src/HOL/simpdata.ML
author wenzelm
Tue Oct 18 17:59:25 2005 +0200 (2005-10-18)
changeset 17892 62c397c17d18
parent 17875 d81094515061
child 17959 8db36a108213
permissions -rw-r--r--
Simplifier.theory_context;
     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 *)
    10 
    11 val Eq_FalseI = thm "Eq_FalseI";
    12 val Eq_TrueI = thm "Eq_TrueI";
    13 val all_conj_distrib = thm "all_conj_distrib";
    14 val all_simps = thms "all_simps";
    15 val cases_simp = thm "cases_simp";
    16 val conj_assoc = thm "conj_assoc";
    17 val conj_comms = thms "conj_comms";
    18 val conj_commute = thm "conj_commute";
    19 val conj_cong = thm "conj_cong";
    20 val conj_disj_distribL = thm "conj_disj_distribL";
    21 val conj_disj_distribR = thm "conj_disj_distribR";
    22 val conj_left_commute = thm "conj_left_commute";
    23 val de_Morgan_conj = thm "de_Morgan_conj";
    24 val de_Morgan_disj = thm "de_Morgan_disj";
    25 val disj_assoc = thm "disj_assoc";
    26 val disj_comms = thms "disj_comms";
    27 val disj_commute = thm "disj_commute";
    28 val disj_cong = thm "disj_cong";
    29 val disj_conj_distribL = thm "disj_conj_distribL";
    30 val disj_conj_distribR = thm "disj_conj_distribR";
    31 val disj_left_commute = thm "disj_left_commute";
    32 val disj_not1 = thm "disj_not1";
    33 val disj_not2 = thm "disj_not2";
    34 val eq_ac = thms "eq_ac";
    35 val eq_assoc = thm "eq_assoc";
    36 val eq_commute = thm "eq_commute";
    37 val eq_left_commute = thm "eq_left_commute";
    38 val eq_sym_conv = thm "eq_sym_conv";
    39 val eta_contract_eq = thm "eta_contract_eq";
    40 val ex_disj_distrib = thm "ex_disj_distrib";
    41 val ex_simps = thms "ex_simps";
    42 val if_False = thm "if_False";
    43 val if_P = thm "if_P";
    44 val if_True = thm "if_True";
    45 val if_bool_eq_conj = thm "if_bool_eq_conj";
    46 val if_bool_eq_disj = thm "if_bool_eq_disj";
    47 val if_cancel = thm "if_cancel";
    48 val if_def2 = thm "if_def2";
    49 val if_eq_cancel = thm "if_eq_cancel";
    50 val if_not_P = thm "if_not_P";
    51 val if_splits = thms "if_splits";
    52 val iff_conv_conj_imp = thm "iff_conv_conj_imp";
    53 val imp_all = thm "imp_all";
    54 val imp_cong = thm "imp_cong";
    55 val imp_conjL = thm "imp_conjL";
    56 val imp_conjR = thm "imp_conjR";
    57 val imp_conv_disj = thm "imp_conv_disj";
    58 val imp_disj1 = thm "imp_disj1";
    59 val imp_disj2 = thm "imp_disj2";
    60 val imp_disjL = thm "imp_disjL";
    61 val imp_disj_not1 = thm "imp_disj_not1";
    62 val imp_disj_not2 = thm "imp_disj_not2";
    63 val imp_ex = thm "imp_ex";
    64 val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
    65 val neq_commute = thm "neq_commute";
    66 val not_all = thm "not_all";
    67 val not_ex = thm "not_ex";
    68 val not_iff = thm "not_iff";
    69 val not_imp = thm "not_imp";
    70 val not_not = thm "not_not";
    71 val rev_conj_cong = thm "rev_conj_cong";
    72 val simp_impliesE = thm "simp_impliesI";
    73 val simp_impliesI = thm "simp_impliesI";
    74 val simp_implies_cong = thm "simp_implies_cong";
    75 val simp_implies_def = thm "simp_implies_def";
    76 val simp_thms = thms "simp_thms";
    77 val split_if = thm "split_if";
    78 val split_if_asm = thm "split_if_asm";
    79 val atomize_not = thm"atomize_not";
    80 
    81 local
    82 val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
    83               (fn prems => [cut_facts_tac prems 1, Blast_tac 1]);
    84 
    85 val iff_allI = allI RS
    86     prove_goal (the_context()) "!x. P x = Q x ==> (!x. P x) = (!x. Q x)"
    87                (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
    88 val iff_exI = allI RS
    89     prove_goal (the_context()) "!x. P x = Q x ==> (? x. P x) = (? x. Q x)"
    90                (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
    91 
