src/HOL/Wellfounded_Recursion.ML
author oheimb
Wed Jan 31 10:15:55 2001 +0100 (2001-01-31)
changeset 11008 f7333f055ef6
parent 10832 e33b47e4246d
child 11141 0d4ca3b3741f
permissions -rw-r--r--
improved theory reference in comment
     1 (*  Title:      HOL/Wellfounded_Recursion.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 Goal "x = y ==> H x z = H y z";
    10 by (Asm_simp_tac 1);
    11 val H_cong2 = (*freeze H!*)
    12 	      read_instantiate [("H","H")] (result());
    13 
    14 val [prem] = Goalw [wf_def]
    15  "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)";
    16 by (Clarify_tac 1);
    17 by (rtac prem 1);
    18 by (assume_tac 1);
    19 qed "wfUNIVI";
    20 
    21 (*Restriction to domain A.  If r is well-founded over A then wf(r)*)
    22 val [prem1,prem2] = Goalw [wf_def]
    23  "[| r <= A <*> A;  \
    24 \    !!x P. [| ALL x. (ALL y. (y,x) : r --> P y) --> P x;  x:A |] ==> P x |]  \
    25 \ ==>  wf r";
    26 by (cut_facts_tac [prem1] 1);
    27 by (blast_tac (claset() addIs [prem2]) 1);
    28 qed "wfI";
    29 
    30 val major::prems = Goalw [wf_def]
    31     "[| wf(r);          \
    32 \       !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x) \
    33 \    |]  ==>  P(a)";
    34 by (rtac (major RS spec RS mp RS spec) 1);
    35 by (blast_tac (claset() addIs prems) 1);
    36 qed "wf_induct";
    37 
    38 (*Perform induction on i, then prove the wf(r) subgoal using prems. *)
    39 fun wf_ind_tac a prems i = 
    40     EVERY [res_inst_tac [("a",a)] wf_induct i,
    41            rename_last_tac a ["1"] (i+1),
    42            ares_tac prems i];
    43 
    44 Goal "wf(r) ==> ALL x. (a,x):r --> (x,a)~:r";
    45 by (wf_ind_tac "a" [] 1);
    46 by (Blast_tac 1);
    47 qed_spec_mp "wf_not_sym";
    48 
    49 (* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)
    50 bind_thm ("wf_asym", cla_make_elim wf_not_sym);
    51 
    52 Goal "wf(r) ==> (a,a) ~: r";
    53 by (blast_tac (claset() addEs [wf_asym]) 1);
    54 qed "wf_not_refl";
    55 
    56 (* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)
    57 bind_thm ("wf_irrefl", make_elim wf_not_refl);
    58 
    59 (*transitive closure of a wf relation is wf! *)
    60 Goal "wf(r) ==> wf(r^+)";
    61 by (stac wf_def 1);
    62 by (Clarify_tac 1);
    63 (*must retain the universal formula for later use!*)
    64 by (rtac allE 1 THEN assume_tac 1);
    65 by (etac mp 1);
    66 by (eres_inst_tac [("a","x")] wf_induct 1);
    67 by (blast_tac (claset() addEs [tranclE]) 1);
    68 qed "wf_trancl";
    69 
    70 Goal "wf (r^-1) ==> wf ((r^+)^-1)";
    71 by (stac (trancl_converse RS sym) 1);
    72 by (etac wf_trancl 1);
    73 qed "wf_converse_trancl";
    74 
    75 
    76 (*----------------------------------------------------------------------------
    77  * Minimal-element characterization of well-foundedness
    78  *---------------------------------------------------------------------------*)
    79 
    80 Goalw [wf_def] "wf r ==> x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)";
    81 by (dtac spec 1);
    82 by (etac (mp RS spec) 1);
    83 by (Blast_tac 1);
    84 val lemma1 = result();
    85 
    86 Goalw [wf_def] "(ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)) ==> wf r";
    87 by (Clarify_tac 1);
    88 by (dres_inst_tac [("x", "{x. ~ P x}")] spec 1);
    89 by (Blast_tac 1);
    90 val lemma2 = result();
    91 
    92 Goal "wf r = (ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q))";
    93 by (blast_tac (claset() addSIs [lemma1, lemma2]) 1);
    94 qed "wf_eq_minimal";
    95 
    96 (*---------------------------------------------------------------------------
    97  * Wellfoundedness of subsets
    98  *---------------------------------------------------------------------------*)
    99 
   100 Goal "[| wf(r);  p<=r |] ==> wf(p)";
   101 by (full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
   102 by (Fast_tac 1);
   103 qed "wf_subset";
   104 
   105 (*---------------------------------------------------------------------------
   106  * Wellfoundedness of the empty relation.
