src/HOL/simpdata.ML
author nipkow
Wed Dec 12 12:40:02 2001 +0100 (2001-12-12)
changeset 12475 18ba10cc782f
parent 12281 3bd113b8f7a6
child 12524 66eb843b1d35
permissions -rw-r--r--
Removed pointless backtracking from arith_tac
     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_thms = thms "simp_thms";
    73 val split_if = thm "split_if";
    74 val split_if_asm = thm "split_if_asm";
    75 
    76 
    77 local
    78 val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
    79               (fn prems => [cut_facts_tac prems 1, Blast_tac 1]);
    80 
    81 val iff_allI = allI RS
    82     prove_goal (the_context()) "!x. P x = Q x ==> (!x. P x) = (!x. Q x)"
    83                (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
    84 in
    85 
    86 (*** make simplification procedures for quantifier elimination ***)
    87 
    88 structure Quantifier1 = Quantifier1Fun
    89 (struct
    90   (*abstract syntax*)
    91   fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
    92     | dest_eq _ = None;
    93   fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
    94     | dest_conj _ = None;
    95   fun dest_imp((c as Const("op -->",_)) $ s $ t) = Some(c,s,t)
    96     | dest_imp _ = None;
    97   val conj = HOLogic.conj
    98   val imp  = HOLogic.imp
    99   (*rules*)
   100   val iff_reflection = eq_reflection
   101   val iffI = iffI
   102   val conjI= conjI
   103   val conjE= conjE
   104   val impI = impI
   105   val mp   = mp
   106   val uncurry = uncurry
   107   val exI  = exI
   108   val exE  = exE
   109   val iff_allI = iff_allI
   110 end);
   111 
   112 end;
   113 
   114 local
   115 val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
   116     ("EX x. P(x) & Q(x)",HOLogic.boolT)
   117 val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
   118     ("ALL x. P(x) --> Q(x)",HOLogic.boolT)
   119 in
   120 val defEX_regroup = mk_simproc "defined EX" [ex_pattern]
   121       Quantifier1.rearrange_ex
   122 val defALL_regroup = mk_simproc "defined ALL" [all_pattern]
   123       Quantifier1.rearrange_all
   124 end;
   125 
   126 
   127 (*** Case splitting ***)
   128 
   129 (*Make meta-equalities.  The operator below is Trueprop*)
   130 
   131 fun mk_meta_eq r = r RS eq_reflection;
   132 fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
   133 
   134 fun mk_eq th = case concl_of th of
   135         Const("==",_)$_$_       => th
   136     |   _$(Const("op =",_)$_$_) => mk_meta_eq th
   137     |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
   138     |   _                       => th RS Eq_TrueI;
   139 (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
   140 
   141 fun mk_eq_True r =
   142   Some (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => None;
   143 
   144 (*Congruence rules for = (instead of ==)*)
   145 fun mk_meta_cong rl =
   146   standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
   147   handle THM _ =>
   148   error("Premises and conclusion of congruence rules must be =-equalities");
   149 
   150 (* Elimination of True from asumptions: *)
   151 
   152 local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
   153 in val True_implies_equals = standard' (equal_intr
   154   (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
   155   (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
   156 end;
   157 
   158 
   159 structure SplitterData =
   160   struct
   161   structure Simplifier = Simplifier
   162   val mk_eq          = mk_eq
   163   val meta_eq_to_iff = meta_eq_to_obj_eq
   164   val iffD           = iffD2
   165   val disjE          = disjE
   166   val conjE          = conjE
   167   val exE            = exE
   168   val contrapos      = contrapos_nn
   169   val contrapos2     = contrapos_pp
   170   val notnotD        = notnotD
   171   end;
   172 
   173 structure Splitter = SplitterFun(SplitterData);
   174 
   175 val split_tac        = Splitter.split_tac;
   176 val split_inside_tac = Splitter.split_inside_tac;
   177 val split_asm_tac    = Splitter.split_asm_tac;
   178 val op addsplits     = Splitter.addsplits;
   179 val op delsplits     = Splitter.delsplits;
   180 val Addsplits        = Splitter.Addsplits;
   181 val Delsplits        = Splitter.Delsplits;
   182 
   183 (*In general it seems wrong to add distributive laws by default: they
   184   might cause exponential blow-up.  But imp_disjL has been in for a while
   185   and cannot be removed without affecting existing proofs.  Moreover,
   186   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   187   grounds that it allows simplification of R in the two cases.*)
   188 
   189 val mksimps_pairs =
   190   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   191    ("All", [spec]), ("True", []), ("False", []),
   192    ("If", [if_bool_eq_conj RS iffD1])];
   193 
   194 (* ###FIXME: move to Provers/simplifier.ML
   195 val mk_atomize:      (string * thm list) list -> thm -> thm list
   196 *)
   197 (* ###FIXME: move to Provers/simplifier.ML *)
   198 fun mk_atomize pairs =
   199   let fun atoms th =
   200         (case concl_of th of
   201            Const("Trueprop",_) $ p =>
   202              (case head_of p of
   203                 Const(a,_) =>
   204                   (case assoc(pairs,a) of
   205                      Some(rls) => flat (map atoms ([th] RL rls))
   206                    | None => [th])
   207               | _ => [th])
   208          | _ => [th])
   209   in atoms end;
