src/HOL/simpdata.ML
author webertj
Wed Jul 26 19:23:04 2006 +0200 (2006-07-26)
changeset 20217 25b068a99d2b
parent 20070 3f31fb81b83a
child 20872 528054ca23e3
permissions -rw-r--r--
linear arithmetic splits certain operators (e.g. min, max, abs)
     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 (*** Simproc for Let ***)
   170 
   171 val use_let_simproc = ref true;
   172 
   173 local
   174 val Let_folded = thm "Let_folded";
   175 val Let_unfold = thm "Let_unfold";
   176 
   177 val (f_Let_unfold,x_Let_unfold) =
   178       let val [(_$(f$x)$_)] = prems_of Let_unfold
   179       in (cterm_of (sign_of (the_context ())) f,cterm_of (sign_of (the_context ())) x) end
   180 val (f_Let_folded,x_Let_folded) =
   181       let val [(_$(f$x)$_)] = prems_of Let_folded
   182       in (cterm_of (sign_of (the_context ())) f, cterm_of (sign_of (the_context ())) x) end;
   183 val g_Let_folded =
   184       let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of (sign_of (the_context ())) g end;
   185 in
   186 val let_simproc =
   187   Simplifier.simproc (Theory.sign_of (the_context ())) "let_simp" ["Let x f"]
   188    (fn sg => fn ss => fn t =>
   189      let val ctxt = Simplifier.the_context ss;
   190          val ([t'],ctxt') = Variable.import_terms false [t] ctxt;
   191      in Option.map (hd o Variable.export ctxt' ctxt o single)
   192       (case t' of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
   193          if not (!use_let_simproc) then NONE
   194          else if is_Free x orelse is_Bound x orelse is_Const x
   195          then SOME Let_def
   196          else
   197           let
   198              val n = case f of (Abs (x,_,_)) => x | _ => "x";
   199              val cx = cterm_of sg x;
   200              val {T=xT,...} = rep_cterm cx;
   201              val cf = cterm_of sg f;
   202              val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
   203              val (_$_$g) = prop_of fx_g;
   204              val g' = abstract_over (x,g);
   205            in (if (g aconv g')
   206                then
   207                   let
   208                     val rl = cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] Let_unfold;
   209                   in SOME (rl OF [fx_g]) end
   210                else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
   211                else let
   212                      val abs_g'= Abs (n,xT,g');
   213                      val g'x = abs_g'$x;
   214                      val g_g'x = symmetric (beta_conversion false (cterm_of sg g'x));
   215                      val rl = cterm_instantiate
   216                                [(f_Let_folded,cterm_of sg f),(x_Let_folded,cx),
   217                                 (g_Let_folded,cterm_of sg abs_g')]
   218                                Let_folded;
   219                    in SOME (rl OF [transitive fx_g g_g'x])
   220                    end)
   221            end
   222         | _ => NONE)
   223      end)
   224 end
   225 
   226 (*** Case splitting ***)
   227 
   228 (*Make meta-equalities.  The operator below is Trueprop*)
   229 
   230 fun mk_meta_eq r = r RS eq_reflection;
   231 fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
   232 
   233 fun mk_eq th = case concl_of th of
   234         Const("==",_)$_$_       => th
   235     |   _$(Const("op =",_)$_$_) => mk_meta_eq th
   236     |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
   237     |   _                       => th RS Eq_TrueI;
   238 (* Expects Trueprop(.) if not == *)
   239 
   240 fun mk_eq_True r =
   241   SOME (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => NONE;
   242 
   243 (* Produce theorems of the form
   244   (P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
   245 *)
   246 fun lift_meta_eq_to_obj_eq i st =
   247   let
   248     val {sign, ...} = rep_thm st;
   249     fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q
   250       | count_imp _ = 0;
   251     val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1)))
   252   in if j = 0 then meta_eq_to_obj_eq
   253     else
   254       let
   255         val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
   256         fun mk_simp_implies Q = foldr (fn (R, S) =>
   257           Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps
   258         val aT = TFree ("'a", HOLogic.typeS);
   259         val x = Free ("x", aT);
   260         val y = Free ("y", aT)
