src/HOL/WF.ML
author paulson
Thu Aug 06 15:48:13 1998 +0200 (1998-08-06)
changeset 5278 a903b66822e2
parent 5148 74919e8f221c
child 5281 f4d16517b360
permissions -rw-r--r--
even more tidying of Goal commands
     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 val wf_converse_trancl = prove_goal thy 
    67 "!!X. wf (r^-1) ==> wf ((r^+)^-1)" (K [
    68 	stac (trancl_converse RS sym) 1,
    69 	etac wf_trancl 1]);
    70 
    71 (*----------------------------------------------------------------------------
    72  * Minimal-element characterization of well-foundedness
    73  *---------------------------------------------------------------------------*)
    74 
    75 val wfr::_ = goalw WF.thy [wf_def]
    76     "wf r ==> x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q)";
    77 by (rtac (wfr RS spec RS mp RS spec) 1);
    78 by (Blast_tac 1);
    79 val lemma1 = result();
    80 
    81 Goalw [wf_def]
    82     "(! Q x. x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q)) ==> wf r";
    83 by (Clarify_tac 1);
    84 by (dres_inst_tac [("x", "{x. ~ P x}")] spec 1);
    85 by (Blast_tac 1);
    86 val lemma2 = result();
    87 
    88 Goal "wf r = (! Q x. x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q))";
    89 by (blast_tac (claset() addSIs [lemma1, lemma2]) 1);
    90 qed "wf_eq_minimal";
    91 
    92 (*---------------------------------------------------------------------------
    93  * Wellfoundedness of subsets
    94  *---------------------------------------------------------------------------*)
    95 
    96 Goal "[| wf(r);  p<=r |] ==> wf(p)";
    97 by (full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
    98 by (Fast_tac 1);
    99 qed "wf_subset";
   100 
   101 (*---------------------------------------------------------------------------
   102  * Wellfoundedness of the empty relation.
   103  *---------------------------------------------------------------------------*)
   104 
   105 Goal "wf({})";
   106 by (simp_tac (simpset() addsimps [wf_def]) 1);
   107 qed "wf_empty";
   108 AddSIs [wf_empty];
   109 
   110 (*---------------------------------------------------------------------------
   111  * Wellfoundedness of `insert'
   112  *---------------------------------------------------------------------------*)
   113 
   114 Goal "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)";
   115 by (rtac iffI 1);
   116  by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl] 
   117 	addIs [rtrancl_into_trancl1,wf_subset,impOfSubs rtrancl_mono]) 1);
   118 by (asm_full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
   119 by Safe_tac;
   120 by (EVERY1[rtac allE, atac, etac impE, Blast_tac]);
   121 by (etac bexE 1);
   122 by (rename_tac "a" 1);
   123 by (case_tac "a = x" 1);
   124  by (res_inst_tac [("x","a")]bexI 2);
   125   by (assume_tac 3);
   126  by (Blast_tac 2);
   127 by (case_tac "y:Q" 1);
   128  by (Blast_tac 2);
   129 by (res_inst_tac [("x","{z. z:Q & (z,y) : r^*}")] allE 1);
   130  by (assume_tac 1);
   131 by (thin_tac "! Q. (? x. x : Q) --> ?P Q" 1);	(*essential for speed*)
   132 (*Blast_tac with new substOccur fails*)
   133 by (best_tac (claset() addIs [rtrancl_into_rtrancl2]) 1);
   134 qed "wf_insert";
   135 AddIffs [wf_insert];
   136 
   137 (*** acyclic ***)
   138 
   139 val acyclicI = prove_goalw WF.thy [acyclic_def] 
   140 "!!r. !x. (x, x) ~: r^+ ==> acyclic r" (K [atac 1]);
   141 
   142 Goalw [acyclic_def]
   143  "wf r ==> acyclic r";
   144 by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]) 1);
   145 qed "wf_acyclic";
   146 
   147 Goalw [acyclic_def]
   148   "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)";
   149 by (simp_tac (simpset() addsimps [trancl_insert]) 1);
