src/HOL/simpdata.ML
author haftmann
Tue Oct 10 10:35:24 2006 +0200 (2006-10-10)
changeset 20944 34b2c1bb7178
parent 20932 e65e1234c9d3
child 20973 0b8e436ed071
permissions -rw-r--r--
cleanup basic HOL bootstrap
     1 (*  Title:      HOL/simpdata.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1991  University of Cambridge
     5 
     6 Instantiation of the generic simplifier for HOL.
     7 *)
     8 
     9 (* legacy ML bindings - FIXME get rid of this *)
    10 
    11 val Eq_FalseI = thm "Eq_FalseI";
    12 val Eq_TrueI = thm "Eq_TrueI";
    13 val de_Morgan_conj = thm "de_Morgan_conj";
    14 val de_Morgan_disj = thm "de_Morgan_disj";
    15 val iff_conv_conj_imp = thm "iff_conv_conj_imp";
    16 val imp_cong = thm "imp_cong";
    17 val imp_conv_disj = thm "imp_conv_disj";
    18 val imp_disj1 = thm "imp_disj1";
    19 val imp_disj2 = thm "imp_disj2";
    20 val imp_disjL = thm "imp_disjL";
    21 val simp_impliesI = thm "simp_impliesI";
    22 val simp_implies_cong = thm "simp_implies_cong";
    23 val simp_implies_def = thm "simp_implies_def";
    24 
    25 local
    26   val uncurry = thm "uncurry"
    27   val iff_allI = thm "iff_allI"
    28   val iff_exI = thm "iff_exI"
    29   val all_comm = thm "all_comm"
    30   val ex_comm = thm "ex_comm"
    31 in
    32 
    33 (*** make simplification procedures for quantifier elimination ***)
    34 
    35 structure Quantifier1 = Quantifier1Fun
    36 (struct
    37   (*abstract syntax*)
    38   fun dest_eq((c as Const("op =",_)) $ s $ t) = SOME(c,s,t)
    39     | dest_eq _ = NONE;
    40   fun dest_conj((c as Const("op &",_)) $ s $ t) = SOME(c,s,t)
    41     | dest_conj _ = NONE;
    42   fun dest_imp((c as Const("op -->",_)) $ s $ t) = SOME(c,s,t)
    43     | dest_imp _ = NONE;
    44   val conj = HOLogic.conj
    45   val imp  = HOLogic.imp
    46   (*rules*)
    47   val iff_reflection = HOL.eq_reflection
    48   val iffI = HOL.iffI
    49   val iff_trans = HOL.trans
    50   val conjI= HOL.conjI
    51   val conjE= HOL.conjE
    52   val impI = HOL.impI
    53   val mp   = HOL.mp
    54   val uncurry = uncurry
    55   val exI  = HOL.exI
    56   val exE  = HOL.exE
    57   val iff_allI = iff_allI
    58   val iff_exI = iff_exI
    59   val all_comm = all_comm
    60   val ex_comm = ex_comm
    61 end);
    62 
    63 end;
    64 
    65 val defEX_regroup =
    66   Simplifier.simproc (the_context ())
    67     "defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
    68 
    69 val defALL_regroup =
    70   Simplifier.simproc (the_context ())
    71     "defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
    72 
    73 
    74 (* simproc for proving "(y = x) == False" from premise "~(x = y)" *)
    75 
    76 val use_neq_simproc = ref true;
    77 
    78 local
    79   val neq_to_EQ_False = thm "not_sym" RS Eq_FalseI;
    80   fun neq_prover sg ss (eq $ lhs $ rhs) =
    81     let
    82       fun test thm = (case #prop (rep_thm thm) of
    83                     _ $ (Not $ (eq' $ l' $ r')) =>
    84                       Not = HOLogic.Not andalso eq' = eq andalso
    85                       r' aconv lhs andalso l' aconv rhs
    86                   | _ => false)
    87     in if !use_neq_simproc then case find_first test (prems_of_ss ss)
    88      of NONE => NONE
    89       | SOME thm => SOME (thm RS neq_to_EQ_False)
    90      else NONE
    91     end
    92 in
    93 
    94 val neq_simproc = Simplifier.simproc (the_context ())
    95   "neq_simproc" ["x = y"] neq_prover;
    96 
    97 end;
    98 
    99 
   100 (* Simproc for Let *)
   101 
   102 val use_let_simproc = ref true;
   103 
   104 local
   105   val Let_folded = thm "Let_folded";
   106   val Let_unfold = thm "Let_unfold";
   107   val (f_Let_unfold,x_Let_unfold) =
   108       let val [(_$(f$x)$_)] = prems_of Let_unfold
   109       in (cterm_of (the_context ()) f,cterm_of (the_context ()) x) end
