src/HOL/simpdata.ML
author berghofe
Fri Sep 28 17:19:46 2001 +0200 (2001-09-28)
changeset 11624 8a45c7abef04
parent 11534 0494d0347f18
child 11684 abd396ca7ef9
permissions -rw-r--r--
mksimps and mk_eq_True no longer raise THM exception.
     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 section "Simplifier";
    10 
    11 val [prem] = goal (the_context ()) "x==y ==> x=y";
    12 by (rewtac prem);
    13 by (rtac refl 1);
    14 qed "meta_eq_to_obj_eq";
    15 
    16 Goal "(%s. f s) = f";
    17 br refl 1;
    18 qed "eta_contract_eq";
    19 
    20 local
    21 
    22   fun prover s = prove_goal (the_context ()) s (fn _ => [(Blast_tac 1)]);
    23 
    24 in
    25 
    26 (*Make meta-equalities.  The operator below is Trueprop*)
    27 
    28 fun mk_meta_eq r = r RS eq_reflection;
    29 fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
    30 
    31 val Eq_TrueI  = mk_meta_eq(prover  "P --> (P = True)"  RS mp);
    32 val Eq_FalseI = mk_meta_eq(prover "~P --> (P = False)" RS mp);
    33 
    34 fun mk_eq th = case concl_of th of
    35         Const("==",_)$_$_       => th
    36     |   _$(Const("op =",_)$_$_) => mk_meta_eq th
    37     |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
    38     |   _                       => th RS Eq_TrueI;
    39 (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
    40 
    41 fun mk_eq_True r =
    42   Some (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => None;
    43 
    44 (*Congruence rules for = (instead of ==)*)
    45 fun mk_meta_cong rl =
    46   standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
    47   handle THM _ =>
    48   error("Premises and conclusion of congruence rules must be =-equalities");
    49 
    50 val not_not = prover "(~ ~ P) = P";
    51 
    52 val simp_thms = [not_not] @ map prover
    53  [ "(x=x) = True",
    54    "(~True) = False", "(~False) = True",
    55    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
    56    "(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)",
    57    "(True --> P) = P", "(False --> P) = True",
    58    "(P --> True) = True", "(P --> P) = True",
    59    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
    60    "(P & True) = P", "(True & P) = P",
    61    "(P & False) = False", "(False & P) = False",
    62    "(P & P) = P", "(P & (P & Q)) = (P & Q)",
    63    "(P & ~P) = False",    "(~P & P) = False",
    64    "(P | True) = True", "(True | P) = True",
    65    "(P | False) = P", "(False | P) = P",
    66    "(P | P) = P", "(P | (P | Q)) = (P | Q)",
    67    "(P | ~P) = True",    "(~P | P) = True",
    68    "((~P) = (~Q)) = (P=Q)",
    69    "(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x",
    70 (* needed for the one-point-rule quantifier simplification procs*)
    71 (*essential for termination!!*)
    72    "(? x. x=t & P(x)) = P(t)",
    73    "(? x. t=x & P(x)) = P(t)",
    74    "(! x. x=t --> P(x)) = P(t)",
    75    "(! x. t=x --> P(x)) = P(t)" ];
    76 
    77 val imp_cong = standard(impI RSN
    78     (2, prove_goal (the_context ()) "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
    79         (fn _=> [(Blast_tac 1)]) RS mp RS mp));
    80 
    81 (*Miniscoping: pushing in existential quantifiers*)
    82 val ex_simps = map prover
    83                 ["(EX x. P x & Q)   = ((EX x. P x) & Q)",
    84                  "(EX x. P & Q x)   = (P & (EX x. Q x))",
    85                  "(EX x. P x | Q)   = ((EX x. P x) | Q)",
    86                  "(EX x. P | Q x)   = (P | (EX x. Q x))",
    87                  "(EX x. P x --> Q) = ((ALL x. P x) --> Q)",
    88                  "(EX x. P --> Q x) = (P --> (EX x. Q x))"];
    89 
    90 (*Miniscoping: pushing in universal quantifiers*)
    91 val all_simps = map prover
    92                 ["(ALL x. P x & Q)   = ((ALL x. P x) & Q)",
    93                  "(ALL x. P & Q x)   = (P & (ALL x. Q x))",
    94                  "(ALL x. P x | Q)   = ((ALL x. P x) | Q)",
    95                  "(ALL x. P | Q x)   = (P | (ALL x. Q x))",
    96                  "(ALL x. P x --> Q) = ((EX x. P x) --> Q)",
    97                  "(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];
    98 
    99 
   100 end;
   101 
   102 bind_thms ("ex_simps", ex_simps);
   103 bind_thms ("all_simps", all_simps);
   104 bind_thm ("not_not", not_not);
   105 bind_thm ("imp_cong", imp_cong);
   106 
   107 (* Elimination of True from asumptions: *)
   108 
   109 local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
   110 in val True_implies_equals = standard' (equal_intr
   111   (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
   112   (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
   113 end;
   114 
   115 fun prove nm thm  = qed_goal nm (the_context ()) thm (fn _ => [(Blast_tac 1)]);
   116 
   117 prove "eq_commute" "(a=b) = (b=a)";
   118 prove "eq_left_commute" "(P=(Q=R)) = (Q=(P=R))";
