src/HOL/simpdata.ML
author wenzelm
Sat Apr 04 12:29:07 1998 +0200 (1998-04-04)
changeset 4794 9db0916ecdae
parent 4769 bb60149fe21b
child 4830 bd73675adbed
permissions -rw-r--r--
no open Simplifier;
     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.
     7 *)
     8 
     9 section "Simplifier";
    10 
    11 (*** Addition of rules to simpsets and clasets simultaneously ***)
    12 
    13 (*Takes UNCONDITIONAL theorems of the form A<->B to 
    14         the Safe Intr     rule B==>A and 
    15         the Safe Destruct rule A==>B.
    16   Also ~A goes to the Safe Elim rule A ==> ?R
    17   Failing other cases, A is added as a Safe Intr rule*)
    18 local
    19   val iff_const = HOLogic.eq_const HOLogic.boolT;
    20 
    21   fun addIff th = 
    22       (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
    23                 (Const("Not",_) $ A) =>
    24                     AddSEs [zero_var_indexes (th RS notE)]
    25               | (con $ _ $ _) =>
    26                     if con=iff_const
    27                     then (AddSIs [zero_var_indexes (th RS iffD2)];  
    28                           AddSDs [zero_var_indexes (th RS iffD1)])
    29                     else  AddSIs [th]
    30               | _ => AddSIs [th];
    31        Addsimps [th])
    32       handle _ => error ("AddIffs: theorem must be unconditional\n" ^ 
    33                          string_of_thm th)
    34 
    35   fun delIff th = 
    36       (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
    37                 (Const("Not",_) $ A) =>
    38                     Delrules [zero_var_indexes (th RS notE)]
    39               | (con $ _ $ _) =>
    40                     if con=iff_const
    41                     then Delrules [zero_var_indexes (th RS iffD2),
    42                                    make_elim (zero_var_indexes (th RS iffD1))]
    43                     else Delrules [th]
    44               | _ => Delrules [th];
    45        Delsimps [th])
    46       handle _ => warning("DelIffs: ignoring conditional theorem\n" ^ 
    47                           string_of_thm th)
    48 in
    49 val AddIffs = seq addIff
    50 val DelIffs = seq delIff
    51 end;
    52 
    53 qed_goal "meta_eq_to_obj_eq" HOL.thy "x==y ==> x=y"
    54   (fn [prem] => [rewtac prem, rtac refl 1]);
    55 
    56 local
    57 
    58   fun prover s = prove_goal HOL.thy s (K [Blast_tac 1]);
    59 
    60   val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
    61   val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
    62 
    63   val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
    64   val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
    65 
    66   fun atomize pairs =
    67     let fun atoms th =
    68           (case concl_of th of
    69              Const("Trueprop",_) $ p =>
    70                (case head_of p of
    71                   Const(a,_) =>
    72                     (case assoc(pairs,a) of
    73                        Some(rls) => flat (map atoms ([th] RL rls))
    74                      | None => [th])
    75                 | _ => [th])
    76            | _ => [th])
    77     in atoms end;
    78 
    79   fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
    80 
    81 in
    82 
    83   fun mk_meta_eq r = r RS eq_reflection;
    84   fun mk_meta_eq_True r = Some(r RS meta_eq_to_obj_eq RS P_imp_P_eq_True);
    85 
    86   fun mk_meta_eq_simp r = case concl_of r of
    87           Const("==",_)$_$_ => r
    88       |   _$(Const("op =",_)$lhs$rhs) => mk_meta_eq r
    89       |   _$(Const("Not",_)$_) => r RS not_P_imp_P_eq_False
    90       |   _ => r RS P_imp_P_eq_True;
    91   (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
    92 
    93 val simp_thms = map prover
    94  [ "(x=x) = True",
    95    "(~True) = False", "(~False) = True", "(~ ~ P) = P",
    96    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
    97    "(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)",
    98    "(True --> P) = P", "(False --> P) = True", 
    99    "(P --> True) = True", "(P --> P) = True",
   100    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
   101    "(P & True) = P", "(True & P) = P", 
   102    "(P & False) = False", "(False & P) = False",
   103    "(P & P) = P", "(P & (P & Q)) = (P & Q)",
   104    "(P & ~P) = False",    "(~P & P) = False",
   105    "(P | True) = True", "(True | P) = True", 
   106    "(P | False) = P", "(False | P) = P",
   107    "(P | P) = P", "(P | (P | Q)) = (P | Q)",
   108    "(P | ~P) = True",    "(~P | P) = True",
   109    "((~P) = (~Q)) = (P=Q)",
   110    "(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x", 
   111 (*two needed for the one-point-rule quantifier simplification procs*)
   112    "(? x. x=t & P(x)) = P(t)",		(*essential for termination!!*)
   113    "(! x. t=x --> P(x)) = P(t)" ];      (*covers a stray case*)
   114 
   115 (*Add congruence rules for = (instead of ==) *)