   110   val (f_Let_folded,x_Let_folded) =
   111       let val [(_$(f$x)$_)] = prems_of Let_folded
   112       in (cterm_of (the_context ()) f, cterm_of (the_context ()) x) end;
   113   val g_Let_folded =
   114       let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of (the_context ()) g end;
   115 in
   116 
   117 val let_simproc =
   118   Simplifier.simproc (the_context ()) "let_simp" ["Let x f"]
   119    (fn sg => fn ss => fn t =>
   120      let val ctxt = Simplifier.the_context ss;
   121          val ([t'],ctxt') = Variable.import_terms false [t] ctxt;
   122      in Option.map (hd o Variable.export ctxt' ctxt o single)
   123       (case t' of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
   124          if not (!use_let_simproc) then NONE
   125          else if is_Free x orelse is_Bound x orelse is_Const x
   126          then SOME (thm "Let_def")
   127          else
   128           let
   129              val n = case f of (Abs (x,_,_)) => x | _ => "x";
   130              val cx = cterm_of sg x;
   131              val {T=xT,...} = rep_cterm cx;
   132              val cf = cterm_of sg f;
   133              val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
   134              val (_$_$g) = prop_of fx_g;
   135              val g' = abstract_over (x,g);
   136            in (if (g aconv g')
   137                then
   138                   let
   139                     val rl = cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] Let_unfold;
   140                   in SOME (rl OF [fx_g]) end
   141                else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
   142                else let
   143                      val abs_g'= Abs (n,xT,g');
   144                      val g'x = abs_g'$x;
   145                      val g_g'x = symmetric (beta_conversion false (cterm_of sg g'x));
   146                      val rl = cterm_instantiate
   147                                [(f_Let_folded,cterm_of sg f),(x_Let_folded,cx),
   148                                 (g_Let_folded,cterm_of sg abs_g')]
   149                                Let_folded;
   150                    in SOME (rl OF [transitive fx_g g_g'x])
   151                    end)
   152            end
   153         | _ => NONE)
   154      end)
   155 
   156 end
   157 
   158 (*** Case splitting ***)
   159 
   160 (*Make meta-equalities.  The operator below is Trueprop*)
   161 
   162 fun mk_meta_eq r = r RS HOL.eq_reflection;
   163 fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
   164 
   165 fun mk_eq th = case concl_of th of
   166         Const("==",_)$_$_       => th
   167     |   _$(Const("op =",_)$_$_) => mk_meta_eq th
   168     |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
   169     |   _                       => th RS Eq_TrueI;
   170 (* Expects Trueprop(.) if not == *)
   171 
   172 fun mk_eq_True r =
   173   SOME (r RS HOL.meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => NONE;
   174 
   175 (* Produce theorems of the form
   176   (P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
   177 *)
   178 fun lift_meta_eq_to_obj_eq i st =
   179   let
   180     val {sign, ...} = rep_thm st;
   181     fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q
   182       | count_imp _ = 0;
   183     val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1)))
   184   in if j = 0 then HOL.meta_eq_to_obj_eq
   185     else
   186       let
   187         val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
   188         fun mk_simp_implies Q = foldr (fn (R, S) =>
   189           Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps
   190         val aT = TFree ("'a", HOLogic.typeS);
   191         val x = Free ("x", aT);
   192         val y = Free ("y", aT)
   193       in Goal.prove_global (Thm.theory_of_thm st) []
