src/HOL/simpdata.ML
author paulson
Thu Sep 12 10:36:51 1996 +0200 (1996-09-12)
changeset 1984 5cf82dc3ce67
parent 1968 daa97cc96feb
child 2022 9d47e2962edd
permissions -rw-r--r--
Installed AddIffs, and some code from HOL.ML
     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 (*** Integration of simplifier with classical reasoner ***)
    14 
    15 (*Add a simpset to a classical set!*)
    16 infix 4 addss;
    17 fun cs addss ss = cs addbefore asm_full_simp_tac ss 1;
    18 
    19 fun Addss ss = (claset := !claset addbefore asm_full_simp_tac ss 1);
    20 
    21 (*Designed to be idempotent, except if best_tac instantiates variables
    22   in some of the subgoals*)
    23 fun auto_tac (cs,ss) = 
    24     ALLGOALS (asm_full_simp_tac ss) THEN
    25     REPEAT (safe_tac cs THEN ALLGOALS (asm_full_simp_tac ss)) THEN
    26     REPEAT (FIRSTGOAL (best_tac (cs addss ss)));
    27 
    28 fun Auto_tac() = auto_tac (!claset, !simpset);
    29 
    30 fun auto() = by (Auto_tac());
    31 
    32 
    33 (*** Addition of rules to simpsets and clasets simultaneously ***)
    34 
    35 (*Takes UNCONDITIONAL theorems of the form A<->B to 
    36 	the Safe Intr     rule B==>A and 
    37 	the Safe Destruct rule A==>B.
    38   Also ~A goes to the Safe Elim rule A ==> ?R
    39   Failing other cases, A is added as a Safe Intr rule*)
    40 local
    41   val iff_const = HOLogic.eq_const HOLogic.boolT;
    42 
    43   fun addIff th = 
    44       (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
    45 		(Const("not",_) $ A) =>
    46 		    AddSEs [zero_var_indexes (th RS notE)]
    47 	      | (con $ _ $ _) =>
    48 		    if con=iff_const
    49 		    then (AddSIs [zero_var_indexes (th RS iffD2)];  
    50 			  AddSDs [zero_var_indexes (th RS iffD1)])
    51 		    else  AddSIs [th]
    52 	      | _ => AddSIs [th];
    53        Addsimps [th])
    54       handle _ => error ("AddIffs: theorem must be unconditional\n" ^ 
    55 			 string_of_thm th)
    56 
    57   fun delIff th = 
    58       (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
    59 		(Const("not",_) $ A) =>
    60 		    Delrules [zero_var_indexes (th RS notE)]
    61 	      | (con $ _ $ _) =>
    62 		    if con=iff_const
    63 		    then Delrules [zero_var_indexes (th RS iffD2),
    64 				   zero_var_indexes (th RS iffD1)]
    65 		    else Delrules [th]
    66 	      | _ => Delrules [th];
    67        Delsimps [th])
    68       handle _ => warning("DelIffs: ignoring conditional theorem\n" ^ 
    69 			  string_of_thm th)
    70 in
    71 val AddIffs = seq addIff
    72 val DelIffs = seq delIff
    73 end;
    74 
    75 
    76 local
    77 
    78   fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
    79 
    80   val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
    81   val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
    82 
    83   val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
    84   val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
    85 
    86   fun atomize pairs =
    87     let fun atoms th =
    88 	  (case concl_of th of
    89 	     Const("Trueprop",_) $ p =>
    90 	       (case head_of p of
    91 		  Const(a,_) =>
    92 		    (case assoc(pairs,a) of
    93 		       Some(rls) => flat (map atoms ([th] RL rls))
    94 		     | None => [th])
    95 		| _ => [th])
    96 	   | _ => [th])
    97     in atoms end;
    98 
    99   fun mk_meta_eq r = case concl_of r of
   100 	  Const("==",_)$_$_ => r
   101       |   _$(Const("op =",_)$_$_) => r RS eq_reflection
   102       |   _$(Const("not",_)$_) => r RS not_P_imp_P_eq_False
   103       |   _ => r RS P_imp_P_eq_True;
   104   (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
   105 
   106   fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
   107 
   108   val simp_thms = map prover
   109    [ "(x=x) = True",
   110      "(~True) = False", "(~False) = True", "(~ ~ P) = P",
   111      "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
   112      "(True=P) = P", "(P=True) = P",
   113      "(True --> P) = P", "(False --> P) = True", 
   114      "(P --> True) = True", "(P --> P) = True",
   115      "(P --> False) = (~P)", "(P --> ~P) = (~P)",
   116      "(P & True) = P", "(True & P) = P", 
   117      "(P & False) = False", "(False & P) = False", "(P & P) = P",
   118      "(P | True) = True", "(True | P) = True", 
   119      "(P | False) = P", "(False | P) = P", "(P | P) = P",
   120      "((~P) = (~Q)) = (P=Q)",
   121      "(!x.P) = P", "(? x.P) = P", "? x. x=t", 
   122      "(? x. x=t & P(x)) = P(t)", "(! x. x=t --> P(x)) = P(t)" ];
   123 
   124 in
   125 
   126 val meta_eq_to_obj_eq = prove_goal HOL.thy "x==y ==> x=y"
   127   (fn [prem] => [rewtac prem, rtac refl 1]);
   128 
   129 val eq_sym_conv = prover "(x=y) = (y=x)";
