src/HOL/ex/meson.ML
 author wenzelm Mon Nov 03 12:13:18 1997 +0100 (1997-11-03) changeset 4089 96fba19bcbe2 parent 3842 b55686a7b22c child 4153 e534c4c32d54 permissions -rw-r--r--
isatool fixclasimp;
```     1 (*  Title:      HOL/ex/meson
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1992  University of Cambridge
```
```     5
```
```     6 The MESON resolution proof procedure for HOL
```
```     7
```
```     8 When making clauses, avoids using the rewriter -- instead uses RS recursively
```
```     9
```
```    10 NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E.  ELIMINATES NEED FOR
```
```    11 FUNCTION nodups -- if done to goal clauses too!
```
```    12 *)
```
```    13
```
```    14 writeln"File HOL/ex/meson.";
```
```    15
```
```    16 (*Prove theorems using fast_tac*)
```
```    17 fun prove_fun s =
```
```    18     prove_goal HOL.thy s
```
```    19          (fn prems => [ cut_facts_tac prems 1, Fast_tac 1 ]);
```
```    20
```
```    21 (**** Negation Normal Form ****)
```
```    22
```
```    23 (*** de Morgan laws ***)
```
```    24
```
```    25 val not_conjD = prove_fun "~(P&Q) ==> ~P | ~Q";
```
```    26 val not_disjD = prove_fun "~(P|Q) ==> ~P & ~Q";
```
```    27 val not_notD = prove_fun "~~P ==> P";
```
```    28 val not_allD = prove_fun  "~(! x. P(x)) ==> ? x. ~P(x)";
```
```    29 val not_exD = prove_fun   "~(? x. P(x)) ==> ! x. ~P(x)";
```
```    30
```
```    31
```
```    32 (*** Removal of --> and <-> (positive and negative occurrences) ***)
```
```    33
```
```    34 val imp_to_disjD = prove_fun "P-->Q ==> ~P | Q";
```
```    35 val not_impD = prove_fun   "~(P-->Q) ==> P & ~Q";
```
```    36
```
```    37 val iff_to_disjD = prove_fun "P=Q ==> (~P | Q) & (~Q | P)";
```
```    38
```
```    39 (*Much more efficient than (P & ~Q) | (Q & ~P) for computing CNF*)
```
```    40 val not_iffD = prove_fun "~(P=Q) ==> (P | Q) & (~P | ~Q)";
```
```    41
```
```    42
```
```    43 (**** Pulling out the existential quantifiers ****)
```
```    44
```
```    45 (*** Conjunction ***)
```
```    46
```
```    47 val conj_exD1 = prove_fun "(? x. P(x)) & Q ==> ? x. P(x) & Q";
```
```    48 val conj_exD2 = prove_fun "P & (? x. Q(x)) ==> ? x. P & Q(x)";
```
```    49
```
```    50 (*** Disjunction ***)
```
```    51
```
```    52 (*DO NOT USE with forall-Skolemization: makes fewer schematic variables!!
