src/HOL/simpdata.ML
author nipkow
Tue Mar 18 08:42:18 1997 +0100 (1997-03-18)
changeset 2800 9741c4c6b62b
parent 2748 3ae9ccdd701e
child 2805 6e5b2d6503eb
permissions -rw-r--r--
Added P&P&Q = P&Q and P|P|Q = P|Q.
     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                                    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 
    56 local
    57 
    58   fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 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 = case concl_of r of
    84           Const("==",_)$_$_ => r
    85       |   _$(Const("op =",_)$_$_) => r RS eq_reflection
    86       |   _$(Const("Not",_)$_) => r RS not_P_imp_P_eq_False
    87       |   _ => r RS P_imp_P_eq_True;
    88   (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
    89 
    90 val simp_thms = map prover
    91  [ "(x=x) = True",
    92    "(~True) = False", "(~False) = True", "(~ ~ P) = P",
    93    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
    94    "(True=P) = P", "(P=True) = P",
    95    "(True --> P) = P", "(False --> P) = True", 
    96    "(P --> True) = True", "(P --> P) = True",
    97    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
    98    "(P & True) = P", "(True & P) = P", 
    99    "(P & False) = False", "(False & P) = False",
   100    "(P & P) = P", "(P & (P & Q)) = (P & Q)",
   101    "(P | True) = True", "(True | P) = True", 
   102    "(P | False) = P", "(False | P) = P",
   103    "(P | P) = P", "(P | (P | Q)) = (P | Q)",
   104    "((~P) = (~Q)) = (P=Q)",
   105    "(!x.P) = P", "(? x.P) = P", "? x. x=t", "? x. t=x", 
   106    "(? x. x=t & P(x)) = P(t)", "(? x. t=x & P(x)) = P(t)", 
   107    "(! x. x=t --> P(x)) = P(t)", "(! x. t=x --> P(x)) = P(t)" ];
   108 
   109 (*Add congruence rules for = (instead of ==) *)
   110 infix 4 addcongs delcongs;
   111 fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection]);
   112 fun ss delcongs congs = ss deleqcongs (congs RL [eq_reflection]);
   113 
   114 fun Addcongs congs = (simpset := !simpset addcongs congs);
   115 fun Delcongs congs = (simpset := !simpset delcongs congs);
   116 
   117 fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
   118 
   119 val imp_cong = impI RSN
   120     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
   121         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
   122 
   123 (*Miniscoping: pushing in existential quantifiers*)
   124 val ex_simps = map prover 
   125                 ["(EX x. P x & Q)   = ((EX x.P x) & Q)",
   126                  "(EX x. P & Q x)   = (P & (EX x.Q x))",
   127                  "(EX x. P x | Q)   = ((EX x.P x) | Q)",
   128                  "(EX x. P | Q x)   = (P | (EX x.Q x))",
   129                  "(EX x. P x --> Q) = ((ALL x.P x) --> Q)",
   130                  "(EX x. P --> Q x) = (P --> (EX x.Q x))"];
   131 
   132 (*Miniscoping: pushing in universal quantifiers*)
   133 val all_simps = map prover
   134                 ["(ALL x. P x & Q)   = ((ALL x.P x) & Q)",
   135                  "(ALL x. P & Q x)   = (P & (ALL x.Q x))",
   136                  "(ALL x. P x | Q)   = ((ALL x.P x) | Q)",
   137                  "(ALL x. P | Q x)   = (P | (ALL x.Q x))",
   138                  "(ALL x. P x --> Q) = ((EX x.P x) --> Q)",
   139                  "(ALL x. P --> Q x) = (P --> (ALL x.Q x))"];
   140 
   141 
   142 
   143 (* elimination of existential quantifiers in assumptions *)
   144 
   145 val ex_all_equiv =
   146   let val lemma1 = prove_goal HOL.thy
   147         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
   148         (fn prems => [resolve_tac prems 1, etac exI 1]);
   149       val lemma2 = prove_goalw HOL.thy [Ex_def]
   150         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
   151         (fn prems => [REPEAT(resolve_tac prems 1)])
   152   in equal_intr lemma1 lemma2 end;
   153 
   154 end;
   155 
   156 fun prove nm thm  = qed_goal nm HOL.thy thm (fn _ => [fast_tac HOL_cs 1]);
   157 
   158 prove "conj_commute" "(P&Q) = (Q&P)";
   159 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
   160 val conj_comms = [conj_commute, conj_left_commute];
   161 prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
   162 
   163 prove "disj_commute" "(P|Q) = (Q|P)";
   164 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
   165 val disj_comms = [disj_commute, disj_left_commute];
   166 prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
