src/HOL/WF.ML
author paulson
Wed Nov 05 13:23:46 1997 +0100 (1997-11-05)
changeset 4153 e534c4c32d54
parent 4089 96fba19bcbe2
child 4350 1983e4054fd8
permissions -rw-r--r--
Ran expandshort, especially to introduce Safe_tac
     1 (*  Title:      HOL/wf.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, with minor changes by Konrad Slind
     4     Copyright   1992  University of Cambridge/1995 TU Munich
     5 
     6 Wellfoundedness, induction, and  recursion
     7 *)
     8 
     9 open WF;
    10 
    11 val H_cong = read_instantiate [("f","H")] (standard(refl RS cong RS cong));
    12 val H_cong1 = refl RS H_cong;
    13 
    14 (*Restriction to domain A.  If r is well-founded over A then wf(r)*)
    15 val [prem1,prem2] = goalw WF.thy [wf_def]
    16  "[| r <= A Times A;  \
    17 \    !!x P. [| ! x. (! y. (y,x) : r --> P(y)) --> P(x);  x:A |] ==> P(x) |]  \
    18 \ ==>  wf(r)";
    19 by (Clarify_tac 1);
    20 by (rtac allE 1);
    21 by (assume_tac 1);
    22 by (best_tac (claset() addSEs [prem1 RS subsetD RS SigmaE2] addIs [prem2]) 1);
    23 qed "wfI";
    24 
    25 val major::prems = goalw WF.thy [wf_def]
    26     "[| wf(r);          \
    27 \       !!x.[| ! y. (y,x): r --> P(y) |] ==> P(x) \
    28 \    |]  ==>  P(a)";
    29 by (rtac (major RS spec RS mp RS spec) 1);
    30 by (blast_tac (claset() addIs prems) 1);
    31 qed "wf_induct";
    32 
    33 (*Perform induction on i, then prove the wf(r) subgoal using prems. *)
    34 fun wf_ind_tac a prems i = 
    35     EVERY [res_inst_tac [("a",a)] wf_induct i,
    36            rename_last_tac a ["1"] (i+1),
    37            ares_tac prems i];
    38 
    39 val prems = goal WF.thy "[| wf(r);  (a,x):r;  (x,a):r |] ==> P";
    40 by (subgoal_tac "! x. (a,x):r --> (x,a):r --> P" 1);
    41 by (blast_tac (claset() addIs prems) 1);
    42 by (wf_ind_tac "a" prems 1);
    43 by (Blast_tac 1);
    44 qed "wf_asym";
    45 
    46 val prems = goal WF.thy "[| wf(r);  (a,a): r |] ==> P";
    47 by (rtac wf_asym 1);
    48 by (REPEAT (resolve_tac prems 1));
    49 qed "wf_irrefl";
    50 
    51 (*transitive closure of a wf relation is wf! *)
    52 val [prem] = goal WF.thy "wf(r) ==> wf(r^+)";
    53 by (rewtac wf_def);
    54 by (Clarify_tac 1);
    55 (*must retain the universal formula for later use!*)
    56 by (rtac allE 1 THEN assume_tac 1);
    57 by (etac mp 1);
    58 by (res_inst_tac [("a","x")] (prem RS wf_induct) 1);
    59 by (rtac (impI RS allI) 1);
    60 by (etac tranclE 1);
    61 by (Blast_tac 1);
    62 by (Blast_tac 1);
    63 qed "wf_trancl";
    64 
    65 
    66 (*----------------------------------------------------------------------------
    67  * Minimal-element characterization of well-foundedness
    68  *---------------------------------------------------------------------------*)
    69 
    70 val wfr::_ = goalw WF.thy [wf_def]
    71     "wf r ==> x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q)";
    72 by (rtac (wfr RS spec RS mp RS spec) 1);
    73 by (Blast_tac 1);
    74 val lemma1 = result();
    75 
    76 goalw WF.thy [wf_def]
    77     "!!r. (! Q x. x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q)) ==> wf r";
    78 by (Clarify_tac 1);
    79 by (dres_inst_tac [("x", "{x. ~ P x}")] spec 1);
    80 by (Blast_tac 1);
    81 val lemma2 = result();
    82 
    83 goal WF.thy "wf r = (! Q x. x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q))";
    84 by (blast_tac (claset() addSIs [lemma1, lemma2]) 1);
    85 qed "wf_eq_minimal";
    86 
    87 (*---------------------------------------------------------------------------
    88  * Wellfoundedness of subsets
    89  *---------------------------------------------------------------------------*)
    90 
    91 goal thy "!!r. [| wf(r);  p<=r |] ==> wf(p)";
    92 by (full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
    93 by (Fast_tac 1);
    94 qed "wf_subset";
    95 
    96 (*---------------------------------------------------------------------------
    97  * Wellfoundedness of the empty relation.