    92 val all_comm = prove_goal (the_context()) "(!x y. P x y) = (!y x. P x y)"
    93                (fn _ => [Blast_tac 1])
    94 val ex_comm = prove_goal (the_context()) "(? x y. P x y) = (? y x. P x y)"
    95                (fn _ => [Blast_tac 1])
    96 in
    97 
    98 (*** make simplification procedures for quantifier elimination ***)
    99 
   100 structure Quantifier1 = Quantifier1Fun
   101 (struct
   102   (*abstract syntax*)
   103   fun dest_eq((c as Const("op =",_)) $ s $ t) = SOME(c,s,t)
   104     | dest_eq _ = NONE;
   105   fun dest_conj((c as Const("op &",_)) $ s $ t) = SOME(c,s,t)
   106     | dest_conj _ = NONE;
   107   fun dest_imp((c as Const("op -->",_)) $ s $ t) = SOME(c,s,t)
   108     | dest_imp _ = NONE;
   109   val conj = HOLogic.conj
   110   val imp  = HOLogic.imp
   111   (*rules*)
   112   val iff_reflection = eq_reflection
   113   val iffI = iffI
   114   val iff_trans = trans
   115   val conjI= conjI
   116   val conjE= conjE
   117   val impI = impI
   118   val mp   = mp
   119   val uncurry = uncurry
   120   val exI  = exI
   121   val exE  = exE
   122   val iff_allI = iff_allI
   123   val iff_exI = iff_exI
   124   val all_comm = all_comm
   125   val ex_comm = ex_comm
   126 end);
   127 
   128 end;
   129 
   130 val defEX_regroup =
   131   Simplifier.simproc (Theory.sign_of (the_context ()))
   132     "defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
   133 
   134 val defALL_regroup =
   135   Simplifier.simproc (Theory.sign_of (the_context ()))
   136     "defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
   137 
   138 
   139 (*** simproc for proving "(y = x) == False" from prmise "~(x = y)" ***)
   140 
   141 val use_neq_simproc = ref true;
   142 
   143 local
   144 
   145 val neq_to_EQ_False = thm "not_sym" RS Eq_FalseI;
   146 
   147 fun neq_prover sg ss (eq $ lhs $ rhs) =
   148 let
   149   fun test thm = (case #prop(rep_thm thm) of
   150                     _ $ (Not $ (eq' $ l' $ r')) =>
   151                       Not = HOLogic.Not andalso eq' = eq andalso
   152                       r' aconv lhs andalso l' aconv rhs
   153                   | _ => false)
   154 in
   155   if !use_neq_simproc then
   156     case Library.find_first test (prems_of_ss ss) of NONE => NONE
   157     | SOME thm => SOME (thm RS neq_to_EQ_False)
   158   else NONE
   159 end
   160 
   161 in
   162 
   163 val neq_simproc =
   164   Simplifier.simproc (the_context ()) "neq_simproc" ["x = y"] neq_prover;
   165 
   166 end;
   167 
   168 
   169 
   170 
   171 (*** Simproc for Let ***)
   172 
   173 val use_let_simproc = ref true;
   174 
   175 local
   176 val Let_folded = thm "Let_folded";
   177 val Let_unfold = thm "Let_unfold";
   178 
   179 val f_Let_unfold = 
   180       let val [(_$(f$_)$_)] = prems_of Let_unfold in cterm_of (sign_of (the_context ())) f end
   181 val f_Let_folded = 
   182       let val [(_$(f$_)$_)] = prems_of Let_folded in cterm_of (sign_of (the_context ())) f end;
   183 val g_Let_folded = 
   184       let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of (sign_of (the_context ())) g end;
   185 in
   186 val let_simproc =
   187   Simplifier.simproc (Theory.sign_of (the_context ())) "let_simp" ["Let x f"] 
   188    (fn sg => fn ss => fn t =>
   189       (case t of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
   190          if not (!use_let_simproc) then NONE
   191          else if is_Free x orelse is_Bound x orelse is_Const x 
   192          then SOME Let_def  
   193          else
   194           let
   195              val n = case f of (Abs (x,_,_)) => x | _ => "x";
   196              val cx = cterm_of sg x;
   197              val {T=xT,...} = rep_cterm cx;
   198              val cf = cterm_of sg f;
   199              val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
   200              val (_$_$g) = prop_of fx_g;