   167 
   168 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
   169 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
   170 
   171 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
   172 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
   173 
   174 prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
   175 prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";
   176 prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
   177 
   178 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
   179 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
   180 prove "not_iff" "(P~=Q) = (P = (~Q))";
   181 
   182 (*Avoids duplication of subgoals after expand_if, when the true and false 
   183   cases boil down to the same thing.*) 
   184 prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
   185 
   186 prove "not_all" "(~ (! x.P(x))) = (? x.~P(x))";
   187 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
   188 prove "not_ex"  "(~ (? x.P(x))) = (! x.~P(x))";
   189 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
   190 
   191 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
   192 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
   193 
   194 (* '&' congruence rule: not included by default!
   195    May slow rewrite proofs down by as much as 50% *)
   196 
   197 let val th = prove_goal HOL.thy 
   198                 "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
   199                 (fn _=> [fast_tac HOL_cs 1])
   200 in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   201 
   202 let val th = prove_goal HOL.thy 
   203                 "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
   204                 (fn _=> [fast_tac HOL_cs 1])
   205 in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   206 
   207 (* '|' congruence rule: not included by default! *)
   208 
   209 let val th = prove_goal HOL.thy 
   210                 "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
   211                 (fn _=> [fast_tac HOL_cs 1])
   212 in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
   213 
   214 prove "eq_sym_conv" "(x=y) = (y=x)";
   215 
   216 qed_goalw "o_apply" HOL.thy [o_def] "(f o g) x = f (g x)"
   217  (fn _ => [rtac refl 1]);
   218 
   219 qed_goal "meta_eq_to_obj_eq" HOL.thy "x==y ==> x=y"
   220   (fn [prem] => [rewtac prem, rtac refl 1]);
   221 
   222 qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x"
   223  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
   224 
   225 qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y"
   226  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
   227 
   228 qed_goal "if_P" HOL.thy "P ==> (if P then x else y) = x"
   229  (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
   230 (*
   231 qed_goal "if_not_P" HOL.thy "~P ==> (if P then x else y) = y"
   232  (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
   233 *)
   234 qed_goalw "if_not_P" HOL.thy [if_def] "!!P. ~P ==> (if P then x else y) = y"
   235  (fn _ => [fast_tac (HOL_cs addIs [select_equality]) 1]);
   236 
   237 qed_goal "expand_if" HOL.thy
   238     "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
   239  (fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
   240          stac if_P 2,
   241          stac if_not_P 1,
   242          REPEAT(fast_tac HOL_cs 1) ]);
   243 
   244 qed_goal "if_bool_eq" HOL.thy
   245                    "(if P then Q else R) = ((P-->Q) & (~P-->R))"
   246                    (fn _ => [rtac expand_if 1]);
   247 
   248 local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
   249 in
   250 fun split_tac splits = mktac (map mk_meta_eq splits)
   251 end;
   252 
   253 local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2)
   254 in
   255 fun split_inside_tac splits = mktac (map mk_meta_eq splits)
   256 end;
   257 
   258 
   259 qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
   260   (fn _ => [split_tac [expand_if] 1, fast_tac HOL_cs 1]);
   261 
   262 (** 'if' congruence rules: neither included by default! *)
   263 
   264 (*Simplifies x assuming c and y assuming ~c*)
   265 qed_goal "if_cong" HOL.thy
   266   "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
   267 \  (if b then x else y) = (if c then u else v)"
   268   (fn rew::prems =>
   269    [stac rew 1, stac expand_if 1, stac expand_if 1,
   270     fast_tac (HOL_cs addDs prems) 1]);
