src/HOL/simpdata.ML
author oheimb
Wed Feb 25 20:25:27 1998 +0100 (1998-02-25)
changeset 4652 d24cca140eeb
parent 4651 70dd492a1698
child 4669 06f3c56dcba8
permissions -rw-r--r--
factored out common code of HOL/simpdata.ML and FOL/simpdata.ML concerning
combination of classical reasoner and simplifier auto_tac into Provers/clasimp.ML
explicitly introducing combined type clasimpset
     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 open Simplifier;
    12 
    13 (*** Addition of rules to simpsets and clasets simultaneously ***)
    14 
    15 (*Takes UNCONDITIONAL theorems of the form A<->B to 
    16         the Safe Intr     rule B==>A and 
    17         the Safe Destruct rule A==>B.
    18   Also ~A goes to the Safe Elim rule A ==> ?R
    19   Failing other cases, A is added as a Safe Intr rule*)
    20 local
    21   val iff_const = HOLogic.eq_const HOLogic.boolT;
    22 
    23   fun addIff th = 
    24       (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
    25                 (Const("Not",_) $ A) =>
    26                     AddSEs [zero_var_indexes (th RS notE)]
    27               | (con $ _ $ _) =>
    28                     if con=iff_const
    29                     then (AddSIs [zero_var_indexes (th RS iffD2)];  
    30                           AddSDs [zero_var_indexes (th RS iffD1)])
    31                     else  AddSIs [th]
    32               | _ => AddSIs [th];
    33        Addsimps [th])
    34       handle _ => error ("AddIffs: theorem must be unconditional\n" ^ 
    35                          string_of_thm th)
    36 
    37   fun delIff th = 
    38       (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
    39                 (Const("Not",_) $ A) =>
    40                     Delrules [zero_var_indexes (th RS notE)]
    41               | (con $ _ $ _) =>
    42                     if con=iff_const
    43                     then Delrules [zero_var_indexes (th RS iffD2),
    44                                    make_elim (zero_var_indexes (th RS iffD1))]
    45                     else Delrules [th]
    46               | _ => Delrules [th];
    47        Delsimps [th])
    48       handle _ => warning("DelIffs: ignoring conditional theorem\n" ^ 
    49                           string_of_thm th)
    50 in
    51 val AddIffs = seq addIff
    52 val DelIffs = seq delIff
    53 end;
    54 
    55 qed_goal "meta_eq_to_obj_eq" HOL.thy "x==y ==> x=y"
    56   (fn [prem] => [rewtac prem, rtac refl 1]);
    57 
    58 local
    59 
    60   fun prover s = prove_goal HOL.thy s (K [blast_tac HOL_cs 1]);
    61 
    62   val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
    63   val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
    64 
    65   val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
    66   val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
    67 
    68   fun atomize pairs =
    69     let fun atoms th =
    70           (case concl_of th of
    71              Const("Trueprop",_) $ p =>
    72                (case head_of p of
    73                   Const(a,_) =>
    74                     (case assoc(pairs,a) of
    75                        Some(rls) => flat (map atoms ([th] RL rls))
    76                      | None => [th])
    77                 | _ => [th])
    78            | _ => [th])
    79     in atoms end;
    80 
    81   fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
    82 
    83 in
    84 
    85   fun mk_meta_eq r = r RS eq_reflection;
    86 
    87   fun mk_meta_eq_simp r = case concl_of r of
    88           Const("==",_)$_$_ => r
    89       |   _$(Const("op =",_)$lhs$rhs) =>
    90              (case fst(Logic.rewrite_rule_ok (#sign(rep_thm r)) (prems_of r) lhs rhs) of
    91                 None => mk_meta_eq r
    92               | Some _ => r RS P_imp_P_eq_True)
    93       |   _$(Const("Not",_)$_) => r RS not_P_imp_P_eq_False
    94       |   _ => r RS P_imp_P_eq_True;
    95   (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
    96 
    97 val simp_thms = map prover
    98  [ "(x=x) = True",
    99    "(~True) = False", "(~False) = True", "(~ ~ P) = P",
   100    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
   101    "(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)",
   102    "(True --> P) = P", "(False --> P) = True", 
   103    "(P --> True) = True", "(P --> P) = True",
   104    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
   105    "(P & True) = P", "(True & P) = P", 
   106    "(P & False) = False", "(False & P) = False",
   107    "(P & P) = P", "(P & (P & Q)) = (P & Q)",
   108    "(P & ~P) = False",    "(~P & P) = False",
   109    "(P | True) = True", "(True | P) = True", 
   110    "(P | False) = P", "(False | P) = P",
   111    "(P | P) = P", "(P | (P | Q)) = (P | Q)",
   112    "(P | ~P) = True",    "(~P | P) = True",
   113    "((~P) = (~Q)) = (P=Q)",
   114    "(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x", 
   115 (*two needed for the one-point-rule quantifier simplification procs*)