   201              val g' = abstract_over (x,g);
   202            in (if (g aconv g') 
   203                then
   204                   let
   205                     val rl = cterm_instantiate [(f_Let_unfold,cf)] Let_unfold;
   206                   in SOME (rl OF [fx_g]) end 
   207                else if betapply (f,x) aconv g then NONE (* avoid identity conversion *)
   208                else let 
   209                      val abs_g'= Abs (n,xT,g');
   210                      val g'x = abs_g'$x;
   211                      val g_g'x = symmetric (beta_conversion false (cterm_of sg g'x));
   212                      val rl = cterm_instantiate
   213                                [(f_Let_folded,cterm_of sg f),
   214                                 (g_Let_folded,cterm_of sg abs_g')]
   215                                Let_folded; 
   216                    in SOME (rl OF [transitive fx_g g_g'x]) end)
   217            end
   218         | _ => NONE))
   219 end
   220 
   221 (*** Case splitting ***)
   222 
   223 (*Make meta-equalities.  The operator below is Trueprop*)
   224 
   225 fun mk_meta_eq r = r RS eq_reflection;
   226 fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
   227 
   228 fun mk_eq th = case concl_of th of
   229         Const("==",_)$_$_       => th
   230     |   _$(Const("op =",_)$_$_) => mk_meta_eq th
   231     |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
   232     |   _                       => th RS Eq_TrueI;
   233 (* Expects Trueprop(.) if not == *)
   234 
   235 fun mk_eq_True r =
   236   SOME (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => NONE;
   237 
   238 (* Produce theorems of the form
   239   (P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
   240 *)
   241 fun lift_meta_eq_to_obj_eq i st =
   242   let
   243     val {sign, ...} = rep_thm st;
   244     fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q
   245       | count_imp _ = 0;
   246     val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1)))
   247   in if j = 0 then meta_eq_to_obj_eq
   248     else
   249       let
   250         val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
   251         fun mk_simp_implies Q = foldr (fn (R, S) =>
   252           Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps
   253         val aT = TFree ("'a", HOLogic.typeS);
   254         val x = Free ("x", aT);
   255         val y = Free ("y", aT)
   256       in prove_goalw_cterm [simp_implies_def]
   257         (cterm_of sign (Logic.mk_implies
   258           (mk_simp_implies (Logic.mk_equals (x, y)),
   259            mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))))))
   260         (fn hyps => [REPEAT (ares_tac (meta_eq_to_obj_eq :: hyps) 1)])
   261       end
   262   end;
   263   
   264 (*Congruence rules for = (instead of ==)*)
   265 fun mk_meta_cong rl = zero_var_indexes
   266   (let val rl' = Seq.hd (TRYALL (fn i => fn st =>
   267      rtac (lift_meta_eq_to_obj_eq i st) i st) rl)
   268    in mk_meta_eq rl' handle THM _ =>
   269      if Logic.is_equals (concl_of rl') then rl'
   270      else error "Conclusion of congruence rules must be =-equality"
   271    end);
   272 
   273 (* Elimination of True from asumptions: *)
   274 
   275 local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
   276 in val True_implies_equals = standard' (equal_intr
   277   (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
   278   (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
   279 end;
   280 
   281 
   282 structure SplitterData =
   283   struct
   284   structure Simplifier = Simplifier
   285   val mk_eq          = mk_eq
   286   val meta_eq_to_iff = meta_eq_to_obj_eq
   287   val iffD           = iffD2
   288   val disjE          = disjE
   289   val conjE          = conjE
   290   val exE            = exE
