author  paulson 
Thu, 10 Jun 1999 10:50:19 +0200  
changeset 6814  d96d4977f94e 
parent 6433  228237ec56e5 
child 7249  4886664d7033 
permissions  rwrr 
1475  1 
(* Title: HOL/wf.ML 
923  2 
ID: $Id$ 
1475  3 
Author: Tobias Nipkow, with minor changes by Konrad Slind 
4 
Copyright 1992 University of Cambridge/1995 TU Munich 

923  5 

3198  6 
Wellfoundedness, induction, and recursion 
923  7 
*) 
8 

950  9 
val H_cong = read_instantiate [("f","H")] (standard(refl RS cong RS cong)); 
923  10 
val H_cong1 = refl RS H_cong; 
11 

5579  12 
val [prem] = Goalw [wf_def] 
13 
"[ !!P x. [ !x. (!y. (y,x) : r > P(y)) > P(x) ] ==> P(x) ] ==> wf(r)"; 

14 
by (Clarify_tac 1); 

15 
by (rtac prem 1); 

16 
by (assume_tac 1); 

17 
qed "wfUNIVI"; 

18 

923  19 
(*Restriction to domain A. If r is wellfounded over A then wf(r)*) 
5316  20 
val [prem1,prem2] = Goalw [wf_def] 
1642  21 
"[ r <= A Times A; \ 
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\ !!x P. [ ! x. (! y. (y,x) : r > P(y)) > P(x); x:A ] ==> P(x) ] \ 
923  23 
\ ==> wf(r)"; 
3708  24 
by (Clarify_tac 1); 
923  25 
by (rtac allE 1); 
26 
by (assume_tac 1); 

4089  27 
by (best_tac (claset() addSEs [prem1 RS subsetD RS SigmaE2] addIs [prem2]) 1); 
923  28 
qed "wfI"; 
29 

5316  30 
val major::prems = Goalw [wf_def] 
923  31 
"[ wf(r); \ 
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\ !!x.[ ! y. (y,x): r > P(y) ] ==> P(x) \ 
923  33 
\ ] ==> P(a)"; 
34 
by (rtac (major RS spec RS mp RS spec) 1); 

4089  35 
by (blast_tac (claset() addIs prems) 1); 
923  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, 

1465  41 
rename_last_tac a ["1"] (i+1), 
42 
ares_tac prems i]; 

923  43 

5452  44 
Goal "wf(r) ==> ! x. (a,x):r > (x,a)~:r"; 
5316  45 
by (wf_ind_tac "a" [] 1); 
2935  46 
by (Blast_tac 1); 
5452  47 
qed_spec_mp "wf_not_sym"; 
48 

49 
(* [ wf(r); (a,x):r; ~P ==> (x,a):r ] ==> P *) 

50 
bind_thm ("wf_asym", wf_not_sym RS swap); 

923  51 

5316  52 
Goal "[ wf(r); (a,a): r ] ==> P"; 
53 
by (blast_tac (claset() addEs [wf_asym]) 1); 

1618  54 
qed "wf_irrefl"; 
923  55 

1475  56 
(*transitive closure of a wf relation is wf! *) 
5316  57 
Goal "wf(r) ==> wf(r^+)"; 
58 
by (stac wf_def 1); 

3708  59 
by (Clarify_tac 1); 
923  60 
(*must retain the universal formula for later use!*) 
61 
by (rtac allE 1 THEN assume_tac 1); 

62 
by (etac mp 1); 

5316  63 
by (eres_inst_tac [("a","x")] wf_induct 1); 
923  64 
by (rtac (impI RS allI) 1); 
65 
by (etac tranclE 1); 

2935  66 
by (Blast_tac 1); 
67 
by (Blast_tac 1); 

923  68 
qed "wf_trancl"; 
69 

70 

4762  71 
val wf_converse_trancl = prove_goal thy 
72 
"!!X. wf (r^1) ==> wf ((r^+)^1)" (K [ 

73 
stac (trancl_converse RS sym) 1, 

74 
etac wf_trancl 1]); 

75 

3198  76 
(* 
77 
* Minimalelement characterization of wellfoundedness 

78 
**) 