   116    "(? x. x=t & P(x)) = P(t)",		(*essential for termination!!*)
   117    "(! x. t=x --> P(x)) = P(t)" ];      (*covers a stray case*)
   118 
   119 (*Add congruence rules for = (instead of ==) *)
   120 infix 4 addcongs delcongs;
   121 
   122 fun mk_meta_cong rl =
   123   standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
   124   handle THM _ =>
   125   error("Premises and conclusion of congruence rules must be =-equalities");
   126 
   127 fun ss addcongs congs = ss addeqcongs (map mk_meta_cong congs);
   128 
   129 fun ss delcongs congs = ss deleqcongs (map mk_meta_cong congs);
   130 
   131 fun Addcongs congs = (simpset_ref() := simpset() addcongs congs);
   132 fun Delcongs congs = (simpset_ref() := simpset() delcongs congs);
   133 
   134 fun mksimps pairs = map mk_meta_eq_simp o atomize pairs o gen_all;
   135 
   136 val imp_cong = impI RSN
   137     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
   138         (fn _=> [blast_tac HOL_cs 1]) RS mp RS mp);
   139 
   140 (*Miniscoping: pushing in existential quantifiers*)
   141 val ex_simps = map prover 
   142                 ["(EX x. P x & Q)   = ((EX x. P x) & Q)",
   143                  "(EX x. P & Q x)   = (P & (EX x. Q x))",
   144                  "(EX x. P x | Q)   = ((EX x. P x) | Q)",
   145                  "(EX x. P | Q x)   = (P | (EX x. Q x))",
   146                  "(EX x. P x --> Q) = ((ALL x. P x) --> Q)",
   147                  "(EX x. P --> Q x) = (P --> (EX x. Q x))"];
   148 
   149 (*Miniscoping: pushing in universal quantifiers*)
   150 val all_simps = map prover
   151                 ["(ALL x. P x & Q)   = ((ALL x. P x) & Q)",
   152                  "(ALL x. P & Q x)   = (P & (ALL x. Q x))",
   153                  "(ALL x. P x | Q)   = ((ALL x. P x) | Q)",
   154                  "(ALL x. P | Q x)   = (P | (ALL x. Q x))",
   155                  "(ALL x. P x --> Q) = ((EX x. P x) --> Q)",
   156                  "(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];
   157 
   158 
   159 (* elimination of existential quantifiers in assumptions *)
   160 
   161 val ex_all_equiv =
   162   let val lemma1 = prove_goal HOL.thy
   163         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
   164         (fn prems => [resolve_tac prems 1, etac exI 1]);
   165       val lemma2 = prove_goalw HOL.thy [Ex_def]
   166         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
   167         (fn prems => [REPEAT(resolve_tac prems 1)])
   168   in equal_intr lemma1 lemma2 end;
   169 
   170 end;
   171 
   172 (* Elimination of True from asumptions: *)
   173 
   174 val True_implies_equals = prove_goal HOL.thy
   175  "(True ==> PROP P) == PROP P"
   176 (K [rtac equal_intr_rule 1, atac 2,
   177           METAHYPS (fn prems => resolve_tac prems 1) 1,
   178           rtac TrueI 1]);
   179 
   180 fun prove nm thm  = qed_goal nm HOL.thy thm (K [blast_tac HOL_cs 1]);
   181 
   182 prove "conj_commute" "(P&Q) = (Q&P)";
   183 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
   184 val conj_comms = [conj_commute, conj_left_commute];
   185 prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
   186 
   187 prove "disj_commute" "(P|Q) = (Q|P)";
   188 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
   189 val disj_comms = [disj_commute, disj_left_commute];
   190 prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
   191 
   192 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
   193 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
   194 
   195 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
   196 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
   197 
   198 prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
   199 prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";
   200 prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
   201 
   202 (*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
   203 prove "imp_disj_not1" "((P --> Q | R)) = (~Q --> P --> R)";
   204 prove "imp_disj_not2" "((P --> Q | R)) = (~R --> P --> Q)";
   205 
   206 prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";
   207 prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";
   208 
   209 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
   210 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
   211 prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
   212 prove "not_iff" "(P~=Q) = (P = (~Q))";
   213 
   214 (*Avoids duplication of subgoals after expand_if, when the true and false 
   215   cases boil down to the same thing.*) 
   216 prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
   217 
   218 prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))";
   219 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
   220 prove "not_ex"  "(~ (? x. P(x))) = (! x.~P(x))";
   221 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
   222 
   223 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
   224 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
   225 
   226 (* '&' congruence rule: not included by default!