   116 infix 4 addcongs delcongs;
   117 
   118 fun mk_meta_cong rl =
   119   standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
   120   handle THM _ =>
   121   error("Premises and conclusion of congruence rules must be =-equalities");
   122 
   123 fun ss addcongs congs = ss addeqcongs (map mk_meta_cong congs);
   124 
   125 fun ss delcongs congs = ss deleqcongs (map mk_meta_cong congs);
   126 
   127 fun Addcongs congs = (simpset_ref() := simpset() addcongs congs);
   128 fun Delcongs congs = (simpset_ref() := simpset() delcongs congs);
   129 
   130 fun mksimps pairs = map mk_meta_eq_simp o atomize pairs o gen_all;
   131 
   132 val imp_cong = impI RSN
   133     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
   134         (fn _=> [Blast_tac 1]) RS mp RS mp);
   135 
   136 (*Miniscoping: pushing in existential quantifiers*)
   137 val ex_simps = map prover 
   138                 ["(EX x. P x & Q)   = ((EX x. P x) & Q)",
   139                  "(EX x. P & Q x)   = (P & (EX x. Q x))",
   140                  "(EX x. P x | Q)   = ((EX x. P x) | Q)",
   141                  "(EX x. P | Q x)   = (P | (EX x. Q x))",
   142                  "(EX x. P x --> Q) = ((ALL x. P x) --> Q)",
   143                  "(EX x. P --> Q x) = (P --> (EX x. Q x))"];
   144 
   145 (*Miniscoping: pushing in universal quantifiers*)
   146 val all_simps = map prover
   147                 ["(ALL x. P x & Q)   = ((ALL x. P x) & Q)",
   148                  "(ALL x. P & Q x)   = (P & (ALL x. Q x))",
   149                  "(ALL x. P x | Q)   = ((ALL x. P x) | Q)",
   150                  "(ALL x. P | Q x)   = (P | (ALL x. Q x))",
   151                  "(ALL x. P x --> Q) = ((EX x. P x) --> Q)",
   152                  "(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];
   153 
   154 
   155 (* elimination of existential quantifiers in assumptions *)
   156 
   157 val ex_all_equiv =
   158   let val lemma1 = prove_goal HOL.thy
   159         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
   160         (fn prems => [resolve_tac prems 1, etac exI 1]);
   161       val lemma2 = prove_goalw HOL.thy [Ex_def]
   162         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
   163         (fn prems => [REPEAT(resolve_tac prems 1)])
   164   in equal_intr lemma1 lemma2 end;
   165 
   166 end;
   167 
   168 (* Elimination of True from asumptions: *)
   169 
   170 val True_implies_equals = prove_goal HOL.thy
   171  "(True ==> PROP P) == PROP P"
   172 (K [rtac equal_intr_rule 1, atac 2,
   173           METAHYPS (fn prems => resolve_tac prems 1) 1,
   174           rtac TrueI 1]);
   175 
   176 fun prove nm thm  = qed_goal nm HOL.thy thm (K [Blast_tac 1]);
   177 
   178 prove "conj_commute" "(P&Q) = (Q&P)";
   179 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
   180 val conj_comms = [conj_commute, conj_left_commute];
   181 prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
   182 
   183 prove "disj_commute" "(P|Q) = (Q|P)";
   184 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
   185 val disj_comms = [disj_commute, disj_left_commute];
   186 prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
   187 
   188 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
   189 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
   190 
   191 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
   192 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
   193 
   194 prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
   195 prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";
   196 prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
   197 
   198 (*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
   199 prove "imp_disj_not1" "((P --> Q | R)) = (~Q --> P --> R)";
   200 prove "imp_disj_not2" "((P --> Q | R)) = (~R --> P --> Q)";
   201 
   202 prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";
   203 prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";
   204 
   205 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
   206 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
   207 prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
   208 prove "not_iff" "(P~=Q) = (P = (~Q))";
   209 prove "disj_not1" "(~P | Q) = (P --> Q)";
   210 prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *)
   211 
   212 (*Avoids duplication of subgoals after expand_if, when the true and false 
   213   cases boil down to the same thing.*) 
   214 prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
   215 
   216 prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))";
   217 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
   218 prove "not_ex"  "(~ (? x. P(x))) = (! x.~P(x))";
   219 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
   220 
   221 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
   222 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
   223 
   224 (* '&' congruence rule: not included by default!