   261       in Goal.prove_global (Thm.theory_of_thm st) []
   262         [mk_simp_implies (Logic.mk_equals (x, y))]
   263         (mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))))
   264         (fn prems => EVERY
   265          [rewrite_goals_tac [simp_implies_def],
   266           REPEAT (ares_tac (meta_eq_to_obj_eq :: map (rewrite_rule [simp_implies_def]) prems) 1)])
   267       end
   268   end;
   269 
   270 (*Congruence rules for = (instead of ==)*)
   271 fun mk_meta_cong rl = zero_var_indexes
   272   (let val rl' = Seq.hd (TRYALL (fn i => fn st =>
   273      rtac (lift_meta_eq_to_obj_eq i st) i st) rl)
   274    in mk_meta_eq rl' handle THM _ =>
   275      if Logic.is_equals (concl_of rl') then rl'
   276      else error "Conclusion of congruence rules must be =-equality"
   277    end);
   278 
   279 (* Elimination of True from asumptions: *)
   280 
   281 local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
   282 in val True_implies_equals = standard' (equal_intr
   283   (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
   284   (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
   285 end;
   286 
   287 
   288 structure SplitterData =
   289   struct
   290   structure Simplifier = Simplifier
   291   val mk_eq          = mk_eq
   292   val meta_eq_to_iff = meta_eq_to_obj_eq
   293   val iffD           = iffD2
   294   val disjE          = disjE
   295   val conjE          = conjE
   296   val exE            = exE
   297   val contrapos      = contrapos_nn
   298   val contrapos2     = contrapos_pp
   299   val notnotD        = notnotD
   300   end;
   301 
   302 structure Splitter = SplitterFun(SplitterData);
   303 
   304 val split_tac        = Splitter.split_tac;
   305 val split_inside_tac = Splitter.split_inside_tac;
   306 val split_asm_tac    = Splitter.split_asm_tac;
   307 val op addsplits     = Splitter.addsplits;
   308 val op delsplits     = Splitter.delsplits;
   309 val Addsplits        = Splitter.Addsplits;
   310 val Delsplits        = Splitter.Delsplits;
   311 
   312 val mksimps_pairs =
   313   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   314    ("All", [spec]), ("True", []), ("False", []),
   315    ("HOL.If", [if_bool_eq_conj RS iffD1])];
   316 
   317 (*
   318 val mk_atomize:      (string * thm list) list -> thm -> thm list
   319 looks too specific to move it somewhere else
   320 *)
   321 fun mk_atomize pairs =
   322   let fun atoms th =
   323         (case concl_of th of
   324            Const("Trueprop",_) $ p =>
   325              (case head_of p of
   326                 Const(a,_) =>
   327                   (case AList.lookup (op =) pairs a of
   328                      SOME(rls) => List.concat (map atoms ([th] RL rls))
   329                    | NONE => [th])
   330               | _ => [th])
   331          | _ => [th])
   332   in atoms end;
   333 
   334 fun mksimps pairs =
   335   (List.mapPartial (try mk_eq) o mk_atomize pairs o gen_all);
   336 
   337 fun unsafe_solver_tac prems =
   338   (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
   339   FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
   340 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
   341 
   342 (*No premature instantiation of variables during simplification*)
   343 fun safe_solver_tac prems =
   344   (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
   345   FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
   346          eq_assume_tac, ematch_tac [FalseE]];
   347 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
   348 
   349 val HOL_basic_ss =
   350   Simplifier.theory_context (the_context ()) empty_ss
   351     setsubgoaler asm_simp_tac
   352     setSSolver safe_solver
   353     setSolver unsafe_solver
   354     setmksimps (mksimps mksimps_pairs)
   355     setmkeqTrue mk_eq_True
   356     setmkcong mk_meta_cong;
   357 
   358 fun unfold_tac ths =
   359   let val ss0 = Simplifier.clear_ss HOL_basic_ss addsimps ths
   360   in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end;
   361 
   362 (*In general it seems wrong to add distributive laws by default: they
   363   might cause exponential blow-up.  But imp_disjL has been in for a while
   364   and cannot be removed without affecting existing proofs.  Moreover,
   365   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   366   grounds that it allows simplification of R in the two cases.*)
   367 
