2     ID:         $Id$

     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     4     Copyright   1994  University of Cambridge

     6 Simplification data for FOL.

     7 *)

    10 (* Elimination of True from asumptions: *)

    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]));

    19 (*** Rewrite rules ***)

    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) ]));

    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)"]);

    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)"]);

    40 bind_thms ("not_simps", map int_prove_fun

    41  ["~(P|Q)  <-> ~P & ~Q",

    42   "~ False <-> True",   "~ True <-> False"]);

    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"]);

    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"]);

    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", "EX x. t=x",

    61   "(EX x. x=t & P(x)) <-> P(t)",

    62   "(EX x. t=x & P(x)) <-> P(t)"]);

    64 (*These are NOT supplied by default!*)

    65 bind_thms ("distrib_simps", map int_prove_fun

    66  ["P & (Q | R) <-> P&Q | P&R",

    67   "(Q | R) & P <-> Q&P | R&P",

    68   "(P | Q --> R) <-> (P --> R) & (Q --> R)"]);

    70 (** Conversion into rewrite rules **)

    72 bind_thm ("P_iff_F", int_prove_fun "~P ==> (P <-> False)");

    73 bind_thm ("iff_reflection_F", P_iff_F RS iff_reflection);

    75 bind_thm ("P_iff_T", int_prove_fun "P ==> (P <-> True)");

    76 bind_thm ("iff_reflection_T", P_iff_T RS iff_reflection);

    78 (*Make meta-equalities.  The operator below is Trueprop*)

    80 fun mk_meta_eq th = case concl_of th of

    81     _ $ (Const("op =",_)$_$_)   => th RS eq_reflection

    82   | _ $ (Const("op <->",_)$_$_) => th RS iff_reflection

    83   | _                           =>

    84   error("conclusion must be a =-equality or <->");;

    86 fun mk_eq th = case concl_of th of

    87     Const("==",_)$_$_           => th

    88   | _ $ (Const("op =",_)$_$_)   => mk_meta_eq th

    89   | _ $ (Const("op <->",_)$_$_) => mk_meta_eq th

    90   | _ $ (Const("Not",_)$_)      => th RS iff_reflection_F

    91   | _                           => th RS iff_reflection_T;

    93 (*Replace premises x=y, X<->Y by X==Y*)

    94 val mk_meta_prems =

    95     rule_by_tactic

    96       (REPEAT_FIRST (resolve_tac [meta_eq_to_obj_eq, def_imp_iff]));

    98 (*Congruence rules for = or <-> (instead of ==)*)

    99 fun mk_meta_cong rl =

   100   standard(mk_meta_eq (mk_meta_prems rl))

   101   handle THM _ =>

   102   error("Premises and conclusion of congruence rules must use =-equality or <->");

   104 val mksimps_pairs =

   105   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),

   106    ("All", [spec]), ("True", []), ("False", [])];

   108 (* ###FIXME: move to simplifier.ML

   109 val mk_atomize:      (string * thm list) list -> thm -> thm list

   110 *)

   111 (* ###FIXME: move to simplifier.ML *)

   112 fun mk_atomize pairs =

   113   let fun atoms th =

   114         (case concl_of th of

   115            Const("Trueprop",_) $ p =>

   116              (case head_of p of

   117                 Const(a,_) =>

   118                   (case AList.lookup (op =) pairs a of

   119                      SOME(rls) => List.concat (map atoms ([th] RL rls))

   120                    | NONE => [th])

   121               | _ => [th])

   122          | _ => [th])

   123   in atoms end;

   125 fun mksimps pairs = (map mk_eq o mk_atomize pairs o gen_all);

   127 (*** Classical laws ***)

   129 fun prove_fun s =

   130  (writeln s;

   131   prove_goal (the_context ()) s

   132    (fn prems => [ (cut_facts_tac prems 1),

   133                   (Cla.fast_tac FOL_cs 1) ]));

   135 (*Avoids duplication of subgoals after expand_if, when the true and false

   136   cases boil down to the same thing.*)

   137 bind_thm ("cases_simp", prove_fun "(P --> Q) & (~P --> Q) <-> Q");

   140 (*** Miniscoping: pushing quantifiers in

   141      We do NOT distribute of ALL over &, or dually that of EX over |

   142      Baaz and Leitsch, On Skolemization and Proof Complexity (1994)

   143      show that this step can increase proof length!

