src/HOL/Wellfounded.thy
 author noschinl Mon Mar 12 15:12:22 2012 +0100 (2012-03-12) changeset 46883 eec472dae593 parent 46882 6242b4bc05bc child 47433 07f4bf913230 permissions -rw-r--r--
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
1 (*  Title:      HOL/Wellfounded.thy
2     Author:     Tobias Nipkow
3     Author:     Lawrence C Paulson
4     Author:     Konrad Slind
5     Author:     Alexander Krauss
6 *)
8 header {*Well-founded Recursion*}
10 theory Wellfounded
11 imports Transitive_Closure
12 uses ("Tools/Function/size.ML")
13 begin
15 subsection {* Basic Definitions *}
17 definition wf :: "('a * 'a) set => bool" where
18   "wf r \<longleftrightarrow> (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
20 definition wfP :: "('a => 'a => bool) => bool" where
21   "wfP r \<longleftrightarrow> wf {(x, y). r x y}"
23 lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
24   by (simp add: wfP_def)
26 lemma wfUNIVI:
27    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
28   unfolding wf_def by blast
30 lemmas wfPUNIVI = wfUNIVI [to_pred]
32 text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
33     well-founded over their intersection, then @{term "wf r"}*}
34 lemma wfI:
35  "[| r \<subseteq> A <*> B;
36      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
37   ==>  wf r"
38   unfolding wf_def by blast
40 lemma wf_induct:
41     "[| wf(r);
42         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
43      |]  ==>  P(a)"
44   unfolding wf_def by blast
46 lemmas wfP_induct = wf_induct [to_pred]
48 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
50 lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
52 lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
53   by (induct a arbitrary: x set: wf) blast
55 lemma wf_asym:
56   assumes "wf r" "(a, x) \<in> r"
57   obtains "(x, a) \<notin> r"
58   by (drule wf_not_sym[OF assms])
60 lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
61   by (blast elim: wf_asym)
63 lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
64 by (drule wf_not_refl[OF assms])
66 lemma wf_wellorderI:
67   assumes wf: "wf {(x::'a::ord, y). x < y}"
68   assumes lin: "OFCLASS('a::ord, linorder_class)"
69   shows "OFCLASS('a::ord, wellorder_class)"
70 using lin by (rule wellorder_class.intro)
71   (blast intro: class.wellorder_axioms.intro wf_induct_rule [OF wf])
73 lemma (in wellorder) wf:
74   "wf {(x, y). x < y}"
75 unfolding wf_def by (blast intro: less_induct)
78 subsection {* Basic Results *}
80 text {* Point-free characterization of well-foundedness *}
82 lemma wfE_pf:
83   assumes wf: "wf R"
84   assumes a: "A \<subseteq> R `` A"
85   shows "A = {}"
86 proof -
87   { fix x
88     from wf have "x \<notin> A"
89     proof induct
90       fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> y \<notin> A"
91       then have "x \<notin> R `` A" by blast
92       with a show "x \<notin> A" by blast
93     qed
94   } thus ?thesis by auto
95 qed
97 lemma wfI_pf:
98   assumes a: "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}"
99   shows "wf R"
100 proof (rule wfUNIVI)
101   fix P :: "'a \<Rightarrow> bool" and x
102   let ?A = "{x. \<not> P x}"
103   assume "\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
104   then have "?A \<subseteq> R `` ?A" by blast
105   with a show "P x" by blast
106 qed
108 text{*Minimal-element characterization of well-foundedness*}
110 lemma wfE_min:
111   assumes wf: "wf R" and Q: "x \<in> Q"
112   obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
113   using Q wfE_pf[OF wf, of Q] by blast
115 lemma wfI_min:
116   assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
117   shows "wf R"
118 proof (rule wfI_pf)
119   fix A assume b: "A \<subseteq> R `` A"
120   { fix x assume "x \<in> A"
121     from a[OF this] b have "False" by blast
122   }
123   thus "A = {}" by blast
124 qed
126 lemma wf_eq_minimal: "wf r = (\<forall>Q x. x\<in>Q --> (\<exists>z\<in>Q. \<forall>y. (y,z)\<in>r --> y\<notin>Q))"
127 apply auto
128 apply (erule wfE_min, assumption, blast)
129 apply (rule wfI_min, auto)
130 done
132 lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
134 text{* Well-foundedness of transitive closure *}
136 lemma wf_trancl:
137   assumes "wf r"