   130 
   131 val conj_assoc = prover "((P&Q)&R) = (P&(Q&R))";
   132 
   133 val disj_assoc = prover "((P|Q)|R) = (P|(Q|R))";
   134 
   135 val imp_disj   = prover "(P|Q --> R) = ((P-->R)&(Q-->R))";
   136 
   137 (*Avoids duplication of subgoals after expand_if, when the true and false 
   138   cases boil down to the same thing.*) 
   139 val cases_simp = prover "((P --> Q) & (~P --> Q)) = Q";
   140 
   141 val if_True = prove_goalw HOL.thy [if_def] "(if True then x else y) = x"
   142  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
   143 
   144 val if_False = prove_goalw HOL.thy [if_def] "(if False then x else y) = y"
   145  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
   146 
   147 val if_P = prove_goal HOL.thy "P ==> (if P then x else y) = x"
   148  (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
   149 
   150 val if_not_P = prove_goal HOL.thy "~P ==> (if P then x else y) = y"
   151  (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
   152 
   153 val expand_if = prove_goal HOL.thy
   154     "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
   155  (fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
   156          rtac (if_P RS ssubst) 2,
   157          rtac (if_not_P RS ssubst) 1,
   158          REPEAT(fast_tac HOL_cs 1) ]);
   159 
   160 val if_bool_eq = prove_goal HOL.thy
   161                    "(if P then Q else R) = ((P-->Q) & (~P-->R))"
   162                    (fn _ => [rtac expand_if 1]);
   163 
   164 (*Add congruence rules for = (instead of ==) *)
   165 infix 4 addcongs;
   166 fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection]);
   167 
   168 fun Addcongs congs = (simpset := !simpset addcongs congs);
   169 
   170 val mksimps_pairs =
   171   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   172    ("All", [spec]), ("True", []), ("False", []),
   173    ("If", [if_bool_eq RS iffD1])];
   174 
   175 fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
   176 
   177 val imp_cong = impI RSN
   178     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
   179         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
   180 
   181 val o_apply = prove_goalw HOL.thy [o_def] "(f o g)(x) = f(g(x))"
   182  (fn _ => [rtac refl 1]);
   183 
   184 (*Miniscoping: pushing in existential quantifiers*)
   185 val ex_simps = map prover 
   186 		["(EX x. P x & Q)   = ((EX x.P x) & Q)",
   187 		 "(EX x. P & Q x)   = (P & (EX x.Q x))",
   188 		 "(EX x. P x | Q)   = ((EX x.P x) | Q)",
   189 		 "(EX x. P | Q x)   = (P | (EX x.Q x))",
   190 		 "(EX x. P x --> Q) = ((ALL x.P x) --> Q)",
   191 		 "(EX x. P --> Q x) = (P --> (EX x.Q x))"];
   192 
   193 (*Miniscoping: pushing in universal quantifiers*)
   194 val all_simps = map prover
   195 		["(ALL x. P x & Q)   = ((ALL x.P x) & Q)",
   196 		 "(ALL x. P & Q x)   = (P & (ALL x.Q x))",
   197 		 "(ALL x. P x | Q)   = ((ALL x.P x) | Q)",
   198 		 "(ALL x. P | Q x)   = (P | (ALL x.Q x))",
   199 		 "(ALL x. P x --> Q) = ((EX x.P x) --> Q)",
   200 		 "(ALL x. P --> Q x) = (P --> (ALL x.Q x))"];
   201 
   202 val HOL_ss = empty_ss
   203       setmksimps (mksimps mksimps_pairs)
   204       setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac
   205                              ORELSE' etac FalseE)
   206       setsubgoaler asm_simp_tac
   207       addsimps ([if_True, if_False, o_apply, imp_disj, conj_assoc, disj_assoc,
   208 		 cases_simp]
   209         @ ex_simps @ all_simps @ simp_thms)
   210       addcongs [imp_cong];
   211 
   212 
   213 (*In general it seems wrong to add distributive laws by default: they
   214   might cause exponential blow-up.  But imp_disj has been in for a while
   215   and cannot be removed without affecting existing proofs.  Moreover, 
   216   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
   217   grounds that it allows simplification of R in the two cases.*)
   218 
   219 
   220 local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
   221 in
   222 fun split_tac splits = mktac (map mk_meta_eq splits)
   223 end;
   224 
   225 local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2)
   226 in
   227 fun split_inside_tac splits = mktac (map mk_meta_eq splits)
   228 end;
   229 
   230 
   231 (* eliminiation of existential quantifiers in assumptions *)
   232 
   233 val ex_all_equiv =
   234   let val lemma1 = prove_goal HOL.thy
   235         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
   236         (fn prems => [resolve_tac prems 1, etac exI 1]);
   237       val lemma2 = prove_goalw HOL.thy [Ex_def]
   238         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
   239         (fn prems => [REPEAT(resolve_tac prems 1)])
   240   in equal_intr lemma1 lemma2 end;
   241 
   242 (* '&' congruence rule: not included by default!