   271 
   272 (*Prevents simplification of x and y: much faster*)
   273 qed_goal "if_weak_cong" HOL.thy
   274   "b=c ==> (if b then x else y) = (if c then x else y)"
   275   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   276 
   277 (*Prevents simplification of t: much faster*)
   278 qed_goal "let_weak_cong" HOL.thy
   279   "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
   280   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   281 
   282 (*In general it seems wrong to add distributive laws by default: they
   283   might cause exponential blow-up.  But imp_disjL has been in for a while
   284   and cannot be removed without affecting existing proofs.  Moreover, 
   285   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   286   grounds that it allows simplification of R in the two cases.*)
   287 
   288 val mksimps_pairs =
   289   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   290    ("All", [spec]), ("True", []), ("False", []),
   291    ("If", [if_bool_eq RS iffD1])];
   292 
   293 fun unsafe_solver prems = FIRST'[resolve_tac (TrueI::refl::prems),
   294 				 atac, etac FalseE];
   295 (*No premature instantiation of variables during simplification*)
   296 fun   safe_solver prems = FIRST'[match_tac (TrueI::refl::prems),
   297 				 eq_assume_tac, ematch_tac [FalseE]];
   298 
   299 val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
   300 			    setSSolver   safe_solver
   301 			    setSolver  unsafe_solver
   302 			    setmksimps (mksimps mksimps_pairs);
   303 
   304 val HOL_ss = HOL_basic_ss addsimps ([triv_forall_equality, (* prunes params *)
   305 				     if_True, if_False, if_cancel,
   306 				     o_apply, imp_disjL, conj_assoc, disj_assoc,
   307 				     de_Morgan_conj, de_Morgan_disj, 
   308 				     not_all, not_ex, cases_simp]
   309 				    @ ex_simps @ all_simps @ simp_thms)
   310 			  addcongs [imp_cong];
   311 
   312 qed_goal "if_distrib" HOL.thy
   313   "f(if c then x else y) = (if c then f x else f y)" 
   314   (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
   315 
   316 qed_goalw "o_assoc" HOL.thy [o_def] "f o (g o h) = f o g o h"
   317   (fn _ => [rtac ext 1, rtac refl 1]);
   318 
   319 
   320 
   321 
   322 (*** Install simpsets and datatypes in theory structure ***)
   323 
   324 simpset := HOL_ss;
   325 
   326 exception SS_DATA of simpset;
   327 
   328 let fun merge [] = SS_DATA empty_ss
   329       | merge ss = let val ss = map (fn SS_DATA x => x) ss;
   330                    in SS_DATA (foldl merge_ss (hd ss, tl ss)) end;
   331 
   332     fun put (SS_DATA ss) = simpset := ss;
   333 
   334     fun get () = SS_DATA (!simpset);
   335 in add_thydata "HOL"
   336      ("simpset", ThyMethods {merge = merge, put = put, get = get})
   337 end;
   338 
   339 type dtype_info = {case_const:term, case_rewrites:thm list,
   340                    constructors:term list, nchotomy:thm, case_cong:thm};
   341 
   342 exception DT_DATA of (string * dtype_info) list;
   343 val datatypes = ref [] : (string * dtype_info) list ref;
   344 
   345 let fun merge [] = DT_DATA []
   346       | merge ds =
   347           let val ds = map (fn DT_DATA x => x) ds;
   348           in DT_DATA (foldl (gen_union eq_fst) (hd ds, tl ds)) end;
   349 
   350     fun put (DT_DATA ds) = datatypes := ds;
   351 
   352     fun get () = DT_DATA (!datatypes);
   353 in add_thydata "HOL"
   354      ("datatypes", ThyMethods {merge = merge, put = put, get = get})
   355 end;
   356 
   357 
   358 add_thy_reader_file "thy_data.ML";
   359 
   360 
   361 
   362 
   363 (*** Integration of simplifier with classical reasoner ***)
   364 
   365 (* rot_eq_tac rotates the first equality premise of subgoal i to the front,
   366    fails if there is no equaliy or if an equality is already at the front *)
   367 fun rot_eq_tac i = let
   368   fun is_eq (Const ("Trueprop", _) $ (Const("op =",_) $ _ $ _)) = true
   369   |   is_eq _ = false;
   370   fun find_eq n [] = None
   371   |   find_eq n (t :: ts) = if (is_eq t) then Some n else find_eq (n + 1) ts;
   372   fun rot_eq state = let val (_, _, Bi, _) = dest_state (state, i) in
   373 	    (case find_eq 0 (Logic.strip_assums_hyp Bi) of
   374 	      None   => no_tac
   375 	    | Some 0 => no_tac
   376 	    | Some n => rotate_tac n i) end;
   377 in STATE rot_eq end;
   378 
   379 (*an unsatisfactory fix for the incomplete asm_full_simp_tac!