79 

5316  80 
Goalw [wf_def] "wf r ==> x:Q > (? z:Q. ! y. (y,z):r > y~:Q)"; 
5318  81 
by (dtac spec 1); 
5316  82 
by (etac (mp RS spec) 1); 
3198  83 
by (Blast_tac 1); 
84 
val lemma1 = result(); 

85 

5316  86 
Goalw [wf_def] "(! Q x. x:Q > (? z:Q. ! y. (y,z):r > y~:Q)) ==> wf r"; 
3708  87 
by (Clarify_tac 1); 
3198  88 
by (dres_inst_tac [("x", "{x. ~ P x}")] spec 1); 
89 
by (Blast_tac 1); 

90 
val lemma2 = result(); 

91 

5069  92 
Goal "wf r = (! Q x. x:Q > (? z:Q. ! y. (y,z):r > y~:Q))"; 
4089  93 
by (blast_tac (claset() addSIs [lemma1, lemma2]) 1); 
3198  94 
qed "wf_eq_minimal"; 
95 

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(* 
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* Wellfoundedness of subsets 
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**) 
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Goal "[ wf(r); p<=r ] ==> wf(p)"; 
4089  101 
by (full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1); 
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by (Fast_tac 1); 
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qed "wf_subset"; 
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(* 
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* Wellfoundedness of the empty relation. 
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**) 
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5069  109 
Goal "wf({})"; 
4089  110 
by (simp_tac (simpset() addsimps [wf_def]) 1); 
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qed "wf_empty"; 
5281  112 
AddIffs [wf_empty]; 
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(* 
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* Wellfoundedness of `insert' 
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**) 
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5069  118 
Goal "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"; 
3457  119 
by (rtac iffI 1); 
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by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl] 
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addIs [rtrancl_into_trancl1,wf_subset,impOfSubs rtrancl_mono]) 1); 
4089  122 
by (asm_full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1); 
4153  123 
by Safe_tac; 
3457  124 
by (EVERY1[rtac allE, atac, 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); 

4059  133 
by (res_inst_tac [("x","{z. z:Q & (z,y) : r^*}")] allE 1); 
3457  134 
by (assume_tac 1); 
4059  135 
by (thin_tac "! Q. (? x. x : Q) > ?P Q" 1); (*essential for speed*) 
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(*Blast_tac with new substOccur fails*) 
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by (best_tac (claset() addIs [rtrancl_into_rtrancl2]) 1); 
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qed "wf_insert"; 
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AddIffs [wf_insert]; 
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5281  141 
(* 
142 
* Wellfoundedness of `disjoint union' 

143 
**) 

144 

5330  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 Rmin 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 Rmin in the 

152 
subset, z' must be Rmin in A. Because z' has an Rpredecessor, it cannot 

153 
have an Ssuccessor and is thus Smin in A as well. 

154 
2. There is no such step. 

155 
Pick an Smin element of A. In this case it must be an Rmin 

156 
element of A as well. 

157 

158 
*) 

159 

5316  160 
Goal "[ !i:I. wf(r i); \ 
161 
\ !i:I.!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)"; 

5318  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 "? i:I. ? a:A. ? b:A. (b,a) : r i" 1); 

168 
by (Clarify_tac 1); 

169 
by (EVERY1[dtac bspec, atac, 

5281  170 
eres_inst_tac[("x","{a. a:A & (? b:A. (b,a) : r i)}")]allE]); 
5318  171 
by (EVERY1[etac allE,etac impE]); 
172 
by (Blast_tac 1); 

173 
by (Clarify_tac 1); 

174 
by (rename_tac "z'" 1); 

175 
by (res_inst_tac [("x","z'")] bexI 1); 

176 
by (assume_tac 2); 

177 
by (Clarify_tac 1); 

178 
by (rename_tac "j" 1); 

179 
by (case_tac "r j = r i" 1); 

180 
by (EVERY1[etac allE,etac impE,atac]); 

181 
by (Asm_full_simp_tac 1); 

182 
by (Blast_tac 1); 

183 
by (blast_tac (claset() addEs [equalityE]) 1); 

184 
by (Asm_full_simp_tac 1); 