```
```    53   With ex-Skolemization, makes fewer Skolem constants*)
```
```    54 val disj_exD = prove_fun "(? x. P(x)) | (? x. Q(x)) ==> ? x. P(x) | Q(x)";
```
```    55
```
```    56 val disj_exD1 = prove_fun "(? x. P(x)) | Q ==> ? x. P(x) | Q";
```
```    57 val disj_exD2 = prove_fun "P | (? x. Q(x)) ==> ? x. P | Q(x)";
```
```    58
```
```    59
```
```    60 (**** Skolemization -- pulling "?" over "!" ****)
```
```    61
```
```    62 (*"Axiom" of Choice, proved using the description operator*)
```
```    63 val [major] = goal HOL.thy
```
```    64     "! x. ? y. Q x y ==> ? f. ! x. Q x (f x)";
```
```    65 by (cut_facts_tac [major] 1);
```
```    66 by (fast_tac (claset() addEs [selectI]) 1);
```
```    67 qed "choice";
```
```    68
```
```    69
```
```    70 (***** Generating clauses for the Meson Proof Procedure *****)
```
```    71
```
```    72 (*** Disjunctions ***)
```
```    73
```
```    74 val disj_assoc = prove_fun "(P|Q)|R ==> P|(Q|R)";
```
```    75
```
```    76 val disj_comm = prove_fun "P|Q ==> Q|P";
```
```    77
```
```    78 val disj_FalseD1 = prove_fun "False|P ==> P";
```
```    79 val disj_FalseD2 = prove_fun "P|False ==> P";
```
```    80
```
```    81 (*** Generation of contrapositives ***)
```
```    82
```
```    83 (*Inserts negated disjunct after removing the negation; P is a literal*)
```
```    84 val [major,minor] = goal HOL.thy "~P|Q ==> ((~P==>P) ==> Q)";
```
```    85 by (rtac (major RS disjE) 1);
```
```    86 by (rtac notE 1);
```
```    87 by (etac minor 2);
```
```    88 by (ALLGOALS assume_tac);
```
```    89 qed "make_neg_rule";
```
```    90
```
```    91 (*For Plaisted's "Postive refinement" of the MESON procedure*)
```
```    92 val [major,minor] = goal HOL.thy "~P|Q ==> (P ==> Q)";
```
```    93 by (rtac (major RS disjE) 1);
```
```    94 by (rtac notE 1);
```
```    95 by (rtac minor 2);
```
```    96 by (ALLGOALS assume_tac);
```
```    97 qed "make_refined_neg_rule";
```
```    98
```
```    99 (*P should be a literal*)
```
```   100 val [major,minor] = goal HOL.thy "P|Q ==> ((P==>~P) ==> Q)";
```
```   101 by (rtac (major RS disjE) 1);
```
```   102 by (rtac notE 1);
```
```   103 by (etac minor 1);
```
```   104 by (ALLGOALS assume_tac);
```
```   105 qed "make_pos_rule";
```
```   106
```
```   107 (*** Generation of a goal clause -- put away the final literal ***)
```
```   108
```
```   109 val [major,minor] = goal HOL.thy "~P ==> ((~P==>P) ==> False)";
```
```   110 by (rtac notE 1);
```
```   111 by (rtac minor 2);
```
```   112 by (ALLGOALS (rtac major));
```
```   113 qed "make_neg_goal";
```
```   114
```
```   115 val [major,minor] = goal HOL.thy "P ==> ((P==>~P) ==> False)";
```
```   116 by (rtac notE 1);
```
```   117 by (rtac minor 1);
```
```   118 by (ALLGOALS (rtac major));
```
```   119 qed "make_pos_goal";
```
```   120
```
```   121
```
```   122 (**** Lemmas for forward proof (like congruence rules) ****)
```
```   123
```
```   124 (*NOTE: could handle conjunctions (faster?) by
```
```   125     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
```
```   126 val major::prems = goal HOL.thy
```
```   127     "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q";
```
```   128 by (rtac (major RS conjE) 1);
```
```   129 by (rtac conjI 1);
```
```   130 by (ALLGOALS (eresolve_tac prems));
```
```   131 qed "conj_forward";
```
```   132
```
```   133 val major::prems = goal HOL.thy
```
```   134     "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q";
```
```   135 by (rtac (major RS disjE) 1);
```
```   136 by (ALLGOALS (dresolve_tac prems));
```
```   137 by (ALLGOALS (eresolve_tac [disjI1,disjI2]));
```
```   138 qed "disj_forward";
```
```   139
```
```   140 val major::prems = goal HOL.thy
```