   107  *---------------------------------------------------------------------------*)
   108 
   109 Goal "wf({})";
   110 by (simp_tac (simpset() addsimps [wf_def]) 1);
   111 qed "wf_empty";
   112 AddIffs [wf_empty];
   113 
   114 (*---------------------------------------------------------------------------
   115  * Wellfoundedness of `insert'
   116  *---------------------------------------------------------------------------*)
   117 
   118 Goal "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)";
   119 by (rtac iffI 1);
   120  by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl] 
   121 	addIs [rtrancl_into_trancl1,wf_subset,impOfSubs rtrancl_mono]) 1);
   122 by (asm_full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
   123 by Safe_tac;
   124 by (EVERY1[rtac allE, assume_tac, etac impE, Blast_tac]);
   125 by (etac bexE 1);
   126 by (rename_tac "a" 1);
   127 by (case_tac "a = x" 1);
   128  by (res_inst_tac [("x","a")]bexI 2);
   129   by (assume_tac 3);
   130  by (Blast_tac 2);
   131 by (case_tac "y:Q" 1);
   132  by (Blast_tac 2);
   133 by (res_inst_tac [("x","{z. z:Q & (z,y) : r^*}")] allE 1);
   134  by (assume_tac 1);
   135 by (thin_tac "ALL Q. (EX x. x : Q) --> ?P Q" 1);	(*essential for speed*)
   136 (*Blast_tac with new substOccur fails*)
   137 by (best_tac (claset() addIs [rtrancl_into_rtrancl2]) 1);
   138 qed "wf_insert";
   139 AddIffs [wf_insert];
   140 
   141 (*---------------------------------------------------------------------------
   142  * Wellfoundedness of `disjoint union'
   143  *---------------------------------------------------------------------------*)
   144 
   145 (*Intuition behind this proof for the case of binary union:
   146 
   147   Goal: find an (R u S)-min element of a nonempty subset A.
   148   by case distinction:
   149   1. There is a step a -R-> b with a,b : A.
   150      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
   151      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
   152      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
   153      have an S-successor and is thus S-min in A as well.
   154   2. There is no such step.
   155      Pick an S-min element of A. In this case it must be an R-min
   156      element of A as well.
   157 
   158 *)
   159 
   160 Goal "[| ALL i:I. wf(r i); \
   161 \        ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {} & \
   162 \                                         Domain(r j) Int Range(r i) = {} \
   163 \     |] ==> wf(UN i:I. r i)";
   164 by (asm_full_simp_tac (HOL_basic_ss addsimps [wf_eq_minimal]) 1);
   165 by (Clarify_tac 1);
   166 by (rename_tac "A a" 1);
   167 by (case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i" 1);
   168  by (Asm_full_simp_tac 2);
   169  by (Best_tac 2);  (*much faster than Blast_tac*)
   170 by (Clarify_tac 1);
   171 by (EVERY1[dtac bspec, assume_tac,
   172 	   eres_inst_tac [("x","{a. a:A & (EX b:A. (b,a) : r i)}")] allE]);
   173 by (EVERY1[etac allE, etac impE]);
   174  by (ALLGOALS Blast_tac);
   175 qed "wf_UN";
   176 
   177 Goalw [Union_def]
   178  "[| ALL r:R. wf r; \
   179 \    ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {} & \
   180 \                                 Domain s Int Range r = {} \
   181 \ |] ==> wf(Union R)";
   182 by (blast_tac (claset() addIs [wf_UN]) 1);
   183 qed "wf_Union";
   184 
   185 Goal "[| wf r; wf s; Domain r Int Range s = {}; Domain s Int Range r = {} \
   186 \     |] ==> wf(r Un s)";
   187 by (rtac (simplify (simpset()) (read_instantiate[("R","{r,s}")]wf_Union)) 1);
   188 by (Blast_tac 1);