   225    May slow rewrite proofs down by as much as 50% *)
   226 
   227 let val th = prove_goal HOL.thy 
   228                 "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
   229                 (fn _=> [Blast_tac 1])
   230 in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   231 
   232 let val th = prove_goal HOL.thy 
   233                 "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
   234                 (fn _=> [Blast_tac 1])
   235 in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   236 
   237 (* '|' congruence rule: not included by default! *)
   238 
   239 let val th = prove_goal HOL.thy 
   240                 "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
   241                 (fn _=> [Blast_tac 1])
   242 in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   243 
   244 prove "eq_sym_conv" "(x=y) = (y=x)";
   245 
   246 qed_goalw "o_apply" HOL.thy [o_def] "(f o g) x = f (g x)"
   247  (K [rtac refl 1]);
   248 
   249 qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x"
   250  (K [Blast_tac 1]);
   251 
   252 qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y"
   253  (K [Blast_tac 1]);
   254 
   255 qed_goal "if_P" HOL.thy "P ==> (if P then x else y) = x"
   256  (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
   257 (*
   258 qed_goal "if_not_P" HOL.thy "~P ==> (if P then x else y) = y"
   259  (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
   260 *)
   261 qed_goalw "if_not_P" HOL.thy [if_def] "!!P. ~P ==> (if P then x else y) = y"
   262  (K [Blast_tac 1]);
   263 
   264 qed_goal "expand_if" HOL.thy
   265     "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))" (K [
   266 	res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1,
   267          stac if_P 2,
   268          stac if_not_P 1,
   269          ALLGOALS (Blast_tac)]);
   270 
   271 qed_goal "split_if_asm" HOL.thy
   272     "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
   273     (K [stac expand_if 1,
   274 	Blast_tac 1]);
   275 
   276 (*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
   277 qed_goal "if_bool_eq_conj" HOL.thy
   278     "(if P then Q else R) = ((P-->Q) & (~P-->R))"
   279     (K [rtac expand_if 1]);
   280 
   281 (*And this form is useful for expanding IFs on the LEFT*)
   282 qed_goal "if_bool_eq_disj" HOL.thy
   283     "(if P then Q else R) = ((P&Q) | (~P&R))"
   284     (K [stac expand_if 1,
   285 	Blast_tac 1]);
   286 
   287 
   288 (*** make simplification procedures for quantifier elimination ***)
   289 
   290 structure Quantifier1 = Quantifier1Fun(
   291 struct
   292   (*abstract syntax*)
   293   fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
   294     | dest_eq _ = None;
   295   fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
   296     | dest_conj _ = None;
   297   val conj = HOLogic.conj
   298   val imp  = HOLogic.imp
   299   (*rules*)
   300   val iff_reflection = eq_reflection
   301   val iffI = iffI
   302   val sym  = sym
   303   val conjI= conjI
   304   val conjE= conjE
   305   val impI = impI
   306   val impE = impE
   307   val mp   = mp
   308   val exI  = exI
   309   val exE  = exE
   310   val allI = allI
   311   val allE = allE
   312 end);
   313 
   314 local
   315 val ex_pattern =
   316   read_cterm (sign_of HOL.thy) ("EX x. P(x) & Q(x)",HOLogic.boolT)
   317 
   318 val all_pattern =
   319   read_cterm (sign_of HOL.thy) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT)
   320 
   321 in
   322 val defEX_regroup =
   323   mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
   324 val defALL_regroup =
   325   mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
   326 end;
   327 
   328 
   329 (*** Case splitting ***)
   330 
   331 local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
   332 in
   333 fun split_tac splits = mktac (map mk_meta_eq splits)
   334 end;
   335 
   336 local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2)
   337 in
   338 fun split_inside_tac splits = mktac (map mk_meta_eq splits)
   339 end;
   340 
   341 val split_asm_tac = mk_case_split_asm_tac split_tac 
   342 			(disjE,conjE,exE,contrapos,contrapos2,notnotD);
   343 
   344 infix 4 addsplits delsplits;
   345 
   346 fun ss addsplits splits =
   347   let fun addsplit(ss,split) =
   348         let val name = "split " ^ const_of_split_thm split
   349         in ss addloop (name,split_tac [split]) end
   350   in foldl addsplit (ss,splits) end;
   351 
   352 fun ss delsplits splits =
   353   let fun delsplit(ss,split) =
   354         let val name = "split " ^ const_of_split_thm split
   355         in ss delloop name end