138   shows "wf (r^+)"
139 proof -
140   {
141     fix P and x
142     assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
143     have "P x"
144     proof (rule induct_step)
145       fix y assume "(y, x) : r^+"
146       with `wf r` show "P y"
147       proof (induct x arbitrary: y)
148         case (less x)
149         note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
150         from `(y, x) : r^+` show "P y"
151         proof cases
152           case base
153           show "P y"
154           proof (rule induct_step)
155             fix y' assume "(y', y) : r^+"
156             with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
157           qed
158         next
159           case step
160           then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
161           then show "P y" by (rule hyp [of x' y])
162         qed
163       qed
164     qed
165   } then show ?thesis unfolding wf_def by blast
166 qed
168 lemmas wfP_trancl = wf_trancl [to_pred]
170 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
171   apply (subst trancl_converse [symmetric])
172   apply (erule wf_trancl)
173   done
175 text {* Well-foundedness of subsets *}
177 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
178   apply (simp (no_asm_use) add: wf_eq_minimal)
179   apply fast
180   done
182 lemmas wfP_subset = wf_subset [to_pred]
184 text {* Well-foundedness of the empty relation *}
186 lemma wf_empty [iff]: "wf {}"
187   by (simp add: wf_def)
189 lemma wfP_empty [iff]:
190   "wfP (\<lambda>x y. False)"
191 proof -
192   have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
193   then show ?thesis by (simp add: bot_fun_def)
194 qed
196 lemma wf_Int1: "wf r ==> wf (r Int r')"
197   apply (erule wf_subset)
198   apply (rule Int_lower1)
199   done
201 lemma wf_Int2: "wf r ==> wf (r' Int r)"
202   apply (erule wf_subset)
203   apply (rule Int_lower2)
204   done
206 text {* Exponentiation *}
208 lemma wf_exp:
209   assumes "wf (R ^^ n)"
210   shows "wf R"
211 proof (rule wfI_pf)
212   fix A assume "A \<subseteq> R `` A"
213   then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
214   with `wf (R ^^ n)`
215   show "A = {}" by (rule wfE_pf)
216 qed
218 text {* Well-foundedness of insert *}
220 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
221 apply (rule iffI)
222  apply (blast elim: wf_trancl [THEN wf_irrefl]
223               intro: rtrancl_into_trancl1 wf_subset
224                      rtrancl_mono [THEN  rev_subsetD])
225 apply (simp add: wf_eq_minimal, safe)
226 apply (rule allE, assumption, erule impE, blast)
227 apply (erule bexE)
228 apply (rename_tac "a", case_tac "a = x")
229  prefer 2
230 apply blast
231 apply (case_tac "y:Q")
232  prefer 2 apply blast
233 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
234  apply assumption
235 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
236   --{*essential for speed*}
237 txt{*Blast with new substOccur fails*}
238 apply (fast intro: converse_rtrancl_into_rtrancl)
239 done
241 text{*Well-foundedness of image*}
243 lemma wf_map_pair_image: "[| wf r; inj f |] ==> wf(map_pair f f ` r)"
244 apply (simp only: wf_eq_minimal, clarify)
245 apply (case_tac "EX p. f p : Q")
246 apply (erule_tac x = "{p. f p : Q}" in allE)
247 apply (fast dest: inj_onD, blast)
248 done
251 subsection {* Well-Foundedness Results for Unions *}
253 lemma wf_union_compatible:
254   assumes "wf R" "wf S"
255   assumes "R O S \<subseteq> R"
256   shows "wf (R \<union> S)"
257 proof (rule wfI_min)
258   fix x :: 'a and Q
259   let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
260   assume "x \<in> Q"
261   obtain a where "a \<in> ?Q'"
262     by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
263   with `wf S`
264   obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
265   {
266     fix y assume "(y, z) \<in> S"
267     then have "y \<notin> ?Q'" by (rule zmin)
269     have "y \<notin> Q"
270     proof
271       assume "y \<in> Q"
272       with `y \<notin> ?Q'`
273       obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
274       from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> R O S" by (rule rel_compI)
275       with `R O S \<subseteq> R` have "(w, z) \<in> R" ..