   210 
   211 fun mksimps pairs =
   212   (mapfilter (try mk_eq) o mk_atomize pairs o forall_elim_vars_safe);
   213 
   214 fun unsafe_solver_tac prems =
   215   FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
   216 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
   217 
   218 (*No premature instantiation of variables during simplification*)
   219 fun safe_solver_tac prems =
   220   FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
   221          eq_assume_tac, ematch_tac [FalseE]];
   222 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
   223 
   224 val HOL_basic_ss =
   225   empty_ss setsubgoaler asm_simp_tac
   226     setSSolver safe_solver
   227     setSolver unsafe_solver
   228     setmksimps (mksimps mksimps_pairs)
   229     setmkeqTrue mk_eq_True
   230     setmkcong mk_meta_cong;
   231 
   232 val HOL_ss =
   233     HOL_basic_ss addsimps
   234      ([triv_forall_equality, (* prunes params *)
   235        True_implies_equals, (* prune asms `True' *)
   236        eta_contract_eq, (* prunes eta-expansions *)
   237        if_True, if_False, if_cancel, if_eq_cancel,
   238        imp_disjL, conj_assoc, disj_assoc,
   239        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
   240        disj_not1, not_all, not_ex, cases_simp,
   241        thm "the_eq_trivial", the_sym_eq_trivial, thm "plus_ac0.zero", thm "plus_ac0_zero_right"]
   242      @ ex_simps @ all_simps @ simp_thms)
   243      addsimprocs [defALL_regroup,defEX_regroup]
   244      addcongs [imp_cong]
   245      addsplits [split_if];
   246 
   247 fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
   248 
   249 
   250 (*Simplifies x assuming c and y assuming ~c*)
   251 val prems = Goalw [if_def]
   252   "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
   253 \  (if b then x else y) = (if c then u else v)";
   254 by (asm_simp_tac (HOL_ss addsimps prems) 1);
   255 qed "if_cong";
   256 
   257 (*Prevents simplification of x and y: faster and allows the execution
   258   of functional programs. NOW THE DEFAULT.*)
   259 Goal "b=c ==> (if b then x else y) = (if c then x else y)";
   260 by (etac arg_cong 1);
   261 qed "if_weak_cong";
   262 
   263 (*Prevents simplification of t: much faster*)
   264 Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
   265 by (etac arg_cong 1);
   266 qed "let_weak_cong";
   267 
   268 Goal "f(if c then x else y) = (if c then f x else f y)";
   269 by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
   270 qed "if_distrib";
   271 
   272 (*For expand_case_tac*)
   273 val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
   274 by (case_tac "P" 1);
   275 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
   276 qed "expand_case";
   277 
   278 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
   279   during unification.*)
   280 fun expand_case_tac P i =
   281     res_inst_tac [("P",P)] expand_case i THEN
   282     Simp_tac (i+1) THEN
   283     Simp_tac i;
   284 
   285 (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
   286   side of an equality.  Used in {Integ,Real}/simproc.ML*)
   287 Goal "x=y ==> (x=z) = (y=z)";
   288 by (asm_simp_tac HOL_ss 1);
   289 qed "restrict_to_left";
   290 
   291 (* default simpset *)
   292 val simpsetup =
   293   [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong]; thy)];
   294 
   295 
   296 (*** integration of simplifier with classical reasoner ***)
   297 
   298 structure Clasimp = ClasimpFun
   299  (structure Simplifier = Simplifier and Splitter = Splitter
   300   and Classical  = Classical and Blast = Blast
   301   val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
   302   val cla_make_elim = cla_make_elim);
   303 open Clasimp;
   304 
   305 val HOL_css = (HOL_cs, HOL_ss);
   306 
   307 
   308 
   309 (*** A general refutation procedure ***)
   310 
   311 (* Parameters:
   312 
   313    test: term -> bool
   314    tests if a term is at all relevant to the refutation proof;
   315    if not, then it can be discarded. Can improve performance,
   316    esp. if disjunctions can be discarded (no case distinction needed!).
   317 
   318    prep_tac: int -> tactic
   319    A preparation tactic to be applied to the goal once all relevant premises
   320    have been moved to the conclusion.
   321 
   322    ref_tac: int -> tactic
   323    the actual refutation tactic. Should be able to deal with goals
   324    [| A1; ...; An |] ==> False
   325    where the Ai are atomic, i.e. no top-level &, | or EX
   326 *)
   327 
   328 fun refute_tac test prep_tac ref_tac =
   329   let val nnf_simps =
   330         [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
   331          not_all,not_ex,not_not];
   332       val nnf_simpset =
   333         empty_ss setmkeqTrue mk_eq_True
   334                  setmksimps (mksimps mksimps_pairs)
   335                  addsimps nnf_simps;
   336       val prem_nnf_tac = full_simp_tac nnf_simpset;
   337 
   338       val refute_prems_tac =
   339         REPEAT_DETERM
   340               (eresolve_tac [conjE, exE] 1 ORELSE
   341                filter_prems_tac test 1 ORELSE
   342                etac disjE 1) THEN
   343         ((etac notE 1 THEN eq_assume_tac 1) ORELSE
   344          ref_tac 1);
   345   in EVERY'[TRY o filter_prems_tac test,
   346             REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
   347             SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
   348   end;