   144 ***)

   146 (*existential miniscoping*)

   147 bind_thms ("int_ex_simps", map int_prove_fun

   148  ["(EX x. P(x) & Q) <-> (EX x. P(x)) & Q",

   149   "(EX x. P & Q(x)) <-> P & (EX x. Q(x))",

   150   "(EX x. P(x) | Q) <-> (EX x. P(x)) | Q",

   151   "(EX x. P | Q(x)) <-> P | (EX x. Q(x))"]);

   153 (*classical rules*)

   154 bind_thms ("cla_ex_simps", map prove_fun

   155  ["(EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q",

   156   "(EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"]);

   158 bind_thms ("ex_simps", int_ex_simps @ cla_ex_simps);

   160 (*universal miniscoping*)

   161 bind_thms ("int_all_simps", map int_prove_fun

   162  ["(ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q",

   163   "(ALL x. P & Q(x)) <-> P & (ALL x. Q(x))",

   164   "(ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q",

   165   "(ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))"]);

   167 (*classical rules*)

   168 bind_thms ("cla_all_simps", map prove_fun

   169  ["(ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q",

   170   "(ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"]);

   172 bind_thms ("all_simps", int_all_simps @ cla_all_simps);

   175 (*** Named rewrite rules proved for IFOL ***)

   177 fun int_prove nm thm  = qed_goal nm IFOL.thy thm

   178     (fn prems => [ (cut_facts_tac prems 1),

   179                    (IntPr.fast_tac 1) ]);

   181 fun prove nm thm  = qed_goal nm (the_context ()) thm (fn _ => [Blast_tac 1]);

   183 int_prove "conj_commute" "P&Q <-> Q&P";

   184 int_prove "conj_left_commute" "P&(Q&R) <-> Q&(P&R)";

   185 bind_thms ("conj_comms", [conj_commute, conj_left_commute]);

   187 int_prove "disj_commute" "P|Q <-> Q|P";

   188 int_prove "disj_left_commute" "P|(Q|R) <-> Q|(P|R)";

   189 bind_thms ("disj_comms", [disj_commute, disj_left_commute]);

   191 int_prove "conj_disj_distribL" "P&(Q|R) <-> (P&Q | P&R)";

   192 int_prove "conj_disj_distribR" "(P|Q)&R <-> (P&R | Q&R)";

   194 int_prove "disj_conj_distribL" "P|(Q&R) <-> (P|Q) & (P|R)";

   195 int_prove "disj_conj_distribR" "(P&Q)|R <-> (P|R) & (Q|R)";

   197 int_prove "imp_conj_distrib" "(P --> (Q&R)) <-> (P-->Q) & (P-->R)";

   198 int_prove "imp_conj"         "((P&Q)-->R)   <-> (P --> (Q --> R))";

   199 int_prove "imp_disj"         "(P|Q --> R)   <-> (P-->R) & (Q-->R)";

   201 prove "imp_disj1" "(P-->Q) | R <-> (P-->Q | R)";

   202 prove "imp_disj2" "Q | (P-->R) <-> (P-->Q | R)";

   204 int_prove "de_Morgan_disj" "(~(P | Q)) <-> (~P & ~Q)";

   205 prove     "de_Morgan_conj" "(~(P & Q)) <-> (~P | ~Q)";

   207 prove     "not_imp" "~(P --> Q) <-> (P & ~Q)";

   208 prove     "not_iff" "~(P <-> Q) <-> (P <-> ~Q)";

   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)";

   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)))";

   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]);

   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])

   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

   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);

   267 end;

   269 val defEX_regroup =

   270   Simplifier.simproc (the_context ())

   271     "defined EX" ["EX x. P(x)"] Quantifier1.rearrange_ex;

   273 val defALL_regroup =

   274   Simplifier.simproc (the_context ())

   275     "defined ALL" ["ALL x. P(x)"] Quantifier1.rearrange_all;

   278 (*** Case splitting ***)

   280 bind_thm ("meta_eq_to_iff", prove_goal IFOL.thy "x==y ==> x<->y"