   189 by (Blast_tac 1);
   190 qed "wf_Un";
   191 
   192 (*---------------------------------------------------------------------------
   193  * Wellfoundedness of `image'
   194  *---------------------------------------------------------------------------*)
   195 
   196 Goal "[| wf r; inj f |] ==> wf(prod_fun f f ` r)";
   197 by (asm_full_simp_tac (HOL_basic_ss addsimps [wf_eq_minimal]) 1);
   198 by (Clarify_tac 1);
   199 by (case_tac "EX p. f p : Q" 1);
   200 by (eres_inst_tac [("x","{p. f p : Q}")]allE 1);
   201 by (fast_tac (claset() addDs [injD]) 1);
   202 by (Blast_tac 1);
   203 qed "wf_prod_fun_image";
   204 
   205 (*** acyclic ***)
   206 
   207 Goalw [acyclic_def] "ALL x. (x, x) ~: r^+ ==> acyclic r";
   208 by (assume_tac 1);
   209 qed "acyclicI";
   210 
   211 Goalw [acyclic_def] "wf r ==> acyclic r";
   212 by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]) 1);
   213 qed "wf_acyclic";
   214 
   215 Goalw [acyclic_def] "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)";
   216 by (simp_tac (simpset() addsimps [trancl_insert]) 1);
   217 by (blast_tac (claset() addIs [rtrancl_trans]) 1);
   218 qed "acyclic_insert";
   219 AddIffs [acyclic_insert];
   220 
   221 Goalw [acyclic_def] "acyclic(r^-1) = acyclic r";
   222 by (simp_tac (simpset() addsimps [trancl_converse]) 1);
   223 qed "acyclic_converse";
   224 AddIffs [acyclic_converse];
   225 
   226 Goalw [acyclic_def,antisym_def] "acyclic r ==> antisym(r^*)";
   227 by(blast_tac (claset() addEs [rtranclE]
   228      addIs [rtrancl_into_trancl1,rtrancl_trancl_trancl]) 1);
   229 qed "acyclic_impl_antisym_rtrancl";
   230 
   231 (* Other direction:
   232 acyclic = no loops
   233 antisym = only self loops
   234 Goalw [acyclic_def,antisym_def] "antisym(r^* ) ==> acyclic(r - Id)";
   235 ==> "antisym(r^* ) = acyclic(r - Id)";
   236 *)
   237 
   238 Goalw [acyclic_def] "[| acyclic s; r <= s |] ==> acyclic r";
   239 by (blast_tac (claset() addIs [trancl_mono]) 1);
   240 qed "acyclic_subset";
   241 
   242 (** cut **)
   243 
   244 (*This rewrite rule works upon formulae; thus it requires explicit use of
   245   H_cong to expose the equality*)
   246 Goalw [cut_def] "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))";
   247 by (simp_tac (HOL_ss addsimps [expand_fun_eq]) 1);
   248 qed "cuts_eq";
   249 
   250 Goalw [cut_def] "(x,a):r ==> (cut f r a)(x) = f(x)";
   251 by (asm_simp_tac HOL_ss 1);
   252 qed "cut_apply";
   253 
   254 (*** is_recfun ***)
   255 
   256 Goalw [is_recfun_def,cut_def]
   257     "[| is_recfun r H a f;  ~(b,a):r |] ==> f(b) = arbitrary";
   258 by (etac ssubst 1);
   259 by (asm_simp_tac HOL_ss 1);
   260 qed "is_recfun_undef";
   261 
   262 (*** NOTE! some simplifications need a different Solver!! ***)
   263 fun indhyp_tac hyps =
   264     (cut_facts_tac hyps THEN'
   265        DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
   266                         eresolve_tac [transD, mp, allE]));
   267 val wf_super_ss = HOL_ss addSolver (mk_solver "WF" indhyp_tac);
   268 
   269 Goalw [is_recfun_def,cut_def]
   270     "[| wf(r);  trans(r);  is_recfun r H a f;  is_recfun r H b g |] ==> \
   271     \ (x,a):r --> (x,b):r --> f(x)=g(x)";
   272 by (etac wf_induct 1);
   273 by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
   274 by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1);
   275 qed_spec_mp "is_recfun_equal";
   276 
   277 
   278 val prems as [wfr,transr,recfa,recgb,_] = goalw (the_context ()) [cut_def]
   279     "[| wf(r);  trans(r); \
   280 \       is_recfun r H a f;  is_recfun r H b g;  (b,a):r |] ==> \