5521  185 
by (fast_tac (claset() delWrapper "bspec") 1); (*faster than Blast_tac*) 
5281  186 
qed "wf_UN"; 
187 

188 
Goalw [Union_def] 

189 
"[ !r:R. wf r; \ 

190 
\ !r:R.!s:R. r ~= s > Domain r Int Range s = {} & \ 

191 
\ Domain s Int Range r = {} \ 

192 
\ ] ==> wf(Union R)"; 

5318  193 
by (rtac wf_UN 1); 
194 
by (Blast_tac 1); 

195 
by (Blast_tac 1); 

5281  196 
qed "wf_Union"; 
197 

5316  198 
Goal "[ wf r; wf s; Domain r Int Range s = {}; Domain s Int Range r = {} \ 
199 
\ ] ==> wf(r Un s)"; 

5318  200 
by (rtac (simplify (simpset()) (read_instantiate[("R","{r,s}")]wf_Union)) 1); 
201 
by (Blast_tac 1); 

202 
by (Blast_tac 1); 

5281  203 
qed "wf_Un"; 
204 

205 
(* 

206 
* Wellfoundedness of `image' 

207 
**) 

208 

209 
Goal "[ wf r; inj f ] ==> wf(prod_fun f f `` r)"; 

5318  210 
by (asm_full_simp_tac (HOL_basic_ss addsimps [wf_eq_minimal]) 1); 
211 
by (Clarify_tac 1); 

212 
by (case_tac "? p. f p : Q" 1); 

213 
by (eres_inst_tac [("x","{p. f p : Q}")]allE 1); 

214 
by (fast_tac (claset() addDs [injD]) 1); 

215 
by (Blast_tac 1); 

5281  216 
qed "wf_prod_fun_image"; 
217 

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(*** acyclic ***) 
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4750  220 
val acyclicI = prove_goalw WF.thy [acyclic_def] 
221 
"!!r. !x. (x, x) ~: r^+ ==> acyclic r" (K [atac 1]); 

222 

5069  223 
Goalw [acyclic_def] 
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"wf r ==> acyclic r"; 
4089  225 
by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]) 1); 
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qed "wf_acyclic"; 
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5452  228 
Goalw [acyclic_def] "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"; 
4089  229 
by (simp_tac (simpset() addsimps [trancl_insert]) 1); 
5452  230 
by (blast_tac (claset() addIs [rtrancl_trans]) 1); 
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qed "acyclic_insert"; 
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AddIffs [acyclic_insert]; 
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5069  234 
Goalw [acyclic_def] "acyclic(r^1) = acyclic r"; 
4746  235 
by (simp_tac (simpset() addsimps [trancl_converse]) 1); 
236 
qed "acyclic_converse"; 

3198  237 

6433  238 
Goalw [acyclic_def] "[ acyclic s; r <= s ] ==> acyclic r"; 
6814  239 
by (blast_tac (claset() addIs [trancl_mono]) 1); 
6433  240 
qed "acyclic_subset"; 
241 

923  242 
(** cut **) 
243 

244 
(*This rewrite rule works upon formulae; thus it requires explicit use of 

245 
H_cong to expose the equality*) 

5069  246 
Goalw [cut_def] 
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"(cut f r x = cut g r x) = (!y. (y,x):r > f(y)=g(y))"; 
4686  248 
by (simp_tac (HOL_ss addsimps [expand_fun_eq]) 1); 
1475  249 
qed "cuts_eq"; 
923  250 

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Goalw [cut_def] "(x,a):r ==> (cut f r a)(x) = f(x)"; 
1552  252 
by (asm_simp_tac HOL_ss 1); 
923  253 
qed "cut_apply"; 
254 

255 
(*** is_recfun ***) 

256 

5069  257 
Goalw [is_recfun_def,cut_def] 
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"[ is_recfun r H a f; ~(b,a):r ] ==> f(b) = arbitrary"; 
923  259 
by (etac ssubst 1); 
1552  260 
by (asm_simp_tac HOL_ss 1); 
923  261 
qed "is_recfun_undef"; 
262 

263 
(*** NOTE! some simplifications need a different finish_tac!! ***) 