   281   (fn [prem] => [rewtac prem, rtac iffI 1, atac 1, atac 1]));

   283 structure SplitterData =

   284   struct

   285   structure Simplifier = Simplifier

   286   val mk_eq          = mk_eq

   287   val meta_eq_to_iff = meta_eq_to_iff

   288   val iffD           = iffD2

   289   val disjE          = disjE

   290   val conjE          = conjE

   291   val exE            = exE

   292   val contrapos      = contrapos

   293   val contrapos2     = contrapos2

   294   val notnotD        = notnotD

   295   end;

   297 structure Splitter = SplitterFun(SplitterData);

   299 val split_tac        = Splitter.split_tac;

   300 val split_inside_tac = Splitter.split_inside_tac;

   301 val split_asm_tac    = Splitter.split_asm_tac;

   302 val op addsplits     = Splitter.addsplits;

   303 val op delsplits     = Splitter.delsplits;

   304 val Addsplits        = Splitter.Addsplits;

   305 val Delsplits        = Splitter.Delsplits;

   308 (*** Standard simpsets ***)

   310 structure Induction = InductionFun(struct val spec=IFOL.spec end);

   312 open Induction;

   315 bind_thms ("meta_simps",

   316  [triv_forall_equality,   (* prunes params *)

   317   True_implies_equals]);  (* prune asms `True' *)

   319 bind_thms ("IFOL_simps",

   320  [refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @

   321   imp_simps @ iff_simps @ quant_simps);

   323 bind_thm ("notFalseI", int_prove_fun "~False");

   324 bind_thms ("triv_rls", [TrueI,refl,reflexive_thm,iff_refl,notFalseI]);

   326 fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems),

   327                                  atac, etac FalseE];

   328 (*No premature instantiation of variables during simplification*)

   329 fun   safe_solver prems = FIRST'[match_tac (triv_rls@prems),

   330                                  eq_assume_tac, ematch_tac [FalseE]];

   332 (*No simprules, but basic infastructure for simplification*)

   333 val FOL_basic_ss =

   334   Simplifier.theory_context (the_context ()) empty_ss

   335   setsubgoaler asm_simp_tac

   336   setSSolver (mk_solver "FOL safe" safe_solver)

   337   setSolver (mk_solver "FOL unsafe" unsafe_solver)

   338   setmksimps (mksimps mksimps_pairs)

   339   setmkcong mk_meta_cong;

   341 fun unfold_tac ths =

   342   let val ss0 = Simplifier.clear_ss FOL_basic_ss addsimps ths

   343   in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end;

   346 (*intuitionistic simprules only*)

   347 val IFOL_ss = FOL_basic_ss

   348   addsimps (meta_simps @ IFOL_simps @ int_ex_simps @ int_all_simps)

   349   addsimprocs [defALL_regroup, defEX_regroup]

   350   addcongs [imp_cong];

   352 bind_thms ("cla_simps",

   353   [de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2,

   354    not_imp, not_all, not_ex, cases_simp] @

   355   map prove_fun

   356    ["~(P&Q) <-> ~P | ~Q",

   357     "P | ~P",             "~P | P",

   358     "~ ~ P <-> P",        "(~P --> P) <-> P",

   359     "(~P <-> ~Q) <-> (P<->Q)"]);

   361 (*classical simprules too*)

   362 val FOL_ss = IFOL_ss addsimps (cla_simps @ cla_ex_simps @ cla_all_simps);

   364 val simpsetup = [fn thy => (change_simpset_of thy (fn _ => FOL_ss); thy)];

   367 (*** integration of simplifier with classical reasoner ***)

   369 structure Clasimp = ClasimpFun

   370  (structure Simplifier = Simplifier and Splitter = Splitter

   371   and Classical  = Cla and Blast = Blast

   372   val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE);

   373 open Clasimp;

   375 val FOL_css = (FOL_cs, FOL_ss);

author	wenzelm
	Sat Dec 31 21:49:36 2005 +0100 (2005-12-31 ago)
changeset 18529	540da2415751
parent 18324	d1c4b1112e33
child 18708	4b3dadb4fe33
permissions	-rw-r--r--