   243    May slow rewrite proofs down by as much as 50% *)
   244 
   245 val conj_cong = impI RSN
   246     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
   247         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
   248 
   249 val rev_conj_cong = impI RSN
   250     (2, prove_goal HOL.thy "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
   251         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
   252 
   253 (** 'if' congruence rules: neither included by default! *)
   254 
   255 (*Simplifies x assuming c and y assuming ~c*)
   256 val if_cong = prove_goal HOL.thy
   257   "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
   258 \  (if b then x else y) = (if c then u else v)"
   259   (fn rew::prems =>
   260    [stac rew 1, stac expand_if 1, stac expand_if 1,
   261     fast_tac (HOL_cs addDs prems) 1]);
   262 
   263 (*Prevents simplification of x and y: much faster*)
   264 val if_weak_cong = prove_goal HOL.thy
   265   "b=c ==> (if b then x else y) = (if c then x else y)"
   266   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   267 
   268 (*Prevents simplification of t: much faster*)
   269 val let_weak_cong = prove_goal HOL.thy
   270   "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
   271   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   272 
   273 end;
   274 
   275 fun prove nm thm  = qed_goal nm HOL.thy thm (fn _ => [fast_tac HOL_cs 1]);
   276 
   277 prove "conj_commute" "(P&Q) = (Q&P)";
   278 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
   279 val conj_comms = [conj_commute, conj_left_commute];
   280 
   281 prove "disj_commute" "(P|Q) = (Q|P)";
   282 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
   283 val disj_comms = [disj_commute, disj_left_commute];
   284 
   285 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
   286 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
   287 
   288 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
   289 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
   290 
   291 prove "imp_conj_distrib" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
   292 prove "imp_conj"         "((P&Q)-->R)   = (P --> (Q --> R))";
   293 
   294 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
   295 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
   296 prove "not_iff" "(P~=Q) = (P = (~Q))";
   297 
   298 prove "not_all" "(~ (! x.P(x))) = (? x.~P(x))";
   299 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
   300 prove "not_ex"  "(~ (? x.P(x))) = (! x.~P(x))";
   301 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
   302 
   303 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
   304 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
   305 
   306 
   307 qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
   308   (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
   309 
   310 qed_goal "if_distrib" HOL.thy
   311   "f(if c then x else y) = (if c then f x else f y)" 
   312   (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
   313 
   314 qed_goalw "o_assoc" HOL.thy [o_def] "f o (g o h) = (f o g o h)"
   315   (fn _=>[rtac ext 1, rtac refl 1]);
   316 
   317 
   318 
   319 
   320 (*** Install simpsets and datatypes in theory structure ***)
   321 
   322 simpset := HOL_ss;
   323 
   324 exception SS_DATA of simpset;
   325 
   326 let fun merge [] = SS_DATA empty_ss
   327       | merge ss = let val ss = map (fn SS_DATA x => x) ss;
   328                    in SS_DATA (foldl merge_ss (hd ss, tl ss)) end;
   329 
   330     fun put (SS_DATA ss) = simpset := ss;
   331 
   332     fun get () = SS_DATA (!simpset);
   333 in add_thydata "HOL"
   334      ("simpset", ThyMethods {merge = merge, put = put, get = get})
   335 end;
   336 
   337 type dtype_info = {case_const:term, case_rewrites:thm list,
   338                    constructors:term list, nchotomy:thm, case_cong:thm};
   339 
   340 exception DT_DATA of (string * dtype_info) list;
   341 val datatypes = ref [] : (string * dtype_info) list ref;
   342 
   343 let fun merge [] = DT_DATA []
   344       | merge ds =
   345           let val ds = map (fn DT_DATA x => x) ds;
   346           in DT_DATA (foldl (gen_union eq_fst) (hd ds, tl ds)) end;
   347 
   348     fun put (DT_DATA ds) = datatypes := ds;
   349 
   350     fun get () = DT_DATA (!datatypes);
   351 in add_thydata "HOL"
   352      ("datatypes", ThyMethods {merge = merge, put = put, get = get})
   353 end;
   354 
   355 
   356 add_thy_reader_file "thy_data.ML";