264 
fun indhyp_tac hyps = 

265 
(cut_facts_tac hyps THEN' 

266 
DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE' 

1465  267 
eresolve_tac [transD, mp, allE])); 
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268 
val wf_super_ss = HOL_ss addSolver indhyp_tac; 
923  269 

5316  270 
Goalw [is_recfun_def,cut_def] 
1475  271 
"[ wf(r); trans(r); is_recfun r H a f; is_recfun r H b g ] ==> \ 
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\ (x,a):r > (x,b):r > f(x)=g(x)"; 
923  273 
by (etac wf_induct 1); 
274 
by (REPEAT (rtac impI 1 ORELSE etac ssubst 1)); 

275 
by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1); 

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276 
qed_spec_mp "is_recfun_equal"; 
923  277 

278 

279 
val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def] 

280 
"[ wf(r); trans(r); \ 

1475  281 
\ is_recfun r H a f; is_recfun r H b g; (b,a):r ] ==> \ 
923  282 
\ cut f r b = g"; 
283 
val gundef = recgb RS is_recfun_undef 

284 
and fisg = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal))); 

285 
by (cut_facts_tac prems 1); 

286 
by (rtac ext 1); 

4686  287 
by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg]) 1); 
923  288 
qed "is_recfun_cut"; 
289 

290 
(*** Main Existence Lemma  Basic Properties of the_recfun ***) 

291 

5316  292 
Goalw [the_recfun_def] 
1475  293 
"is_recfun r H a f ==> is_recfun r H a (the_recfun r H a)"; 
5316  294 
by (eres_inst_tac [("P", "is_recfun r H a")] selectI 1); 
923  295 
qed "is_the_recfun"; 
296 

5316  297 
Goal "[ wf(r); trans(r) ] ==> is_recfun r H a (the_recfun r H a)"; 
298 
by (wf_ind_tac "a" [] 1); 

4821  299 
by (res_inst_tac [("f","cut (%y. H (the_recfun r H y) y) r a1")] 
300 
is_the_recfun 1); 

301 
by (rewtac is_recfun_def); 

302 
by (stac cuts_eq 1); 

303 
by (Clarify_tac 1); 

304 
by (rtac (refl RSN (2,H_cong)) 1); 

305 
by (subgoal_tac 

1475  306 
"the_recfun r H y = cut(%x. H(cut(the_recfun r H y) r x) x) r y" 1); 
4821  307 
by (etac allE 2); 
308 
by (dtac impE 2); 

309 
by (atac 2); 

1475  310 
by (atac 3); 
4821  311 
by (atac 2); 
312 
by (etac ssubst 1); 

313 
by (simp_tac (HOL_ss addsimps [cuts_eq]) 1); 

314 
by (Clarify_tac 1); 

315 
by (stac cut_apply 1); 

5132  316 
by (fast_tac (claset() addDs [transD]) 1); 
4821  317 
by (rtac (refl RSN (2,H_cong)) 1); 
318 
by (fold_tac [is_recfun_def]); 

319 
by (asm_simp_tac (wf_super_ss addsimps[is_recfun_cut]) 1); 

923  320 
qed "unfold_the_recfun"; 
321 

1475  322 
val unwind1_the_recfun = rewrite_rule[is_recfun_def] unfold_the_recfun; 
923  323 

1475  324 
(*Old proof 
5316  325 
val prems = Goal 
1475  326 
"[ wf(r); trans(r) ] ==> is_recfun r H a (the_recfun r H a)"; 
327 
by (cut_facts_tac prems 1); 

328 
by (wf_ind_tac "a" prems 1); 

329 
by (res_inst_tac [("f", "cut (%y. wftrec r H y) r a1")] is_the_recfun 1); 

330 
by (rewrite_goals_tac [is_recfun_def, wftrec_def]); 

2031  331 
by (stac cuts_eq 1); 
1475  332 
(*Applying the substitution: must keep the quantified assumption!!*) 
3708  333 
by (EVERY1 [Clarify_tac, rtac H_cong1, rtac allE, atac, 
1475  334 
etac (mp RS ssubst), atac]); 
335 
by (fold_tac [is_recfun_def]); 

336 
by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1); 

337 
qed "unfold_the_recfun"; 