   227    May slow rewrite proofs down by as much as 50% *)
   228 
   229 let val th = prove_goal HOL.thy 
   230                 "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
   231                 (fn _=> [blast_tac HOL_cs 1])
   232 in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   233 
   234 let val th = prove_goal HOL.thy 
   235                 "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
   236                 (fn _=> [blast_tac HOL_cs 1])
   237 in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   238 
   239 (* '|' congruence rule: not included by default! *)
   240 
   241 let val th = prove_goal HOL.thy 
   242                 "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
   243                 (fn _=> [blast_tac HOL_cs 1])
   244 in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   245 
   246 prove "eq_sym_conv" "(x=y) = (y=x)";
   247 
   248 qed_goalw "o_apply" HOL.thy [o_def] "(f o g) x = f (g x)"
   249  (K [rtac refl 1]);
   250 
   251 qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x"
   252  (K [Blast_tac 1]);
   253 
   254 qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y"
   255  (K [Blast_tac 1]);
   256 
   257 qed_goal "if_P" HOL.thy "P ==> (if P then x else y) = x"
   258  (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
   259 (*
   260 qed_goal "if_not_P" HOL.thy "~P ==> (if P then x else y) = y"
   261  (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
   262 *)
   263 qed_goalw "if_not_P" HOL.thy [if_def] "!!P. ~P ==> (if P then x else y) = y"
   264  (K [Blast_tac 1]);
   265 
   266 qed_goal "expand_if" HOL.thy
   267     "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))" (K [
   268 	res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1,
   269          stac if_P 2,
   270          stac if_not_P 1,
   271          ALLGOALS (blast_tac HOL_cs)]);
   272 
   273 qed_goal "split_if_asm" HOL.thy
   274     "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))" (K [
   275 	stac expand_if 1,
   276         blast_tac HOL_cs 1]);
   277 
   278 qed_goal "if_bool_eq" HOL.thy
   279                    "(if P then Q else R) = ((P-->Q) & (~P-->R))"
   280                    (K [rtac expand_if 1]);
   281 
   282 
   283 (*** make simplification procedures for quantifier elimination ***)
   284 
   285 structure Quantifier1 = Quantifier1Fun(
   286 struct
   287   (*abstract syntax*)
   288   fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
   289     | dest_eq _ = None;
   290   fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
   291     | dest_conj _ = None;
   292   val conj = HOLogic.conj
   293   val imp  = HOLogic.imp
   294   (*rules*)
   295   val iff_reflection = eq_reflection
   296   val iffI = iffI
   297   val sym  = sym
   298   val conjI= conjI
   299   val conjE= conjE
   300   val impI = impI
   301   val impE = impE
   302   val mp   = mp
   303   val exI  = exI
   304   val exE  = exE
   305   val allI = allI
   306   val allE = allE
   307 end);
   308 
   309 local
   310 val ex_pattern =
   311   read_cterm (sign_of HOL.thy) ("EX x. P(x) & Q(x)",HOLogic.boolT)
   312 
   313 val all_pattern =
   314   read_cterm (sign_of HOL.thy) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT)
   315 
   316 in
   317 val defEX_regroup =
   318   mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
   319 val defALL_regroup =
   320   mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
   321 end;
   322 
   323 
   324 (*** Case splitting ***)
   325 
   326 local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
   327 in
   328 fun split_tac splits = mktac (map mk_meta_eq splits)
   329 end;
   330 
   331 local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2)
   332 in
   333 fun split_inside_tac splits = mktac (map mk_meta_eq splits)
   334 end;
   335 
   336 val split_asm_tac = mk_case_split_asm_tac split_tac 
   337 			(disjE,conjE,exE,contrapos,contrapos2,notnotD);
   338 
   339 infix 4 addsplits;
   340 fun ss addsplits splits = ss addloop (split_tac splits);
   341 
   342 
   343 qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