   291   val contrapos      = contrapos_nn
   292   val contrapos2     = contrapos_pp
   293   val notnotD        = notnotD
   294   end;
   295 
   296 structure Splitter = SplitterFun(SplitterData);
   297 
   298 val split_tac        = Splitter.split_tac;
   299 val split_inside_tac = Splitter.split_inside_tac;
   300 val split_asm_tac    = Splitter.split_asm_tac;
   301 val op addsplits     = Splitter.addsplits;
   302 val op delsplits     = Splitter.delsplits;
   303 val Addsplits        = Splitter.Addsplits;
   304 val Delsplits        = Splitter.Delsplits;
   305 
   306 val mksimps_pairs =
   307   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   308    ("All", [spec]), ("True", []), ("False", []),
   309    ("HOL.If", [if_bool_eq_conj RS iffD1])];
   310 
   311 (*
   312 val mk_atomize:      (string * thm list) list -> thm -> thm list
   313 looks too specific to move it somewhere else
   314 *)
   315 fun mk_atomize pairs =
   316   let fun atoms th =
   317         (case concl_of th of
   318            Const("Trueprop",_) $ p =>
   319              (case head_of p of
   320                 Const(a,_) =>
   321                   (case AList.lookup (op =) pairs a of
   322                      SOME(rls) => List.concat (map atoms ([th] RL rls))
   323                    | NONE => [th])
   324               | _ => [th])
   325          | _ => [th])
   326   in atoms end;
   327 
   328 fun mksimps pairs =
   329   (List.mapPartial (try mk_eq) o mk_atomize pairs o gen_all);
   330 
   331 fun unsafe_solver_tac prems =
   332   (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
   333   FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
   334 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
   335 
   336 (*No premature instantiation of variables during simplification*)
   337 fun safe_solver_tac prems =
   338   (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
   339   FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
   340          eq_assume_tac, ematch_tac [FalseE]];
   341 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
   342 
   343 val HOL_basic_ss =
   344   Simplifier.theory_context (the_context ()) empty_ss
   345     setsubgoaler asm_simp_tac
   346     setSSolver safe_solver
   347     setSolver unsafe_solver
   348     setmksimps (mksimps mksimps_pairs)
   349     setmkeqTrue mk_eq_True
   350     setmkcong mk_meta_cong;
   351 
   352 fun unfold_tac ss ths =
   353   ALLGOALS (full_simp_tac
   354     (Simplifier.inherit_context ss (Simplifier.clear_ss HOL_basic_ss) addsimps ths));
   355 
   356 (*In general it seems wrong to add distributive laws by default: they
   357   might cause exponential blow-up.  But imp_disjL has been in for a while
   358   and cannot be removed without affecting existing proofs.  Moreover,
   359   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   360   grounds that it allows simplification of R in the two cases.*)
   361 
   362 val HOL_ss =
   363     HOL_basic_ss addsimps
   364      ([triv_forall_equality, (* prunes params *)
   365        True_implies_equals, (* prune asms `True' *)
   366        eta_contract_eq, (* prunes eta-expansions *)
   367        if_True, if_False, if_cancel, if_eq_cancel,
   368        imp_disjL, conj_assoc, disj_assoc,
   369        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
   370        disj_not1, not_all, not_ex, cases_simp,
   371        thm "the_eq_trivial", the_sym_eq_trivial]
   372      @ ex_simps @ all_simps @ simp_thms)
   373      addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc]
   374      addcongs [imp_cong, simp_implies_cong]
   375      addsplits [split_if];
   376 
   377 fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
   378 
   379 
   380 (*Simplifies x assuming c and y assuming ~c*)
   381 val prems = Goalw [if_def]
   382   "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