```   141     "[| ! x. P'(x);  !!x. P'(x) ==> P(x) |] ==> ! x. P(x)";
```
```   142 by (rtac allI 1);
```
```   143 by (resolve_tac prems 1);
```
```   144 by (rtac (major RS spec) 1);
```
```   145 qed "all_forward";
```
```   146
```
```   147 val major::prems = goal HOL.thy
```
```   148     "[| ? x. P'(x);  !!x. P'(x) ==> P(x) |] ==> ? x. P(x)";
```
```   149 by (rtac (major RS exE) 1);
```
```   150 by (rtac exI 1);
```
```   151 by (eresolve_tac prems 1);
```
```   152 qed "ex_forward";
```
```   153
```
```   154
```
```   155 (**** Operators for forward proof ****)
```
```   156
```
```   157 (*raises exception if no rules apply -- unlike RL*)
```
```   158 fun tryres (th, rl::rls) = (th RS rl handle THM _ => tryres(th,rls))
```
```   159   | tryres (th, []) = raise THM("tryres", 0, [th]);
```
```   160
```
```   161 val prop_of = #prop o rep_thm;
```
```   162
```
```   163 (*Permits forward proof from rules that discharge assumptions*)
```
```   164 fun forward_res nf st =
```
```   165   case Sequence.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st)
```
```   166   of Some(th,_) => th
```
```   167    | None => raise THM("forward_res", 0, [st]);
```
```   168
```
```   169
```
```   170 (*Negation Normal Form*)
```
```   171 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
```
```   172                not_impD, not_iffD, not_allD, not_exD, not_notD];
```
```   173 fun make_nnf th = make_nnf (tryres(th, nnf_rls))
```
```   174     handle THM _ =>
```
```   175         forward_res make_nnf
```
```   176            (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
```
```   177     handle THM _ => th;
```
```   178
```
```   179
```
```   180 (*Are any of the constants in "bs" present in the term?*)
```
```   181 fun has_consts bs =
```
```   182   let fun has (Const(a,_)) = a mem bs
```
```   183         | has (f\$u) = has f orelse has u
```
```   184         | has (Abs(_,_,t)) = has t
```
```   185         | has _ = false
```
```   186   in  has  end;
```
```   187
```
```   188 (*Pull existential quantifiers (Skolemization)*)
```
```   189 fun skolemize th =
```
```   190   if not (has_consts ["Ex"] (prop_of th)) then th
```
```   191   else skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
```
```   192                               disj_exD, disj_exD1, disj_exD2]))
```
```   193     handle THM _ =>
```
```   194         skolemize (forward_res skolemize
```
```   195                    (tryres (th, [conj_forward, disj_forward, all_forward])))
```
```   196     handle THM _ => forward_res skolemize (th RS ex_forward);
```
```   197
```
```   198
```
```   199 (**** Clause handling ****)
```
```   200
```
```   201 fun literals (Const("Trueprop",_) \$ P) = literals P
```
```   202   | literals (Const("op |",_) \$ P \$ Q) = literals P @ literals Q
```
```   203   | literals (Const("Not",_) \$ P) = [(false,P)]
```
```   204   | literals P = [(true,P)];
```
```   205
```
```   206 (*number of literals in a term*)
```
```   207 val nliterals = length o literals;
```
```   208
```
```   209 (*to detect, and remove, tautologous clauses*)
```
```   210 fun taut_lits [] = false
```
```   211   | taut_lits ((flg,t)::ts) = (not flg,t) mem ts orelse taut_lits ts;
```
```   212
```
```   213 val term_False = term_of (read_cterm (sign_of HOL.thy)
```
```   214                           ("False", Type("bool",[])));
```
```   215
```
```   216 (*Include False as a literal: an occurrence of ~False is a tautology*)
```
```   217 fun is_taut th = taut_lits ((true,term_False) :: literals (prop_of th));
```
```   218
```
```   219 (*Generation of unique names -- maxidx cannot be relied upon to increase!
```
```   220   Cannot rely on "variant", since variables might coincide when literals
```
```   221   are joined to make a clause...