   150 by (blast_tac (claset() addEs [make_elim rtrancl_trans]) 1);
   151 qed "acyclic_insert";
   152 AddIffs [acyclic_insert];
   153 
   154 Goalw [acyclic_def] "acyclic(r^-1) = acyclic r";
   155 by (simp_tac (simpset() addsimps [trancl_converse]) 1);
   156 qed "acyclic_converse";
   157 
   158 (** cut **)
   159 
   160 (*This rewrite rule works upon formulae; thus it requires explicit use of
   161   H_cong to expose the equality*)
   162 Goalw [cut_def]
   163     "(cut f r x = cut g r x) = (!y. (y,x):r --> f(y)=g(y))";
   164 by (simp_tac (HOL_ss addsimps [expand_fun_eq]) 1);
   165 qed "cuts_eq";
   166 
   167 Goalw [cut_def] "(x,a):r ==> (cut f r a)(x) = f(x)";
   168 by (asm_simp_tac HOL_ss 1);
   169 qed "cut_apply";
   170 
   171 (*** is_recfun ***)
   172 
   173 Goalw [is_recfun_def,cut_def]
   174     "[| is_recfun r H a f;  ~(b,a):r |] ==> f(b) = arbitrary";
   175 by (etac ssubst 1);
   176 by (asm_simp_tac HOL_ss 1);
   177 qed "is_recfun_undef";
   178 
   179 (*** NOTE! some simplifications need a different finish_tac!! ***)
   180 fun indhyp_tac hyps =
   181     (cut_facts_tac hyps THEN'
   182        DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
   183                         eresolve_tac [transD, mp, allE]));
   184 val wf_super_ss = HOL_ss addSolver indhyp_tac;
   185 
   186 val prems = goalw WF.thy [is_recfun_def,cut_def]
   187     "[| wf(r);  trans(r);  is_recfun r H a f;  is_recfun r H b g |] ==> \
   188     \ (x,a):r --> (x,b):r --> f(x)=g(x)";
   189 by (cut_facts_tac prems 1);
   190 by (etac wf_induct 1);
   191 by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
   192 by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1);
   193 qed_spec_mp "is_recfun_equal";
   194 
   195 
   196 val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def]
   197     "[| wf(r);  trans(r); \
   198 \       is_recfun r H a f;  is_recfun r H b g;  (b,a):r |] ==> \
   199 \    cut f r b = g";
   200 val gundef = recgb RS is_recfun_undef
   201 and fisg   = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal)));
   202 by (cut_facts_tac prems 1);
   203 by (rtac ext 1);
   204 by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg]) 1);
   205 qed "is_recfun_cut";
   206 
   207 (*** Main Existence Lemma -- Basic Properties of the_recfun ***)
   208 
   209 val prems = goalw WF.thy [the_recfun_def]
   210     "is_recfun r H a f ==> is_recfun r H a (the_recfun r H a)";
   211 by (res_inst_tac [("P", "is_recfun r H a")] selectI 1);
   212 by (resolve_tac prems 1);
   213 qed "is_the_recfun";
   214 
   215 val prems = goal WF.thy
   216  "!!r. [| wf(r);  trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
   217 by (wf_ind_tac "a" prems 1);
   218 by (res_inst_tac [("f","cut (%y. H (the_recfun r H y) y) r a1")]
   219                  is_the_recfun 1);
   220 by (rewtac is_recfun_def);
   221 by (stac cuts_eq 1);
   222 by (Clarify_tac 1);
   223 by (rtac (refl RSN (2,H_cong)) 1);
   224 by (subgoal_tac
   225          "the_recfun r H y = cut(%x. H(cut(the_recfun r H y) r x) x) r y" 1);
   226  by (etac allE 2);
   227  by (dtac impE 2);
   228    by (atac 2);
   229   by (atac 3);
   230  by (atac 2);
   231 by (etac ssubst 1);
   232 by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
   233 by (Clarify_tac 1);
   234 by (stac cut_apply 1);
   235  by (fast_tac (claset() addDs [transD]) 1);
   236 by (rtac (refl RSN (2,H_cong)) 1);
   237 by (fold_tac [is_recfun_def]);
   238 by (asm_simp_tac (wf_super_ss addsimps[is_recfun_cut]) 1);
   239 qed "unfold_the_recfun";
   240 
   241 val unwind1_the_recfun = rewrite_rule[is_recfun_def] unfold_the_recfun;
   242 
   243 (*--------------Old proof-----------------------------------------------------
   244 val prems = goal WF.thy