   281 \    cut f r b = g";
   282 val gundef = recgb RS is_recfun_undef
   283 and fisg   = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal)));
   284 by (cut_facts_tac prems 1);
   285 by (rtac ext 1);
   286 by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg]) 1);
   287 qed "is_recfun_cut";
   288 
   289 (*** Main Existence Lemma -- Basic Properties of the_recfun ***)
   290 
   291 Goalw [the_recfun_def]
   292     "is_recfun r H a f ==> is_recfun r H a (the_recfun r H a)";
   293 by (eres_inst_tac [("P", "is_recfun r H a")] someI 1);
   294 qed "is_the_recfun";
   295 
   296 Goal "[| wf(r);  trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
   297 by (wf_ind_tac "a" [] 1);
   298 by (res_inst_tac [("f","cut (%y. H (the_recfun r H y) y) r a1")]
   299                  is_the_recfun 1);
   300 by (rewtac is_recfun_def);
   301 by (stac cuts_eq 1);
   302 by (Clarify_tac 1);
   303 by (rtac H_cong2 1);
   304 by (subgoal_tac
   305          "the_recfun r H y = cut(%x. H(cut(the_recfun r H y) r x) x) r y" 1);
   306  by (Blast_tac 2);
   307 by (etac ssubst 1);
   308 by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
   309 by (Clarify_tac 1);
   310 by (stac cut_apply 1);
   311  by (fast_tac (claset() addDs [transD]) 1);
   312 by (fold_tac [is_recfun_def]);
   313 by (asm_simp_tac (wf_super_ss addsimps[is_recfun_cut]) 1);
   314 qed "unfold_the_recfun";
   315 
   316 Goal "[| wf r; trans r; (x,a) : r; (x,b) : r |] \
   317 \     ==> the_recfun r H a x = the_recfun r H b x";
   318 by (best_tac (claset() addIs [is_recfun_equal, unfold_the_recfun]) 1);
   319 qed "the_recfun_equal";
   320 
   321 (** Removal of the premise trans(r) **)
   322 val th = rewrite_rule[is_recfun_def]
   323                      (trans_trancl RSN (2,(wf_trancl RS unfold_the_recfun)));
   324 
   325 Goalw [wfrec_def]
   326     "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
   327 by (rtac H_cong2 1);
   328 by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
   329 by (rtac allI 1);
   330 by (rtac impI 1);
   331 by (res_inst_tac [("a1","a")] (th RS ssubst) 1);
   332 by (assume_tac 1);
   333 by (ftac wf_trancl 1);
   334 by (ftac r_into_trancl 1);
   335 by (asm_simp_tac (HOL_ss addsimps [cut_apply]) 1);
   336 by (rtac H_cong2 1);    (*expose the equality of cuts*)
   337 by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
   338 by (blast_tac (claset() addIs [the_recfun_equal, transD, trans_trancl, 
   339 			       r_into_trancl]) 1);
   340 qed "wfrec";
   341 
   342 (*---------------------------------------------------------------------------
   343  * This form avoids giant explosions in proofs.  NOTE USE OF == 
   344  *---------------------------------------------------------------------------*)
   345 Goal "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a";
   346 by Auto_tac;
   347 by (blast_tac (claset() addIs [wfrec]) 1);   
   348 qed "def_wfrec";
   349 
   350 
   351 (**** TFL variants ****)
   352 
   353 Goal "ALL R. wf R --> \
   354 \      (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))";
   355 by (Clarify_tac 1);
   356 by (res_inst_tac [("r","R"),("P","P"), ("a","x")] wf_induct 1);
   357 by (assume_tac 1);
   358 by (Blast_tac 1);
   359 qed"tfl_wf_induct";
   360 
   361 Goal "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)";
   362 by (Clarify_tac 1);
   363 by (rtac cut_apply 1);
   364 by (assume_tac 1);
   365 qed"tfl_cut_apply";
   366 
   367 Goal "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)";
   368 by (Clarify_tac 1);
   369 by (etac wfrec 1);
   370 qed "tfl_wfrec";