```
```   222   19 chooses "U" as the first variable name*)
```
```   223 val name_ref = ref 19;
```
```   224
```
```   225 (*Replaces universally quantified variables by FREE variables -- because
```
```   226   assumptions may not contain scheme variables.  Later, call "generalize". *)
```
```   227 fun freeze_spec th =
```
```   228   let val sth = th RS spec
```
```   229       val newname = (name_ref := !name_ref + 1;
```
```   230                      radixstring(26, "A", !name_ref))
```
```   231   in  read_instantiate [("x", newname)] sth  end;
```
```   232
```
```   233 fun resop nf [prem] = resolve_tac (nf prem) 1;
```
```   234
```
```   235 (*Conjunctive normal form, detecting tautologies early.
```
```   236   Strips universal quantifiers and breaks up conjunctions. *)
```
```   237 fun cnf_aux seen (th,ths) =
```
```   238   if taut_lits (literals(prop_of th) @ seen)  then ths
```
```   239   else if not (has_consts ["All","op &"] (prop_of th))  then th::ths
```
```   240   else (*conjunction?*)
```
```   241         cnf_aux seen (th RS conjunct1,
```
```   242                       cnf_aux seen (th RS conjunct2, ths))
```
```   243   handle THM _ => (*universal quant?*)
```
```   244         cnf_aux  seen (freeze_spec th,  ths)
```
```   245   handle THM _ => (*disjunction?*)
```
```   246     let val tac =
```
```   247         (METAHYPS (resop (cnf_nil seen)) 1) THEN
```
```   248         (fn st' => st' |>
```
```   249                 METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1)
```
```   250     in  Sequence.list_of_s (tac (th RS disj_forward)) @ ths  end
```
```   251 and cnf_nil seen th = cnf_aux seen (th,[]);
```
```   252
```
```   253 (*Top-level call to cnf -- it's safe to reset name_ref*)
```
```   254 fun cnf (th,ths) =
```
```   255    (name_ref := 19;  cnf (th RS conjunct1, cnf (th RS conjunct2, ths))
```
```   256     handle THM _ => (*not a conjunction*) cnf_aux [] (th, ths));
```
```   257
```
```   258 (**** Removal of duplicate literals ****)
```
```   259
```
```   260 (*Version for removal of duplicate literals*)
```
```   261 val major::prems = goal HOL.thy
```
```   262     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q";
```
```   263 by (rtac (major RS disjE) 1);
```
```   264 by (rtac disjI1 1);
```
```   265 by (rtac (disjCI RS disj_comm) 2);
```
```   266 by (ALLGOALS (eresolve_tac prems));
```
```   267 by (etac notE 1);
```
```   268 by (assume_tac 1);
```
```   269 qed "disj_forward2";
```
```   270
```
```   271 (*Forward proof, passing extra assumptions as theorems to the tactic*)
```
```   272 fun forward_res2 nf hyps st =
```
```   273   case Sequence.pull
```
```   274         (REPEAT
```
```   275          (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
```
```   276          st)
```
```   277   of Some(th,_) => th
```
```   278    | None => raise THM("forward_res2", 0, [st]);
```
```   279
```
```   280 (*Remove duplicates in P|Q by assuming ~P in Q
```
```   281   rls (initially []) accumulates assumptions of the form P==>False*)
```
```   282 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
```
```   283     handle THM _ => tryres(th,rls)
```
```   284     handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
```
```   285                            [disj_FalseD1, disj_FalseD2, asm_rl])
```
```   286     handle THM _ => th;
```
```   287
```
```   288 (*Remove duplicate literals, if there are any*)
```
```   289 fun nodups th =
```
```   290     if null(findrep(literals(prop_of th))) then th
```
```   291     else nodups_aux [] th;
```
```   292
```
```   293
```
```   294 (**** Generation of contrapositives ****)
```
```   295
```
```   296 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
```
```   297 fun assoc_right th = assoc_right (th RS disj_assoc)
```
```   298         handle THM _ => th;
```
```   299
```
```   300 (*Must check for negative literal first!*)
```
```   301 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
```
```   302
```