   383 \  (if b then x else y) = (if c then u else v)";
   384 by (asm_simp_tac (HOL_ss addsimps prems) 1);
   385 qed "if_cong";
   386 
   387 (*Prevents simplification of x and y: faster and allows the execution
   388   of functional programs. NOW THE DEFAULT.*)
   389 Goal "b=c ==> (if b then x else y) = (if c then x else y)";
   390 by (etac arg_cong 1);
   391 qed "if_weak_cong";
   392 
   393 (*Prevents simplification of t: much faster*)
   394 Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
   395 by (etac arg_cong 1);
   396 qed "let_weak_cong";
   397 
   398 (*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
   399 Goal "u = u' ==> (t==u) == (t==u')";
   400 by (asm_simp_tac HOL_ss 1);
   401 qed "eq_cong2";
   402 
   403 Goal "f(if c then x else y) = (if c then f x else f y)";
   404 by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
   405 qed "if_distrib";
   406 
   407 (*For expand_case_tac*)
   408 val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
   409 by (case_tac "P" 1);
   410 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
   411 qed "expand_case";
   412 
   413 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
   414   during unification.*)
   415 fun expand_case_tac P i =
   416     res_inst_tac [("P",P)] expand_case i THEN
   417     Simp_tac (i+1) THEN
   418     Simp_tac i;
   419 
   420 (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
   421   side of an equality.  Used in {Integ,Real}/simproc.ML*)
   422 Goal "x=y ==> (x=z) = (y=z)";
   423 by (asm_simp_tac HOL_ss 1);
   424 qed "restrict_to_left";
   425 
   426 (* default simpset *)
   427 val simpsetup =
   428   [fn thy => (change_simpset_of thy (fn _ => HOL_ss addcongs [if_weak_cong]); thy)];
   429 
   430 
   431 (*** integration of simplifier with classical reasoner ***)
   432 
   433 structure Clasimp = ClasimpFun
   434  (structure Simplifier = Simplifier and Splitter = Splitter
   435   and Classical  = Classical and Blast = Blast
   436   val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
   437   val cla_make_elim = cla_make_elim);
   438 open Clasimp;
   439 
   440 val HOL_css = (HOL_cs, HOL_ss);
   441 
   442 
   443 
   444 (*** A general refutation procedure ***)
   445 
   446 (* Parameters:
   447 
   448    test: term -> bool
   449    tests if a term is at all relevant to the refutation proof;
   450    if not, then it can be discarded. Can improve performance,
   451    esp. if disjunctions can be discarded (no case distinction needed!).
   452 
   453    prep_tac: int -> tactic
   454    A preparation tactic to be applied to the goal once all relevant premises
   455    have been moved to the conclusion.
   456 
   457    ref_tac: int -> tactic
   458    the actual refutation tactic. Should be able to deal with goals
   459    [| A1; ...; An |] ==> False
   460    where the Ai are atomic, i.e. no top-level &, | or EX
   461 *)
   462 
   463 local
   464   val nnf_simpset =
   465     empty_ss setmkeqTrue mk_eq_True
   466     setmksimps (mksimps mksimps_pairs)
   467     addsimps [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
   468       not_all,not_ex,not_not];
   469   fun prem_nnf_tac i st =
   470     full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
   471 in
   472 fun refute_tac test prep_tac ref_tac =
   473   let val refute_prems_tac =
   474         REPEAT_DETERM
   475               (eresolve_tac [conjE, exE] 1 ORELSE
   476                filter_prems_tac test 1 ORELSE
   477                etac disjE 1) THEN
   478         ((etac notE 1 THEN eq_assume_tac 1) ORELSE
   479          ref_tac 1);
   480   in EVERY'[TRY o filter_prems_tac test,
   481             REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
   482             SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
   483   end;
   484 end;