    98  *---------------------------------------------------------------------------*)
    99 
   100 goal thy "wf({})";
   101 by (simp_tac (simpset() addsimps [wf_def]) 1);
   102 qed "wf_empty";
   103 AddSIs [wf_empty];
   104 
   105 (*---------------------------------------------------------------------------
   106  * Wellfoundedness of `insert'
   107  *---------------------------------------------------------------------------*)
   108 
   109 goal WF.thy "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)";
   110 by (rtac iffI 1);
   111  by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl] addIs
   112       [rtrancl_into_trancl1,wf_subset,impOfSubs rtrancl_mono]) 1);
   113 by (asm_full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
   114 by Safe_tac;
   115 by (EVERY1[rtac allE, atac, etac impE, Blast_tac]);
   116 by (etac bexE 1);
   117 by (rename_tac "a" 1);
   118 by (case_tac "a = x" 1);
   119  by (res_inst_tac [("x","a")]bexI 2);
   120   by (assume_tac 3);
   121  by (Blast_tac 2);
   122 by (case_tac "y:Q" 1);
   123  by (Blast_tac 2);
   124 by (res_inst_tac [("x","{z. z:Q & (z,y) : r^*}")] allE 1);
   125  by (assume_tac 1);
   126 by (thin_tac "! Q. (? x. x : Q) --> ?P Q" 1);	(*essential for speed*)
   127 by (blast_tac (claset() addIs [rtrancl_into_rtrancl2]) 1);
   128 qed "wf_insert";
   129 AddIffs [wf_insert];
   130 
   131 (*** acyclic ***)
   132 
   133 goalw WF.thy [acyclic_def]
   134  "!!r. wf r ==> acyclic r";
   135 by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]) 1);
   136 qed "wf_acyclic";
   137 
   138 goalw WF.thy [acyclic_def]
   139   "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)";
   140 by (simp_tac (simpset() addsimps [trancl_insert]) 1);
   141 by (blast_tac (claset() addEs [make_elim rtrancl_trans]) 1);
   142 qed "acyclic_insert";
   143 AddIffs [acyclic_insert];
   144 
   145 goalw WF.thy [acyclic_def] "acyclic(r^-1) = acyclic r";
   146 by (simp_tac (simpset() addsimps [trancl_inverse]) 1);
   147 qed "acyclic_inverse";
   148 
   149 (** cut **)
   150 
   151 (*This rewrite rule works upon formulae; thus it requires explicit use of
   152   H_cong to expose the equality*)
   153 goalw WF.thy [cut_def]
   154     "(cut f r x = cut g r x) = (!y. (y,x):r --> f(y)=g(y))";
   155 by (simp_tac (HOL_ss addsimps [expand_fun_eq] addsplits [expand_if]) 1);
   156 qed "cuts_eq";
   157 
   158 goalw WF.thy [cut_def] "!!x. (x,a):r ==> (cut f r a)(x) = f(x)";
   159 by (asm_simp_tac HOL_ss 1);
   160 qed "cut_apply";
   161 
   162 (*** is_recfun ***)
   163 
   164 goalw WF.thy [is_recfun_def,cut_def]
   165     "!!f. [| is_recfun r H a f;  ~(b,a):r |] ==> f(b) = arbitrary";
   166 by (etac ssubst 1);
   167 by (asm_simp_tac HOL_ss 1);
   168 qed "is_recfun_undef";
   169 
   170 (*** NOTE! some simplifications need a different finish_tac!! ***)
   171 fun indhyp_tac hyps =
   172     (cut_facts_tac hyps THEN'
   173        DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
   174                         eresolve_tac [transD, mp, allE]));
   175 val wf_super_ss = HOL_ss addSolver indhyp_tac;
   176 
   177 val prems = goalw WF.thy [is_recfun_def,cut_def]