   344   (K [split_tac [expand_if] 1, blast_tac HOL_cs 1]);
   345 
   346 (** 'if' congruence rules: neither included by default! *)
   347 
   348 (*Simplifies x assuming c and y assuming ~c*)
   349 qed_goal "if_cong" HOL.thy
   350   "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
   351 \  (if b then x else y) = (if c then u else v)"
   352   (fn rew::prems =>
   353    [stac rew 1, stac expand_if 1, stac expand_if 1,
   354     blast_tac (HOL_cs addDs prems) 1]);
   355 
   356 (*Prevents simplification of x and y: much faster*)
   357 qed_goal "if_weak_cong" HOL.thy
   358   "b=c ==> (if b then x else y) = (if c then x else y)"
   359   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   360 
   361 (*Prevents simplification of t: much faster*)
   362 qed_goal "let_weak_cong" HOL.thy
   363   "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
   364   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   365 
   366 (*In general it seems wrong to add distributive laws by default: they
   367   might cause exponential blow-up.  But imp_disjL has been in for a while
   368   and cannot be removed without affecting existing proofs.  Moreover, 
   369   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   370   grounds that it allows simplification of R in the two cases.*)
   371 
   372 val mksimps_pairs =
   373   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   374    ("All", [spec]), ("True", []), ("False", []),
   375    ("If", [if_bool_eq RS iffD1])];
   376 
   377 fun unsafe_solver prems = FIRST'[resolve_tac (reflexive_thm::TrueI::refl::prems),
   378 				 atac, etac FalseE];
   379 (*No premature instantiation of variables during simplification*)
   380 fun   safe_solver prems = FIRST'[match_tac (reflexive_thm::TrueI::prems),
   381 				 eq_assume_tac, ematch_tac [FalseE]];
   382 
   383 val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
   384 			    setSSolver   safe_solver
   385 			    setSolver  unsafe_solver
   386 			    setmksimps (mksimps mksimps_pairs);
   387 
   388 val HOL_ss = 
   389     HOL_basic_ss addsimps 
   390      ([triv_forall_equality, (* prunes params *)
   391        True_implies_equals, (* prune asms `True' *)
   392        if_True, if_False, if_cancel,
   393        o_apply, imp_disjL, conj_assoc, disj_assoc,
   394        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
   395        not_all, not_ex, cases_simp]
   396      @ ex_simps @ all_simps @ simp_thms)
   397      addsimprocs [defALL_regroup,defEX_regroup]
   398      addcongs [imp_cong];
   399 
   400 qed_goal "if_distrib" HOL.thy
   401   "f(if c then x else y) = (if c then f x else f y)" 
   402   (K [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
   403 
   404 qed_goalw "o_assoc" HOL.thy [o_def] "f o (g o h) = f o g o h"
   405   (K [rtac ext 1, rtac refl 1]);
   406 
   407 
   408 (*For expand_case_tac*)
   409 val prems = goal HOL.thy "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
   410 by (case_tac "P" 1);
   411 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
   412 val expand_case = result();
   413 
   414 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
   415   during unification.*)
   416 fun expand_case_tac P i =
   417     res_inst_tac [("P",P)] expand_case i THEN
   418     Simp_tac (i+1) THEN 
   419     Simp_tac i;
   420 
   421 
   422 (* install implicit simpset *)
   423 
   424 simpset_ref() := HOL_ss;
   425 
   426 
   427 
   428 
   429 
   430 
   431 (*** Integration of simplifier with classical reasoner ***)
   432 
   433 (* rot_eq_tac rotates the first equality premise of subgoal i to the front,
   434    fails if there is no equaliy or if an equality is already at the front *)
   435 local
   436   fun is_eq (Const ("Trueprop", _) $ (Const("op ="  ,_) $ _ $ _)) = true
   437     | is_eq _ = false;
   438   val find_eq = find_index is_eq;
   439 in
   440 val rot_eq_tac = 
   441      SUBGOAL (fn (Bi,i) => let val n = find_eq (Logic.strip_assums_hyp Bi) in
   442 		if n>0 then rotate_tac n i else no_tac end)
   443 end;
   444 
   445 use "$ISABELLE_HOME/src/Provers/clasimp.ML";
   446 open Clasimp;
   447 
   448 val HOL_css = (HOL_cs, HOL_ss);