   194         [mk_simp_implies (Logic.mk_equals (x, y))]
   195         (mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))))
   196         (fn prems => EVERY
   197          [rewrite_goals_tac [simp_implies_def],
   198           REPEAT (ares_tac (HOL.meta_eq_to_obj_eq :: map (rewrite_rule [simp_implies_def]) prems) 1)])
   199       end
   200   end;
   201 
   202 (*Congruence rules for = (instead of ==)*)
   203 fun mk_meta_cong rl = zero_var_indexes
   204   (let val rl' = Seq.hd (TRYALL (fn i => fn st =>
   205      rtac (lift_meta_eq_to_obj_eq i st) i st) rl)
   206    in mk_meta_eq rl' handle THM _ =>
   207      if can Logic.dest_equals (concl_of rl') then rl'
   208      else error "Conclusion of congruence rules must be =-equality"
   209    end);
   210 
   211 structure SplitterData =
   212 struct
   213   structure Simplifier = Simplifier
   214   val mk_eq          = mk_eq
   215   val meta_eq_to_iff = HOL.meta_eq_to_obj_eq
   216   val iffD           = HOL.iffD2
   217   val disjE          = HOL.disjE
   218   val conjE          = HOL.conjE
   219   val exE            = HOL.exE
   220   val contrapos      = HOL.contrapos_nn
   221   val contrapos2     = HOL.contrapos_pp
   222   val notnotD        = HOL.notnotD
   223 end;
   224 
   225 structure Splitter = SplitterFun(SplitterData);
   226 
   227 val split_tac        = Splitter.split_tac;
   228 val split_inside_tac = Splitter.split_inside_tac;
   229 val split_asm_tac    = Splitter.split_asm_tac;
   230 val op addsplits     = Splitter.addsplits;
   231 val op delsplits     = Splitter.delsplits;
   232 val Addsplits        = Splitter.Addsplits;
   233 val Delsplits        = Splitter.Delsplits;
   234 
   235 val mksimps_pairs =
   236   [("op -->", [HOL.mp]), ("op &", [thm "conjunct1", thm "conjunct2"]),
   237    ("All", [HOL.spec]), ("True", []), ("False", []),
   238    ("HOL.If", [thm "if_bool_eq_conj" RS HOL.iffD1])];
   239 
   240 (*
   241 val mk_atomize:      (string * thm list) list -> thm -> thm list
   242 looks too specific to move it somewhere else
   243 *)
   244 fun mk_atomize pairs =
   245   let fun atoms th =
   246         (case concl_of th of
   247            Const("Trueprop",_) $ p =>
   248              (case head_of p of
   249                 Const(a,_) =>
   250                   (case AList.lookup (op =) pairs a of
   251                      SOME(rls) => List.concat (map atoms ([th] RL rls))
   252                    | NONE => [th])
   253               | _ => [th])
   254          | _ => [th])
   255   in atoms end;
   256 
   257 fun mksimps pairs =
   258   (List.mapPartial (try mk_eq) o mk_atomize pairs o gen_all);
   259 
   260 fun unsafe_solver_tac prems =
   261   (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
   262   FIRST'[resolve_tac(reflexive_thm :: HOL.TrueI :: HOL.refl :: prems), atac, etac HOL.FalseE];
   263 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
   264 
   265 (*No premature instantiation of variables during simplification*)
   266 fun safe_solver_tac prems =
   267   (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
   268   FIRST'[match_tac(reflexive_thm :: HOL.TrueI :: HOL.refl :: prems),
   269          eq_assume_tac, ematch_tac [HOL.FalseE]];
   270 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
   271 
   272 val HOL_basic_ss =
   273   Simplifier.theory_context (the_context ()) empty_ss
   274     setsubgoaler asm_simp_tac
   275     setSSolver safe_solver
   276     setSolver unsafe_solver
   277     setmksimps (mksimps mksimps_pairs)
   278     setmkeqTrue mk_eq_True
   279     setmkcong mk_meta_cong;
   280 
   281 fun unfold_tac ths =
   282   let val ss0 = Simplifier.clear_ss HOL_basic_ss addsimps ths
   283   in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end;
   284 
   285 (*In general it seems wrong to add distributive laws by default: they