   380   better: asm_really_full_simp_tac, a yet to be implemented version of
   381 			asm_full_simp_tac that applies all equalities in the
   382 			premises to all the premises *)
   383 fun safe_asm_more_full_simp_tac ss = TRY o rot_eq_tac THEN' 
   384 				     safe_asm_full_simp_tac ss;
   385 
   386 (*Add a simpset to a classical set!*)
   387 infix 4 addss;
   388 fun cs addss ss = cs addSaltern (CHANGED o (safe_asm_more_full_simp_tac ss));
   389 (*old version, for compatibility with unstable old proofs*)
   390 infix 4 unsafe_addss;
   391 fun cs unsafe_addss ss = cs addbefore asm_full_simp_tac ss;
   392 
   393 fun Addss ss = (claset := !claset addss ss);
   394 (*old version, for compatibility with unstable old proofs*)
   395 fun Unsafe_Addss ss = (claset := !claset unsafe_addss ss);
   396 
   397 (*Designed to be idempotent, except if best_tac instantiates variables
   398   in some of the subgoals*)
   399 (*old version, for compatibility with unstable old proofs*)
   400 fun unsafe_auto_tac (cs,ss) = 
   401     ALLGOALS (asm_full_simp_tac ss) THEN
   402     REPEAT   (safe_tac cs THEN ALLGOALS (asm_full_simp_tac ss)) THEN
   403     REPEAT   (FIRSTGOAL (best_tac (cs addss ss))) THEN
   404     prune_params_tac;
   405 
   406 type clasimpset = (claset * simpset);
   407 
   408 val HOL_css = (HOL_cs, HOL_ss);
   409 
   410 fun pair_upd1 f ((a,b),x) = (f(a,x), b);
   411 fun pair_upd2 f ((a,b),x) = (a, f(b,x));
   412 
   413 infix 4 addSIs2 addSEs2 addSDs2 addIs2 addEs2 addDs2
   414 	addsimps2 delsimps2 addcongs2 delcongs2;
   415 fun op addSIs2   arg = pair_upd1 (op addSIs) arg;
   416 fun op addSEs2   arg = pair_upd1 (op addSEs) arg;
   417 fun op addSDs2   arg = pair_upd1 (op addSDs) arg;
   418 fun op addIs2    arg = pair_upd1 (op addIs ) arg;
   419 fun op addEs2    arg = pair_upd1 (op addEs ) arg;
   420 fun op addDs2    arg = pair_upd1 (op addDs ) arg;
   421 fun op addsimps2 arg = pair_upd2 (op addsimps) arg;
   422 fun op delsimps2 arg = pair_upd2 (op delsimps) arg;
   423 fun op addcongs2 arg = pair_upd2 (op addcongs) arg;
   424 fun op delcongs2 arg = pair_upd2 (op delcongs) arg;
   425 
   426 fun auto_tac (cs,ss) = let val cs' = cs addss ss in
   427 EVERY [	TRY (safe_tac cs'),
   428 	REPEAT (FIRSTGOAL (fast_tac cs')),
   429 	prune_params_tac] end;
   430 
   431 fun Auto_tac () = auto_tac (!claset, !simpset);
   432 
   433 fun auto () = by (Auto_tac ());