   356   in foldl delsplit (ss,splits) end;
   357 
   358 fun Addsplits splits = (simpset_ref() := simpset() addsplits splits);
   359 fun Delsplits splits = (simpset_ref() := simpset() delsplits splits);
   360 
   361 qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
   362   (K [split_tac [expand_if] 1, Blast_tac 1]);
   363 
   364 qed_goal "if_eq_cancel" HOL.thy "(if x = y then y else x) = x"
   365   (K [split_tac [expand_if] 1, Blast_tac 1]);
   366 
   367 (** 'if' congruence rules: neither included by default! *)
   368 
   369 (*Simplifies x assuming c and y assuming ~c*)
   370 qed_goal "if_cong" HOL.thy
   371   "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
   372 \  (if b then x else y) = (if c then u else v)"
   373   (fn rew::prems =>
   374    [stac rew 1, stac expand_if 1, stac expand_if 1,
   375     blast_tac (HOL_cs addDs prems) 1]);
   376 
   377 (*Prevents simplification of x and y: much faster*)
   378 qed_goal "if_weak_cong" HOL.thy
   379   "b=c ==> (if b then x else y) = (if c then x else y)"
   380   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   381 
   382 (*Prevents simplification of t: much faster*)
   383 qed_goal "let_weak_cong" HOL.thy
   384   "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
   385   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   386 
   387 (*In general it seems wrong to add distributive laws by default: they
   388   might cause exponential blow-up.  But imp_disjL has been in for a while
   389   and cannot be removed without affecting existing proofs.  Moreover, 
   390   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   391   grounds that it allows simplification of R in the two cases.*)
   392 
   393 val mksimps_pairs =
   394   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   395    ("All", [spec]), ("True", []), ("False", []),
   396    ("If", [if_bool_eq_conj RS iffD1])];
   397 
   398 fun unsafe_solver prems = FIRST'[resolve_tac (reflexive_thm::TrueI::refl::prems),
   399 				 atac, etac FalseE];
   400 (*No premature instantiation of variables during simplification*)
   401 fun   safe_solver prems = FIRST'[match_tac (reflexive_thm::TrueI::prems),
   402 				 eq_assume_tac, ematch_tac [FalseE]];
   403 
   404 val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
   405 			    setSSolver   safe_solver
   406 			    setSolver  unsafe_solver
   407 			    setmksimps (mksimps mksimps_pairs)
   408 			    setmkeqTrue mk_meta_eq_True;
   409 
   410 val HOL_ss = 
   411     HOL_basic_ss addsimps 
   412      ([triv_forall_equality, (* prunes params *)
   413        True_implies_equals, (* prune asms `True' *)
   414        if_True, if_False, if_cancel, if_eq_cancel,
   415        o_apply, imp_disjL, conj_assoc, disj_assoc,
   416        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
   417        disj_not1, not_all, not_ex, cases_simp]
   418      @ ex_simps @ all_simps @ simp_thms)
   419      addsimprocs [defALL_regroup,defEX_regroup]
   420      addcongs [imp_cong]
   421      addsplits [expand_if];
   422 
   423 qed_goal "if_distrib" HOL.thy
   424   "f(if c then x else y) = (if c then f x else f y)" 
   425   (K [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
   426 
   427 qed_goalw "o_assoc" HOL.thy [o_def] "f o (g o h) = f o g o h"
   428   (K [rtac ext 1, rtac refl 1]);
   429 
   430 
   431 (*For expand_case_tac*)
   432 val prems = goal HOL.thy "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
   433 by (case_tac "P" 1);
   434 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
   435 val expand_case = result();
   436 
   437 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
   438   during unification.*)
   439 fun expand_case_tac P i =
   440     res_inst_tac [("P",P)] expand_case i THEN
   441     Simp_tac (i+1) THEN 
   442     Simp_tac i;
   443 
   444 
   445 (* install implicit simpset *)
   446 
   447 simpset_ref() := HOL_ss;
   448 
   449 
   450 
   451 (*** Integration of simplifier with classical reasoner ***)
   452 
   453 (* rot_eq_tac rotates the first equality premise of subgoal i to the front,
   454    fails if there is no equaliy or if an equality is already at the front *)
   455 local
   456   fun is_eq (Const ("Trueprop", _) $ (Const("op ="  ,_) $ _ $ _)) = true
   457     | is_eq _ = false;
   458   val find_eq = find_index is_eq;
   459 in
   460 val rot_eq_tac = 
   461      SUBGOAL (fn (Bi,i) => let val n = find_eq (Logic.strip_assums_hyp Bi) in
   462 		if n>0 then rotate_tac n i else no_tac end)
   463 end;
   464 
   465 use "$ISABELLE_HOME/src/Provers/clasimp.ML";
   466 open Clasimp;
   467 
   468 val HOL_css = (HOL_cs, HOL_ss);