   286   might cause exponential blow-up.  But imp_disjL has been in for a while
   287   and cannot be removed without affecting existing proofs.  Moreover,
   288   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   289   grounds that it allows simplification of R in the two cases.*)
   290 
   291 local
   292   val ex_simps = thms "ex_simps";
   293   val all_simps = thms "all_simps";
   294   val simp_thms = thms "simp_thms";
   295   val cases_simp = thm "cases_simp";
   296   val conj_assoc = thm "conj_assoc";
   297   val if_False = thm "if_False";
   298   val if_True = thm "if_True";
   299   val disj_assoc = thm "disj_assoc";
   300   val disj_not1 = thm "disj_not1";
   301   val if_cancel = thm "if_cancel";
   302   val if_eq_cancel = thm "if_eq_cancel";
   303   val True_implies_equals = thm "True_implies_equals";
   304 in
   305 
   306 val HOL_ss =
   307     HOL_basic_ss addsimps
   308      ([triv_forall_equality, (* prunes params *)
   309        True_implies_equals, (* prune asms `True' *)
   310        if_True, if_False, if_cancel, if_eq_cancel,
   311        imp_disjL, conj_assoc, disj_assoc,
   312        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, thm "not_imp",
   313        disj_not1, thm "not_all", thm "not_ex", cases_simp,
   314        thm "the_eq_trivial", HOL.the_sym_eq_trivial]
   315      @ ex_simps @ all_simps @ simp_thms)
   316      addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc]
   317      addcongs [imp_cong, simp_implies_cong]
   318      addsplits [thm "split_if"];
   319 
   320 end;
   321 
   322 fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
   323 
   324 (* default simpset *)
   325 val simpsetup =
   326   (fn thy => (change_simpset_of thy (fn _ => HOL_ss); thy));
   327 
   328 
   329 (*** integration of simplifier with classical reasoner ***)
   330 
   331 structure Clasimp = ClasimpFun
   332  (structure Simplifier = Simplifier and Splitter = Splitter
   333   and Classical  = Classical and Blast = Blast
   334   val iffD1 = HOL.iffD1 val iffD2 = HOL.iffD2 val notE = HOL.notE);
   335 open Clasimp;
   336 
   337 val HOL_css = (HOL_cs, HOL_ss);
   338 
   339 
   340 
   341 (*** A general refutation procedure ***)
   342 
   343 (* Parameters:
   344 
   345    test: term -> bool
   346    tests if a term is at all relevant to the refutation proof;
   347    if not, then it can be discarded. Can improve performance,
   348    esp. if disjunctions can be discarded (no case distinction needed!).
   349 
   350    prep_tac: int -> tactic
   351    A preparation tactic to be applied to the goal once all relevant premises
   352    have been moved to the conclusion.
   353 
   354    ref_tac: int -> tactic
   355    the actual refutation tactic. Should be able to deal with goals
   356    [| A1; ...; An |] ==> False
   357    where the Ai are atomic, i.e. no top-level &, | or EX
   358 *)
   359 
   360 local
   361   val nnf_simpset =
   362     empty_ss setmkeqTrue mk_eq_True
   363     setmksimps (mksimps mksimps_pairs)
   364     addsimps [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
   365       thm "not_all", thm "not_ex", thm "not_not"];
   366   fun prem_nnf_tac i st =
   367     full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
   368 in
   369 fun refute_tac test prep_tac ref_tac =
   370   let val refute_prems_tac =
   371         REPEAT_DETERM
   372               (eresolve_tac [HOL.conjE, HOL.exE] 1 ORELSE
   373                filter_prems_tac test 1 ORELSE
   374                etac HOL.disjE 1) THEN
   375         ((etac HOL.notE 1 THEN eq_assume_tac 1) ORELSE
   376          ref_tac 1);
   377   in EVERY'[TRY o filter_prems_tac test,
   378             REPEAT_DETERM o etac HOL.rev_mp, prep_tac, rtac HOL.ccontr, prem_nnf_tac,
   379             SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
   380   end;
   381 end;