```   303 (*For Plaisted's postive refinement.  [currently unused] *)
```
```   304 val refined_clause_rules = [disj_assoc, make_refined_neg_rule, make_pos_rule];
```
```   305
```
```   306 (*Create a goal or support clause, conclusing False*)
```
```   307 fun make_goal th =   (*Must check for negative literal first!*)
```
```   308     make_goal (tryres(th, clause_rules))
```
```   309   handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
```
```   310
```
```   311 (*Sort clauses by number of literals*)
```
```   312 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
```
```   313
```
```   314 (*TAUTOLOGY CHECK SHOULD NOT BE NECESSARY!*)
```
```   315 fun sort_clauses ths = sort fewerlits (filter (not o is_taut) ths);
```
```   316
```
```   317 (*Convert all suitable free variables to schematic variables*)
```
```   318 fun generalize th = forall_elim_vars 0 (forall_intr_frees th);
```
```   319
```
```   320 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
```
```   321   The resulting clauses are HOL disjunctions.*)
```
```   322 fun make_clauses ths =
```
```   323     sort_clauses (map (generalize o nodups) (foldr cnf (ths,[])));
```
```   324
```
```   325 (*Create a meta-level Horn clause*)
```
```   326 fun make_horn crules th = make_horn crules (tryres(th,crules))
```
```   327                           handle THM _ => th;
```
```   328
```
```   329 (*Generate Horn clauses for all contrapositives of a clause*)
```
```   330 fun add_contras crules (th,hcs) =
```
```   331   let fun rots (0,th) = hcs
```
```   332         | rots (k,th) = zero_var_indexes (make_horn crules th) ::
```
```   333                         rots(k-1, assoc_right (th RS disj_comm))
```
```   334   in case nliterals(prop_of th) of
```
```   335         1 => th::hcs
```
```   336       | n => rots(n, assoc_right th)
```
```   337   end;
```
```   338
```
```   339 (*Use "theorem naming" to label the clauses*)
```
```   340 fun name_thms label =
```
```   341     let fun name1 (th, (k,ths)) =
```
```   342           (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
```
```   343
```
```   344     in  fn ths => #2 (foldr name1 (ths, (length ths, [])))  end;
```
```   345
```
```   346 (*Convert a list of clauses to (contrapositive) Horn clauses*)
```
```   347 fun make_horns ths =
```
```   348     name_thms "Horn#"
```
```   349       (gen_distinct eq_thm (foldr (add_contras clause_rules) (ths,[])));
```
```   350
```
```   351 (*Find an all-negative support clause*)
```
```   352 fun is_negative th = forall (not o #1) (literals (prop_of th));
```
```   353
```
```   354 val neg_clauses = filter is_negative;
```
```   355
```
```   356
```
```   357 (***** MESON PROOF PROCEDURE *****)
```
```   358
```
```   359 fun rhyps (Const("==>",_) \$ (Const("Trueprop",_) \$ A) \$ phi,
```
```   360            As) = rhyps(phi, A::As)
```
```   361   | rhyps (_, As) = As;
```
```   362
```
```   363 (** Detecting repeated assumptions in a subgoal **)
```
```   364
```
```   365 (*The stringtree detects repeated assumptions.*)
```
```   366 fun ins_term (net,t) = Net.insert_term((t,t), net, op aconv);
```
```   367
```
```   368 (*detects repetitions in a list of terms*)
```
```   369 fun has_reps [] = false
```
```   370   | has_reps [_] = false
```
```   371   | has_reps [t,u] = (t aconv u)
```
```   372   | has_reps ts = (foldl ins_term (Net.empty, ts);  false)
```
```   373                   handle INSERT => true;
```
```   374
```
```   375 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
```
```   376 fun TRYALL_eq_assume_tac 0 st = Sequence.single st
```
```   377   | TRYALL_eq_assume_tac i st = TRYALL_eq_assume_tac (i-1) (eq_assumption i st)
```
```   378                                 handle THM _ => TRYALL_eq_assume_tac (i-1) st;
```
```   379
```
```   380 (*Loop checking: FAIL if trying to prove the same thing twice
```
```   381   -- if *ANY* subgoal has repeated literals*)
```
```   382 fun check_tac st =
```
```   383   if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
```