   119 prove "eq_assoc" "((P=Q)=R) = (P=(Q=R))";
   120 val eq_ac = [eq_commute, eq_left_commute, eq_assoc];
   121 
   122 prove "neq_commute" "(a~=b) = (b~=a)";
   123 
   124 prove "conj_commute" "(P&Q) = (Q&P)";
   125 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
   126 val conj_comms = [conj_commute, conj_left_commute];
   127 prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
   128 
   129 prove "disj_commute" "(P|Q) = (Q|P)";
   130 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
   131 val disj_comms = [disj_commute, disj_left_commute];
   132 prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
   133 
   134 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
   135 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
   136 
   137 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
   138 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
   139 
   140 prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
   141 prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";
   142 prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
   143 
   144 (*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
   145 prove "imp_disj_not1" "(P --> Q | R) = (~Q --> P --> R)";
   146 prove "imp_disj_not2" "(P --> Q | R) = (~R --> P --> Q)";
   147 
   148 prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";
   149 prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";
   150 
   151 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
   152 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
   153 prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
   154 prove "not_iff" "(P~=Q) = (P = (~Q))";
   155 prove "disj_not1" "(~P | Q) = (P --> Q)";
   156 prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *)
   157 prove "imp_conv_disj" "(P --> Q) = ((~P) | Q)";
   158 
   159 prove "iff_conv_conj_imp" "(P = Q) = ((P --> Q) & (Q --> P))";
   160 
   161 
   162 (*Avoids duplication of subgoals after split_if, when the true and false
   163   cases boil down to the same thing.*)
   164 prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
   165 
   166 prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))";
   167 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
   168 prove "not_ex"  "(~ (? x. P(x))) = (! x.~P(x))";
   169 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
   170 
   171 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
   172 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
   173 
   174 (* '&' congruence rule: not included by default!
   175    May slow rewrite proofs down by as much as 50% *)
   176 
   177 let val th = prove_goal (the_context ())
   178                 "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
   179                 (fn _=> [(Blast_tac 1)])
   180 in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   181 
   182 let val th = prove_goal (the_context ())
   183                 "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
   184                 (fn _=> [(Blast_tac 1)])
   185 in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   186 
   187 (* '|' congruence rule: not included by default! *)
   188 
   189 let val th = prove_goal (the_context ())
   190                 "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
   191                 (fn _=> [(Blast_tac 1)])
   192 in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   193 
   194 prove "eq_sym_conv" "(x=y) = (y=x)";
   195 
   196 
   197 (** if-then-else rules **)
   198 
   199 Goalw [if_def] "(if True then x else y) = x";
   200 by (Blast_tac 1);
   201 qed "if_True";
   202 
   203 Goalw [if_def] "(if False then x else y) = y";
   204 by (Blast_tac 1);
   205 qed "if_False";
   206 
   207 Goalw [if_def] "P ==> (if P then x else y) = x";
   208 by (Blast_tac 1);
   209 qed "if_P";
   210 
   211 Goalw [if_def] "~P ==> (if P then x else y) = y";
   212 by (Blast_tac 1);
   213 qed "if_not_P";
   214 
   215 Goal "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))";
   216 by (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1);
   217 by (stac if_P 2);
   218 by (stac if_not_P 1);
   219 by (ALLGOALS (Blast_tac));
   220 qed "split_if";
   221 
   222 Goal "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))";
   223 by (stac split_if 1);
   224 by (Blast_tac 1);
   225 qed "split_if_asm";
   226 
   227 bind_thms ("if_splits", [split_if, split_if_asm]);
   228 
   229 bind_thm ("if_def2", read_instantiate [("P","\\<lambda>x. x")] split_if);
   230 
   231 Goal "(if c then x else x) = x";
   232 by (stac split_if 1);
   233 by (Blast_tac 1);
   234 qed "if_cancel";
   235 
   236 Goal "(if x = y then y else x) = x";
   237 by (stac split_if 1);
   238 by (Blast_tac 1);
   239 qed "if_eq_cancel";
   240 
   241 (*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