   178     "[| wf(r);  trans(r);  is_recfun r H a f;  is_recfun r H b g |] ==> \
   179     \ (x,a):r --> (x,b):r --> f(x)=g(x)";
   180 by (cut_facts_tac prems 1);
   181 by (etac wf_induct 1);
   182 by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
   183 by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1);
   184 qed_spec_mp "is_recfun_equal";
   185 
   186 
   187 val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def]
   188     "[| wf(r);  trans(r); \
   189 \       is_recfun r H a f;  is_recfun r H b g;  (b,a):r |] ==> \
   190 \    cut f r b = g";
   191 val gundef = recgb RS is_recfun_undef
   192 and fisg   = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal)));
   193 by (cut_facts_tac prems 1);
   194 by (rtac ext 1);
   195 by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg]
   196                               addsplits [expand_if]) 1);
   197 qed "is_recfun_cut";
   198 
   199 (*** Main Existence Lemma -- Basic Properties of the_recfun ***)
   200 
   201 val prems = goalw WF.thy [the_recfun_def]
   202     "is_recfun r H a f ==> is_recfun r H a (the_recfun r H a)";
   203 by (res_inst_tac [("P", "is_recfun r H a")] selectI 1);
   204 by (resolve_tac prems 1);
   205 qed "is_the_recfun";
   206 
   207 val prems = goal WF.thy
   208  "[| wf(r);  trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
   209   by (cut_facts_tac prems 1);
   210   by (wf_ind_tac "a" prems 1);
   211   by (res_inst_tac [("f","cut (%y. H (the_recfun r H y) y) r a1")]
   212                    is_the_recfun 1);
   213   by (rewtac is_recfun_def);
   214   by (stac cuts_eq 1);
   215   by (rtac allI 1);
   216   by (rtac impI 1);
   217   by (res_inst_tac [("f1","H"),("x","y")](arg_cong RS fun_cong) 1);
   218   by (subgoal_tac
   219          "the_recfun r H y = cut(%x. H(cut(the_recfun r H y) r x) x) r y" 1);
   220   by (etac allE 2);
   221   by (dtac impE 2);
   222   by (atac 2);
   223   by (atac 3);
   224   by (atac 2);
   225   by (etac ssubst 1);
   226   by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
   227   by (rtac allI 1);
   228   by (rtac impI 1);
   229   by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1);
   230   by (res_inst_tac [("f1","H"),("x","ya")](arg_cong RS fun_cong) 1);
   231   by (fold_tac [is_recfun_def]);
   232   by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1);
   233 qed "unfold_the_recfun";
   234 
   235 val unwind1_the_recfun = rewrite_rule[is_recfun_def] unfold_the_recfun;
   236 
   237 (*--------------Old proof-----------------------------------------------------
   238 val prems = goal WF.thy
   239     "[| wf(r);  trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
   240 by (cut_facts_tac prems 1);
   241 by (wf_ind_tac "a" prems 1);
   242 by (res_inst_tac [("f", "cut (%y. wftrec r H y) r a1")] is_the_recfun 1); 
   243 by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
   244 by (stac cuts_eq 1);
   245 (*Applying the substitution: must keep the quantified assumption!!*)
   246 by (EVERY1 [Clarify_tac, rtac H_cong1, rtac allE, atac,
   247             etac (mp RS ssubst), atac]); 
   248 by (fold_tac [is_recfun_def]);
   249 by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1);