```   384   then  Sequence.null  else  Sequence.single st;
```
```   385
```
```   386
```
```   387 (* net_resolve_tac actually made it slower... *)
```
```   388 fun prolog_step_tac horns i =
```
```   389     (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
```
```   390     TRYALL eq_assume_tac;
```
```   391
```
```   392
```
```   393 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
```
```   394 local fun addconcl(prem,sz) = size_of_term (Logic.strip_assums_concl prem) + sz
```
```   395 in
```
```   396 fun size_of_subgoals st = foldr addconcl (prems_of st, 0)
```
```   397 end;
```
```   398
```
```   399 (*Could simply use nprems_of, which would count remaining subgoals -- no
```
```   400   discrimination as to their size!  With BEST_FIRST, fails for problem 41.*)
```
```   401
```
```   402 fun best_prolog_tac sizef horns =
```
```   403     BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
```
```   404
```
```   405 fun depth_prolog_tac horns =
```
```   406     DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
```
```   407
```
```   408 (*Return all negative clauses, as possible goal clauses*)
```
```   409 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
```
```   410
```
```   411
```
```   412 fun skolemize_tac prems =
```
```   413     cut_facts_tac (map (skolemize o make_nnf) prems)  THEN'
```
```   414     REPEAT o (etac exE);
```
```   415
```
```   416 (*Shell of all meson-tactics.  Supplies cltac with clauses: HOL disjunctions*)
```
```   417 fun MESON cltac = SELECT_GOAL
```
```   418  (EVERY1 [rtac ccontr,
```
```   419           METAHYPS (fn negs =>
```
```   420                     EVERY1 [skolemize_tac negs,
```
```   421                             METAHYPS (cltac o make_clauses)])]);
```
```   422
```
```   423 (** Best-first search versions **)
```
```   424
```
```   425 fun best_meson_tac sizef =
```
```   426   MESON (fn cls =>
```
```   427          THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
```
```   428                          (has_fewer_prems 1, sizef)
```
```   429                          (prolog_step_tac (make_horns cls) 1));
```
```   430
```
```   431 (*First, breaks the goal into independent units*)
```
```   432 val safe_best_meson_tac =
```
```   433      SELECT_GOAL (TRY (safe_tac (claset())) THEN
```
```   434                   TRYALL (best_meson_tac size_of_subgoals));
```
```   435
```
```   436 (** Depth-first search version **)
```
```   437
```
```   438 val depth_meson_tac =
```
```   439      MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
```
```   440                              depth_prolog_tac (make_horns cls)]);
```
```   441
```
```   442
```
```   443
```
```   444 (** Iterative deepening version **)
```
```   445
```
```   446 (*This version does only one inference per call;
```
```   447   having only one eq_assume_tac speeds it up!*)
```
```   448 fun prolog_step_tac' horns =
```
```   449     let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
```
```   450             take_prefix (fn rl => nprems_of rl=0) horns
```
```   451         val nrtac = net_resolve_tac horns
```
```   452     in  fn i => eq_assume_tac i ORELSE
```
```   453                 match_tac horn0s i ORELSE  (*no backtracking if unit MATCHES*)
```
```   454                 ((assume_tac i APPEND nrtac i) THEN check_tac)
```
```   455     end;
```
```   456
```
```   457 fun iter_deepen_prolog_tac horns =
```
```   458     ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
```
```   459
```
```   460 val iter_deepen_meson_tac =
```
```   461   MESON (fn cls =>
```
```   462          (THEN_ITER_DEEPEN (resolve_tac (gocls cls) 1)
```
```   463                            (has_fewer_prems 1)
```
```   464                            (prolog_step_tac' (make_horns cls))));
```
```   465
```
```   466 val safe_meson_tac =
```
```   467      SELECT_GOAL (TRY (safe_tac (claset())) THEN
```
```   468                   TRYALL (iter_deepen_meson_tac));
```
```   469
```
```   470
```
```   471 writeln"Reached end of file.";
```