338 
*) 

923  339 

340 
(** Removal of the premise trans(r) **) 

1475  341 
val th = rewrite_rule[is_recfun_def] 
342 
(trans_trancl RSN (2,(wf_trancl RS unfold_the_recfun))); 

923  343 

5069  344 
Goalw [wfrec_def] 
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345 
"wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"; 
1475  346 
by (rtac H_cong 1); 
347 
by (rtac refl 2); 

348 
by (simp_tac (HOL_ss addsimps [cuts_eq]) 1); 

349 
by (rtac allI 1); 

350 
by (rtac impI 1); 

351 
by (simp_tac(HOL_ss addsimps [wfrec_def]) 1); 

352 
by (res_inst_tac [("a1","a")] (th RS ssubst) 1); 

353 
by (atac 1); 

354 
by (forward_tac[wf_trancl] 1); 

355 
by (forward_tac[r_into_trancl] 1); 

356 
by (asm_simp_tac (HOL_ss addsimps [cut_apply]) 1); 

357 
by (rtac H_cong 1); (*expose the equality of cuts*) 

358 
by (rtac refl 2); 

359 
by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1); 

3708  360 
by (Clarify_tac 1); 
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Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
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changeset

361 
by (res_inst_tac [("r","r^+")] is_recfun_equal 1); 
1475  362 
by (atac 1); 
363 
by (rtac trans_trancl 1); 

364 
by (rtac unfold_the_recfun 1); 

365 
by (atac 1); 

366 
by (rtac trans_trancl 1); 

367 
by (rtac unfold_the_recfun 1); 

368 
by (atac 1); 

369 
by (rtac trans_trancl 1); 

370 
by (rtac transD 1); 

371 
by (rtac trans_trancl 1); 

4762  372 
by (forw_inst_tac [("p","(ya,y)")] r_into_trancl 1); 
1475  373 
by (atac 1); 
374 
by (atac 1); 

4762  375 
by (forw_inst_tac [("p","(ya,y)")] r_into_trancl 1); 
1475  376 
by (atac 1); 
377 
qed "wfrec"; 

378 

379 
(*Old proof 

5069  380 
Goalw [wfrec_def] 
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More tidying and removal of "\!\!... from Goal commands
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parents:
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diff
changeset

381 
"wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"; 
923  382 
by (etac (wf_trancl RS wftrec RS ssubst) 1); 
383 
by (rtac trans_trancl 1); 

384 
by (rtac (refl RS H_cong) 1); (*expose the equality of cuts*) 

1475  385 
by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1); 
923  386 
qed "wfrec"; 
1475  387 
*) 
923  388 

1475  389 
(* 
390 
* This form avoids giant explosions in proofs. NOTE USE OF == 

391 
**) 

5316  392 
val rew::prems = goal thy 
1475  393 
"[ f==wfrec r H; wf(r) ] ==> f(a) = H (cut f r a) a"; 
923  394 
by (rewtac rew); 
395 
by (REPEAT (resolve_tac (prems@[wfrec]) 1)); 

396 
qed "def_wfrec"; 

1475  397 

3198  398 

399 
(**** TFL variants ****) 

400 

5278  401 
Goal "!R. wf R > (!P. (!x. (!y. (y,x):R > P y) > P x) > (!x. P x))"; 
3708  402 
by (Clarify_tac 1); 
3198  403 
by (res_inst_tac [("r","R"),("P","P"), ("a","x")] wf_induct 1); 
404 
by (assume_tac 1); 

405 
by (Blast_tac 1); 

406 
qed"tfl_wf_induct"; 

407 

5069  408 
Goal "!f R. (x,a):R > (cut f R a)(x) = f(x)"; 
3708  409 
by (Clarify_tac 1); 
3198  410 
by (rtac cut_apply 1); 
411 
by (assume_tac 1); 

412 
qed"tfl_cut_apply"; 

413 

5069  414 
Goal "!M R f. (f=wfrec R M) > wf R > (!x. f x = M (cut f R x) x)"; 
3708  415 
by (Clarify_tac 1); 
4153  416 
by (etac wfrec 1); 
3198  417 
qed "tfl_wfrec"; 