src/HOL/simpdata.ML
author paulson
Thu Mar 04 12:06:07 2004 +0100 (2004-03-04)
changeset 14430 5cb24165a2e1
parent 13743 f8f9393be64c
child 14749 9ccfd0f59e11
permissions -rw-r--r--
new material from Avigad, and simplified treatment of division by 0
     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 val iff_exI = allI RS
    85     prove_goal (the_context()) "!x. P x = Q x ==> (? x. P x) = (? x. Q x)"
    86                (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
    87 
    88 val all_comm = prove_goal (the_context()) "(!x y. P x y) = (!y x. P x y)"
    89                (fn _ => [Blast_tac 1])
    90 val ex_comm = prove_goal (the_context()) "(? x y. P x y) = (? y x. P x y)"
    91                (fn _ => [Blast_tac 1])
    92 in
    93 
    94 (*** make simplification procedures for quantifier elimination ***)
    95 
    96 structure Quantifier1 = Quantifier1Fun
    97 (struct
    98   (*abstract syntax*)
    99   fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
   100     | dest_eq _ = None;
   101   fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
   102     | dest_conj _ = None;
   103   fun dest_imp((c as Const("op -->",_)) $ s $ t) = Some(c,s,t)
   104     | dest_imp _ = None;
   105   val conj = HOLogic.conj
   106   val imp  = HOLogic.imp
   107   (*rules*)
   108   val iff_reflection = eq_reflection
   109   val iffI = iffI
   110   val iff_trans = trans
   111   val conjI= conjI
   112   val conjE= conjE
   113   val impI = impI
   114   val mp   = mp
   115   val uncurry = uncurry
   116   val exI  = exI
   117   val exE  = exE
   118   val iff_allI = iff_allI
   119   val iff_exI = iff_exI
   120   val all_comm = all_comm
   121   val ex_comm = ex_comm
   122 end);
   123 
   124 end;
   125 
   126 val defEX_regroup =
   127   Simplifier.simproc (Theory.sign_of (the_context ()))
   128     "defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
   129 
   130 val defALL_regroup =
   131   Simplifier.simproc (Theory.sign_of (the_context ()))
   132     "defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
   133 
   134 
   135 (*** Case splitting ***)
   136 
   137 (*Make meta-equalities.  The operator below is Trueprop*)
   138 
   139 fun mk_meta_eq r = r RS eq_reflection;
   140 fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
   141 
   142 fun mk_eq th = case concl_of th of
   143         Const("==",_)$_$_       => th
   144     |   _$(Const("op =",_)$_$_) => mk_meta_eq th
   145     |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
   146     |   _                       => th RS Eq_TrueI;
   147 (* Expects Trueprop(.) if not == *)
   148 
   149 fun mk_eq_True r =
   150   Some (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => None;
   151 
   152 (*Congruence rules for = (instead of ==)*)
   153 fun mk_meta_cong rl =
   154   zero_var_indexes(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
   155   handle THM _ =>
   156   error("Premises and conclusion of congruence rules must be =-equalities");
   157 
   158 (* Elimination of True from asumptions: *)
   159 
   160 local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
   161 in val True_implies_equals = standard' (equal_intr
   162   (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
   163   (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
   164 end;
   165 
   166 
   167 structure SplitterData =
   168   struct
   169   structure Simplifier = Simplifier
   170   val mk_eq          = mk_eq
   171   val meta_eq_to_iff = meta_eq_to_obj_eq
   172   val iffD           = iffD2
   173   val disjE          = disjE
   174   val conjE          = conjE
   175   val exE            = exE
   176   val contrapos      = contrapos_nn
   177   val contrapos2     = contrapos_pp
   178   val notnotD        = notnotD
   179   end;
   180 
   181 structure Splitter = SplitterFun(SplitterData);
   182 
   183 val split_tac        = Splitter.split_tac;
   184 val split_inside_tac = Splitter.split_inside_tac;
   185 val split_asm_tac    = Splitter.split_asm_tac;
   186 val op addsplits     = Splitter.addsplits;
   187 val op delsplits     = Splitter.delsplits;
   188 val Addsplits        = Splitter.Addsplits;
   189 val Delsplits        = Splitter.Delsplits;
   190 
   191 val mksimps_pairs =
   192   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   193    ("All", [spec]), ("True", []), ("False", []),
   194    ("If", [if_bool_eq_conj RS iffD1])];
   195 
   196 (*
   197 val mk_atomize:      (string * thm list) list -> thm -> thm list
   198 looks too specific to move it somewhere else
   199 *)
   200 fun mk_atomize pairs =
   201   let fun atoms th =
   202         (case concl_of th of
   203            Const("Trueprop",_) $ p =>
   204              (case head_of p of
   205                 Const(a,_) =>
   206                   (case assoc(pairs,a) of
   207                      Some(rls) => flat (map atoms ([th] RL rls))
   208                    | None => [th])
   209               | _ => [th])
   210          | _ => [th])
   211   in atoms end;
   212 
   213 fun mksimps pairs =
   214   (mapfilter (try mk_eq) o mk_atomize pairs o gen_all);
   215 
   216 fun unsafe_solver_tac prems =
   217   FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
   218 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
   219 
   220 (*No premature instantiation of variables during simplification*)
   221 fun safe_solver_tac prems =
   222   FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