   242 Goal "(if P then Q else R) = ((P-->Q) & (~P-->R))";
   243 by (rtac split_if 1);
   244 qed "if_bool_eq_conj";
   245 
   246 (*And this form is useful for expanding IFs on the LEFT*)
   247 Goal "(if P then Q else R) = ((P&Q) | (~P&R))";
   248 by (stac split_if 1);
   249 by (Blast_tac 1);
   250 qed "if_bool_eq_disj";
   251 
   252 local
   253 val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
   254               (fn prems => [cut_facts_tac prems 1, Blast_tac 1]);
   255 
   256 val iff_allI = allI RS
   257     prove_goal (the_context()) "!x. P x = Q x ==> (!x. P x) = (!x. Q x)"
   258                (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
   259 in
   260 
   261 (*** make simplification procedures for quantifier elimination ***)
   262 
   263 structure Quantifier1 = Quantifier1Fun
   264 (struct
   265   (*abstract syntax*)
   266   fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
   267     | dest_eq _ = None;
   268   fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
   269     | dest_conj _ = None;
   270   fun dest_imp((c as Const("op -->",_)) $ s $ t) = Some(c,s,t)
   271     | dest_imp _ = None;
   272   val conj = HOLogic.conj
   273   val imp  = HOLogic.imp
   274   (*rules*)
   275   val iff_reflection = eq_reflection
   276   val iffI = iffI
   277   val conjI= conjI
   278   val conjE= conjE
   279   val impI = impI
   280   val mp   = mp
   281   val uncurry = uncurry
   282   val exI  = exI
   283   val exE  = exE
   284   val iff_allI = iff_allI
   285 end);
   286 
   287 end;
   288 
   289 local
   290 val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
   291     ("EX x. P(x) & Q(x)",HOLogic.boolT)
   292 val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
   293     ("ALL x. P(x) --> Q(x)",HOLogic.boolT)
   294 in
   295 val defEX_regroup = mk_simproc "defined EX" [ex_pattern]
   296       Quantifier1.rearrange_ex
   297 val defALL_regroup = mk_simproc "defined ALL" [all_pattern]
   298       Quantifier1.rearrange_all
   299 end;
   300 
   301 
   302 (*** Case splitting ***)
   303 
   304 structure SplitterData =
   305   struct
   306   structure Simplifier = Simplifier
   307   val mk_eq          = mk_eq
   308   val meta_eq_to_iff = meta_eq_to_obj_eq
   309   val iffD           = iffD2
   310   val disjE          = disjE
   311   val conjE          = conjE
   312   val exE            = exE
   313   val contrapos      = contrapos_nn
   314   val contrapos2     = contrapos_pp
   315   val notnotD        = notnotD
   316   end;
   317 
   318 structure Splitter = SplitterFun(SplitterData);
   319 
   320 val split_tac        = Splitter.split_tac;
   321 val split_inside_tac = Splitter.split_inside_tac;
   322 val split_asm_tac    = Splitter.split_asm_tac;
   323 val op addsplits     = Splitter.addsplits;
   324 val op delsplits     = Splitter.delsplits;
   325 val Addsplits        = Splitter.Addsplits;
   326 val Delsplits        = Splitter.Delsplits;
   327 
   328 (*In general it seems wrong to add distributive laws by default: they
   329   might cause exponential blow-up.  But imp_disjL has been in for a while
   330   and cannot be removed without affecting existing proofs.  Moreover,
   331   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   332   grounds that it allows simplification of R in the two cases.*)
   333 
   334 val mksimps_pairs =
   335   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   336    ("All", [spec]), ("True", []), ("False", []),
   337    ("If", [if_bool_eq_conj RS iffD1])];
   338 
   339 (* ###FIXME: move to Provers/simplifier.ML
   340 val mk_atomize:      (string * thm list) list -> thm -> thm list
   341 *)
   342 (* ###FIXME: move to Provers/simplifier.ML *)
   343 fun mk_atomize pairs =
   344   let fun atoms th =
   345         (case concl_of th of
   346            Const("Trueprop",_) $ p =>
   347              (case head_of p of
   348                 Const(a,_) =>
   349                   (case assoc(pairs,a) of
   350                      Some(rls) => flat (map atoms ([th] RL rls))
   351                    | None => [th])
   352               | _ => [th])
   353          | _ => [th])
   354   in atoms end;
   355 
   356 fun mksimps pairs =
   357   (mapfilter (try mk_eq) o mk_atomize pairs o forall_elim_vars_safe);
   358 
   359 fun unsafe_solver_tac prems =
   360   FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
   361 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
   362 
   363 (*No premature instantiation of variables during simplification*)
   364 fun safe_solver_tac prems =
   365   FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
   366          eq_assume_tac, ematch_tac [FalseE]];
   367 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
   368 
   369 val HOL_basic_ss =
   370   empty_ss setsubgoaler asm_simp_tac
   371     setSSolver safe_solver
   372     setSolver unsafe_solver