   368 val HOL_ss =
   369     HOL_basic_ss addsimps
   370      ([triv_forall_equality, (* prunes params *)
   371        True_implies_equals, (* prune asms `True' *)
   372        if_True, if_False, if_cancel, if_eq_cancel,
   373        imp_disjL, conj_assoc, disj_assoc,
   374        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
   375        disj_not1, not_all, not_ex, cases_simp,
   376        thm "the_eq_trivial", the_sym_eq_trivial]
   377      @ ex_simps @ all_simps @ simp_thms)
   378      addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc]
   379      addcongs [imp_cong, simp_implies_cong]
   380      addsplits [split_if];
   381 
   382 fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
   383 
   384 
   385 (*Simplifies x assuming c and y assuming ~c*)
   386 val prems = Goalw [if_def]
   387   "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
   388 \  (if b then x else y) = (if c then u else v)";
   389 by (asm_simp_tac (HOL_ss addsimps prems) 1);
   390 qed "if_cong";
   391 
   392 (*Prevents simplification of x and y: faster and allows the execution
   393   of functional programs. NOW THE DEFAULT.*)
   394 Goal "b=c ==> (if b then x else y) = (if c then x else y)";
   395 by (etac arg_cong 1);
   396 qed "if_weak_cong";
   397 
   398 (*Prevents simplification of t: much faster*)
   399 Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
   400 by (etac arg_cong 1);
   401 qed "let_weak_cong";
   402 
   403 (*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
   404 Goal "u = u' ==> (t==u) == (t==u')";
   405 by (asm_simp_tac HOL_ss 1);
   406 qed "eq_cong2";
   407 
   408 Goal "f(if c then x else y) = (if c then f x else f y)";
   409 by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
   410 qed "if_distrib";
   411 
   412 (*For expand_case_tac*)
   413 val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
   414 by (case_tac "P" 1);
   415 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
   416 qed "expand_case";
   417 
   418 (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
   419   side of an equality.  Used in {Integ,Real}/simproc.ML*)
   420 Goal "x=y ==> (x=z) = (y=z)";
   421 by (asm_simp_tac HOL_ss 1);
   422 qed "restrict_to_left";
   423 
   424 (* default simpset *)
   425 val simpsetup =
   426   (fn thy => (change_simpset_of thy (fn _ => HOL_ss addcongs [if_weak_cong]); thy));
   427 
   428 
   429 (*** integration of simplifier with classical reasoner ***)
   430 
   431 structure Clasimp = ClasimpFun
   432  (structure Simplifier = Simplifier and Splitter = Splitter
   433   and Classical  = Classical and Blast = Blast
   434   val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE);
   435 open Clasimp;
   436 
   437 val HOL_css = (HOL_cs, HOL_ss);
   438 
   439 
   440 
   441 (*** A general refutation procedure ***)
   442 
   443 (* Parameters:
   444 
   445    test: term -> bool
   446    tests if a term is at all relevant to the refutation proof;
   447    if not, then it can be discarded. Can improve performance,
   448    esp. if disjunctions can be discarded (no case distinction needed!).
   449 
   450    prep_tac: int -> tactic
   451    A preparation tactic to be applied to the goal once all relevant premises
   452    have been moved to the conclusion.
   453 
   454    ref_tac: int -> tactic
   455    the actual refutation tactic. Should be able to deal with goals
   456    [| A1; ...; An |] ==> False
   457    where the Ai are atomic, i.e. no top-level &, | or EX
   458 *)
   459 
   460 local
   461   val nnf_simpset =
   462     empty_ss setmkeqTrue mk_eq_True
   463     setmksimps (mksimps mksimps_pairs)
   464     addsimps [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
   465       not_all,not_ex,not_not];
   466   fun prem_nnf_tac i st =
   467     full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
   468 in
   469 fun refute_tac test prep_tac ref_tac =
   470   let val refute_prems_tac =
   471         REPEAT_DETERM
   472               (eresolve_tac [conjE, exE] 1 ORELSE
   473                filter_prems_tac test 1 ORELSE
   474                etac disjE 1) THEN
   475         ((etac notE 1 THEN eq_assume_tac 1) ORELSE
   476          ref_tac 1);
   477   in EVERY'[TRY o filter_prems_tac test,
   478             REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
   479             SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
   480   end;
   481 end;