   245     "[| wf(r);  trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
   246 by (cut_facts_tac prems 1);
   247 by (wf_ind_tac "a" prems 1);
   248 by (res_inst_tac [("f", "cut (%y. wftrec r H y) r a1")] is_the_recfun 1); 
   249 by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
   250 by (stac cuts_eq 1);
   251 (*Applying the substitution: must keep the quantified assumption!!*)
   252 by (EVERY1 [Clarify_tac, rtac H_cong1, rtac allE, atac,
   253             etac (mp RS ssubst), atac]); 
   254 by (fold_tac [is_recfun_def]);
   255 by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1);
   256 qed "unfold_the_recfun";
   257 ---------------------------------------------------------------------------*)
   258 
   259 (** Removal of the premise trans(r) **)
   260 val th = rewrite_rule[is_recfun_def]
   261                      (trans_trancl RSN (2,(wf_trancl RS unfold_the_recfun)));
   262 
   263 Goalw [wfrec_def]
   264     "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
   265 by (rtac H_cong 1);
   266 by (rtac refl 2);
   267 by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
   268 by (rtac allI 1);
   269 by (rtac impI 1);
   270 by (simp_tac(HOL_ss addsimps [wfrec_def]) 1);
   271 by (res_inst_tac [("a1","a")] (th RS ssubst) 1);
   272 by (atac 1);
   273 by (forward_tac[wf_trancl] 1);
   274 by (forward_tac[r_into_trancl] 1);
   275 by (asm_simp_tac (HOL_ss addsimps [cut_apply]) 1);
   276 by (rtac H_cong 1);    (*expose the equality of cuts*)
   277 by (rtac refl 2);
   278 by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
   279 by (Clarify_tac 1);
   280 by (res_inst_tac [("r","r^+")] is_recfun_equal 1);
   281 by (atac 1);
   282 by (rtac trans_trancl 1);
   283 by (rtac unfold_the_recfun 1);
   284 by (atac 1);
   285 by (rtac trans_trancl 1);
   286 by (rtac unfold_the_recfun 1);
   287 by (atac 1);
   288 by (rtac trans_trancl 1);
   289 by (rtac transD 1);
   290 by (rtac trans_trancl 1);
   291 by (forw_inst_tac [("p","(ya,y)")] r_into_trancl 1);
   292 by (atac 1);
   293 by (atac 1);
   294 by (forw_inst_tac [("p","(ya,y)")] r_into_trancl 1);
   295 by (atac 1);
   296 qed "wfrec";
   297 
   298 (*--------------Old proof-----------------------------------------------------
   299 Goalw [wfrec_def]
   300     "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
   301 by (etac (wf_trancl RS wftrec RS ssubst) 1);
   302 by (rtac trans_trancl 1);
   303 by (rtac (refl RS H_cong) 1);    (*expose the equality of cuts*)
   304 by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
   305 qed "wfrec";
   306 ---------------------------------------------------------------------------*)
   307 
   308 (*---------------------------------------------------------------------------
   309  * This form avoids giant explosions in proofs.  NOTE USE OF == 
   310  *---------------------------------------------------------------------------*)
   311 val rew::prems = goal WF.thy
   312     "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a";
   313 by (rewtac rew);
   314 by (REPEAT (resolve_tac (prems@[wfrec]) 1));
   315 qed "def_wfrec";
   316 
   317 
   318 (**** TFL variants ****)
   319 
   320 Goal "!R. wf R --> (!P. (!x. (!y. (y,x):R --> P y) --> P x) --> (!x. P x))";
   321 by (Clarify_tac 1);
   322 by (res_inst_tac [("r","R"),("P","P"), ("a","x")] wf_induct 1);
   323 by (assume_tac 1);
   324 by (Blast_tac 1);
   325 qed"tfl_wf_induct";
   326 
   327 Goal "!f R. (x,a):R --> (cut f R a)(x) = f(x)";
   328 by (Clarify_tac 1);
   329 by (rtac cut_apply 1);
   330 by (assume_tac 1);
   331 qed"tfl_cut_apply";
   332 
   333 Goal "!M R f. (f=wfrec R M) --> wf R --> (!x. f x = M (cut f R x) x)";
   334 by (Clarify_tac 1);
   335 by (etac wfrec 1);
   336 qed "tfl_wfrec";