   373     setmksimps (mksimps mksimps_pairs)
   374     setmkeqTrue mk_eq_True
   375     setmkcong mk_meta_cong;
   376 
   377 val HOL_ss =
   378     HOL_basic_ss addsimps
   379      ([triv_forall_equality, (* prunes params *)
   380        True_implies_equals, (* prune asms `True' *)
   381        eta_contract_eq, (* prunes eta-expansions *)
   382        if_True, if_False, if_cancel, if_eq_cancel,
   383        imp_disjL, conj_assoc, disj_assoc,
   384        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
   385        disj_not1, not_all, not_ex, cases_simp,
   386        thm "the_eq_trivial", the_sym_eq_trivial, thm "plus_ac0.zero", thm "plus_ac0_zero_right"]
   387      @ ex_simps @ all_simps @ simp_thms)
   388      addsimprocs [defALL_regroup,defEX_regroup]
   389      addcongs [imp_cong]
   390      addsplits [split_if];
   391 
   392 fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
   393 fun hol_rewrite_cterm rews =
   394   #2 o Thm.dest_comb o #prop o Thm.crep_thm o Simplifier.full_rewrite (HOL_basic_ss addsimps rews);
   395 
   396 
   397 (*Simplifies x assuming c and y assuming ~c*)
   398 val prems = Goalw [if_def]
   399   "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
   400 \  (if b then x else y) = (if c then u else v)";
   401 by (asm_simp_tac (HOL_ss addsimps prems) 1);
   402 qed "if_cong";
   403 
   404 (*Prevents simplification of x and y: faster and allows the execution
   405   of functional programs. NOW THE DEFAULT.*)
   406 Goal "b=c ==> (if b then x else y) = (if c then x else y)";
   407 by (etac arg_cong 1);
   408 qed "if_weak_cong";
   409 
   410 (*Prevents simplification of t: much faster*)
   411 Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
   412 by (etac arg_cong 1);
   413 qed "let_weak_cong";
   414 
   415 Goal "f(if c then x else y) = (if c then f x else f y)";
   416 by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
   417 qed "if_distrib";
   418 
   419 (*For expand_case_tac*)
   420 val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
   421 by (case_tac "P" 1);
   422 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
   423 qed "expand_case";
   424 
   425 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
   426   during unification.*)
   427 fun expand_case_tac P i =
   428     res_inst_tac [("P",P)] expand_case i THEN
   429     Simp_tac (i+1) THEN
   430     Simp_tac i;
   431 
   432 (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
   433   side of an equality.  Used in {Integ,Real}/simproc.ML*)
   434 Goal "x=y ==> (x=z) = (y=z)";
   435 by (asm_simp_tac HOL_ss 1);
   436 qed "restrict_to_left";
   437 
   438 (* default simpset *)
   439 val simpsetup =
   440   [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong]; thy)];
   441 
   442 
   443 (*** integration of simplifier with classical reasoner ***)
   444 
   445 structure Clasimp = ClasimpFun
   446  (structure Simplifier = Simplifier and Splitter = Splitter
   447   and Classical  = Classical and Blast = Blast
   448   val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
   449   val cla_make_elim = cla_make_elim);
   450 open Clasimp;
   451 
   452 val HOL_css = (HOL_cs, HOL_ss);
   453 
   454 
   455 
   456 (*** A general refutation procedure ***)
   457 
   458 (* Parameters:
   459 
   460    test: term -> bool
   461    tests if a term is at all relevant to the refutation proof;
   462    if not, then it can be discarded. Can improve performance,
   463    esp. if disjunctions can be discarded (no case distinction needed!).
   464 
   465    prep_tac: int -> tactic
   466    A preparation tactic to be applied to the goal once all relevant premises
   467    have been moved to the conclusion.
   468 
   469    ref_tac: int -> tactic
   470    the actual refutation tactic. Should be able to deal with goals
   471    [| A1; ...; An |] ==> False
   472    where the Ai are atomic, i.e. no top-level &, | or EX
   473 *)
   474 
   475 fun refute_tac test prep_tac ref_tac =
   476   let val nnf_simps =
   477         [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
   478          not_all,not_ex,not_not];
   479       val nnf_simpset =
   480         empty_ss setmkeqTrue mk_eq_True
   481                  setmksimps (mksimps mksimps_pairs)
   482                  addsimps nnf_simps;
   483       val prem_nnf_tac = full_simp_tac nnf_simpset;
   484 
   485       val refute_prems_tac =
   486         REPEAT(eresolve_tac [conjE, exE] 1 ORELSE
   487                filter_prems_tac test 1 ORELSE
   488                etac disjE 1) THEN
   489         ((etac notE 1 THEN eq_assume_tac 1) ORELSE
   490          ref_tac 1);
   491   in EVERY'[TRY o filter_prems_tac test,
   492             DETERM o REPEAT o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
   493             SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
   494   end;