   223          eq_assume_tac, ematch_tac [FalseE]];
   224 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
   225 
   226 val HOL_basic_ss =
   227   empty_ss setsubgoaler asm_simp_tac
   228     setSSolver safe_solver
   229     setSolver unsafe_solver
   230     setmksimps (mksimps mksimps_pairs)
   231     setmkeqTrue mk_eq_True
   232     setmkcong mk_meta_cong;
   233 
   234 (*In general it seems wrong to add distributive laws by default: they
   235   might cause exponential blow-up.  But imp_disjL has been in for a while
   236   and cannot be removed without affecting existing proofs.  Moreover,
   237   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   238   grounds that it allows simplification of R in the two cases.*)
   239 
   240 val HOL_ss =
   241     HOL_basic_ss addsimps
   242      ([triv_forall_equality, (* prunes params *)
   243        True_implies_equals, (* prune asms `True' *)
   244        eta_contract_eq, (* prunes eta-expansions *)
   245        if_True, if_False, if_cancel, if_eq_cancel,
   246        imp_disjL, conj_assoc, disj_assoc,
   247        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
   248        disj_not1, not_all, not_ex, cases_simp,
   249        thm "the_eq_trivial", the_sym_eq_trivial]
   250      @ ex_simps @ all_simps @ simp_thms)
   251      addsimprocs [defALL_regroup,defEX_regroup]
   252      addcongs [imp_cong]
   253      addsplits [split_if];
   254 
   255 fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
   256 
   257 
   258 (*Simplifies x assuming c and y assuming ~c*)
   259 val prems = Goalw [if_def]
   260   "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
   261 \  (if b then x else y) = (if c then u else v)";
   262 by (asm_simp_tac (HOL_ss addsimps prems) 1);
   263 qed "if_cong";
   264 
   265 (*Prevents simplification of x and y: faster and allows the execution
   266   of functional programs. NOW THE DEFAULT.*)
   267 Goal "b=c ==> (if b then x else y) = (if c then x else y)";
   268 by (etac arg_cong 1);
   269 qed "if_weak_cong";
   270 
   271 (*Prevents simplification of t: much faster*)
   272 Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
   273 by (etac arg_cong 1);
   274 qed "let_weak_cong";
   275 
   276 (*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
   277 Goal "u = u' ==> (t==u) == (t==u')";
   278 by (asm_simp_tac HOL_ss 1);
   279 qed "eq_cong2";
   280 
   281 Goal "f(if c then x else y) = (if c then f x else f y)";
   282 by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
   283 qed "if_distrib";
   284 
   285 (*For expand_case_tac*)
   286 val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
   287 by (case_tac "P" 1);
   288 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
   289 qed "expand_case";
   290 
   291 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
   292   during unification.*)
   293 fun expand_case_tac P i =
   294     res_inst_tac [("P",P)] expand_case i THEN
   295     Simp_tac (i+1) THEN
   296     Simp_tac i;
   297 
   298 (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
   299   side of an equality.  Used in {Integ,Real}/simproc.ML*)
   300 Goal "x=y ==> (x=z) = (y=z)";
   301 by (asm_simp_tac HOL_ss 1);
   302 qed "restrict_to_left";
   303 
   304 (* default simpset *)
   305 val simpsetup =
   306   [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong]; thy)];
   307 
   308 
   309 (*** integration of simplifier with classical reasoner ***)
   310 
   311 structure Clasimp = ClasimpFun
   312  (structure Simplifier = Simplifier and Splitter = Splitter
   313   and Classical  = Classical and Blast = Blast
   314   val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
   315   val cla_make_elim = cla_make_elim);
   316 open Clasimp;
   317 
   318 val HOL_css = (HOL_cs, HOL_ss);
   319 
   320 
   321 
   322 (*** A general refutation procedure ***)
   323 
   324 (* Parameters:
   325 
   326    test: term -> bool
   327    tests if a term is at all relevant to the refutation proof;
   328    if not, then it can be discarded. Can improve performance,
   329    esp. if disjunctions can be discarded (no case distinction needed!).
   330 
   331    prep_tac: int -> tactic
   332    A preparation tactic to be applied to the goal once all relevant premises
   333    have been moved to the conclusion.
   334 
   335    ref_tac: int -> tactic
   336    the actual refutation tactic. Should be able to deal with goals
   337    [| A1; ...; An |] ==> False
   338    where the Ai are atomic, i.e. no top-level &, | or EX
   339 *)
   340 
   341 fun refute_tac test prep_tac ref_tac =
   342   let val nnf_simps =
   343         [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
   344          not_all,not_ex,not_not];
   345       val nnf_simpset =
   346         empty_ss setmkeqTrue mk_eq_True
   347                  setmksimps (mksimps mksimps_pairs)
   348                  addsimps nnf_simps;
   349       val prem_nnf_tac = full_simp_tac nnf_simpset;
   350 
   351       val refute_prems_tac =
   352         REPEAT_DETERM
   353               (eresolve_tac [conjE, exE] 1 ORELSE
   354                filter_prems_tac test 1 ORELSE
   355                etac disjE 1) THEN
   356         ((etac notE 1 THEN eq_assume_tac 1) ORELSE
   357          ref_tac 1);
   358   in EVERY'[TRY o filter_prems_tac test,
   359             REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
   360             SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
   361   end;