   250 qed "unfold_the_recfun";
   251 ---------------------------------------------------------------------------*)
   252 
   253 (** Removal of the premise trans(r) **)
   254 val th = rewrite_rule[is_recfun_def]
   255                      (trans_trancl RSN (2,(wf_trancl RS unfold_the_recfun)));
   256 
   257 goalw WF.thy [wfrec_def]
   258     "!!r. wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
   259 by (rtac H_cong 1);
   260 by (rtac refl 2);
   261 by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
   262 by (rtac allI 1);
   263 by (rtac impI 1);
   264 by (simp_tac(HOL_ss addsimps [wfrec_def]) 1);
   265 by (res_inst_tac [("a1","a")] (th RS ssubst) 1);
   266 by (atac 1);
   267 by (forward_tac[wf_trancl] 1);
   268 by (forward_tac[r_into_trancl] 1);
   269 by (asm_simp_tac (HOL_ss addsimps [cut_apply]) 1);
   270 by (rtac H_cong 1);    (*expose the equality of cuts*)
   271 by (rtac refl 2);
   272 by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
   273 by (Clarify_tac 1);
   274 by (res_inst_tac [("r","r^+")] is_recfun_equal 1);
   275 by (atac 1);
   276 by (rtac trans_trancl 1);
   277 by (rtac unfold_the_recfun 1);
   278 by (atac 1);
   279 by (rtac trans_trancl 1);
   280 by (rtac unfold_the_recfun 1);
   281 by (atac 1);
   282 by (rtac trans_trancl 1);
   283 by (rtac transD 1);
   284 by (rtac trans_trancl 1);
   285 by (forw_inst_tac [("a","ya")] r_into_trancl 1);
   286 by (atac 1);
   287 by (atac 1);
   288 by (forw_inst_tac [("a","ya")] r_into_trancl 1);
   289 by (atac 1);
   290 qed "wfrec";
   291 
   292 (*--------------Old proof-----------------------------------------------------
   293 goalw WF.thy [wfrec_def]
   294     "!!r. wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
   295 by (etac (wf_trancl RS wftrec RS ssubst) 1);
   296 by (rtac trans_trancl 1);
   297 by (rtac (refl RS H_cong) 1);    (*expose the equality of cuts*)
   298 by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
   299 qed "wfrec";
   300 ---------------------------------------------------------------------------*)
   301 
   302 (*---------------------------------------------------------------------------
   303  * This form avoids giant explosions in proofs.  NOTE USE OF == 
   304  *---------------------------------------------------------------------------*)
   305 val rew::prems = goal WF.thy
   306     "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a";
   307 by (rewtac rew);
   308 by (REPEAT (resolve_tac (prems@[wfrec]) 1));
   309 qed "def_wfrec";
   310 
   311 
   312 (**** TFL variants ****)
   313 
   314 goal WF.thy
   315     "!R. wf R --> (!P. (!x. (!y. (y,x):R --> P y) --> P x) --> (!x. P x))";
   316 by (Clarify_tac 1);
   317 by (res_inst_tac [("r","R"),("P","P"), ("a","x")] wf_induct 1);
   318 by (assume_tac 1);
   319 by (Blast_tac 1);
   320 qed"tfl_wf_induct";
   321 
   322 goal WF.thy "!f R. (x,a):R --> (cut f R a)(x) = f(x)";
   323 by (Clarify_tac 1);
   324 by (rtac cut_apply 1);
   325 by (assume_tac 1);
   326 qed"tfl_cut_apply";
   327 
   328 goal WF.thy "!M R f. (f=wfrec R M) --> wf R --> (!x. f x = M (cut f R x) x)";
   329 by (Clarify_tac 1);
   330 by (etac wfrec 1);
   331 qed "tfl_wfrec";