276       with `z \<in> ?Q'` have "w \<notin> Q" by blast
277       with `w \<in> Q` show False by contradiction
278     qed
279   }
280   with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
281 qed
284 text {* Well-foundedness of indexed union with disjoint domains and ranges *}
286 lemma wf_UN: "[| ALL i:I. wf(r i);
287          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
288       |] ==> wf(UN i:I. r i)"
289 apply (simp only: wf_eq_minimal, clarify)
290 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
291  prefer 2
292  apply force
293 apply clarify
294 apply (drule bspec, assumption)
295 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
296 apply (blast elim!: allE)
297 done
299 lemma wfP_SUP:
300   "\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (DomainP (r i)) (RangeP (r j)) = bot \<Longrightarrow> wfP (SUPR UNIV r)"
301   apply (rule wf_UN[to_pred])
302   apply simp_all
303   done
305 lemma wf_Union:
306  "[| ALL r:R. wf r;
307      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}
308   |] ==> wf(Union R)"
309   using wf_UN[of R "\<lambda>i. i"] by (simp add: SUP_def)
311 (*Intuition: we find an (R u S)-min element of a nonempty subset A
312              by case distinction.
313   1. There is a step a -R-> b with a,b : A.
314      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
315      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
316      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
317      have an S-successor and is thus S-min in A as well.
318   2. There is no such step.
319      Pick an S-min element of A. In this case it must be an R-min
320      element of A as well.
322 *)
323 lemma wf_Un:
324      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
325   using wf_union_compatible[of s r]
326   by (auto simp: Un_ac)
328 lemma wf_union_merge:
329   "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)" (is "wf ?A = wf ?B")
330 proof
331   assume "wf ?A"
332   with wf_trancl have wfT: "wf (?A^+)" .
333   moreover have "?B \<subseteq> ?A^+"
334     by (subst trancl_unfold, subst trancl_unfold) blast
335   ultimately show "wf ?B" by (rule wf_subset)
336 next
337   assume "wf ?B"
339   show "wf ?A"
340   proof (rule wfI_min)
341     fix Q :: "'a set" and x
342     assume "x \<in> Q"
344     with `wf ?B`
345     obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q"
346       by (erule wfE_min)
347     then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
348       and A2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
349       and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
350       by auto
352     show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
353     proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
354       case True
355       with `z \<in> Q` A3 show ?thesis by blast
356     next
357       case False
358       then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
360       have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
361       proof (intro allI impI)
362         fix y assume "(y, z') \<in> ?A"
363         then show "y \<notin> Q"
364         proof
365           assume "(y, z') \<in> R"
366           then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
367           with A1 show "y \<notin> Q" .
368         next
369           assume "(y, z') \<in> S"
370           then have "(y, z) \<in> S O R" using  `(z', z) \<in> R` ..
371           with A2 show "y \<notin> Q" .
372         qed
373       qed
374       with `z' \<in> Q` show ?thesis ..
375     qed
376   qed
377 qed
379 lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}
380   by (rule wf_union_merge [where S = "{}", simplified])
383 subsection {* Acyclic relations *}
385 lemma wf_acyclic: "wf r ==> acyclic r"
386 apply (simp add: acyclic_def)
387 apply (blast elim: wf_trancl [THEN wf_irrefl])
388 done
390 lemmas wfP_acyclicP = wf_acyclic [to_pred]
392 text{* Wellfoundedness of finite acyclic relations*}
394 lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
395 apply (erule finite_induct, blast)
396 apply (simp (no_asm_simp) only: split_tupled_all)
397 apply simp
398 done
400 lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
401 apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
402 apply (erule acyclic_converse [THEN iffD2])
403 done
405 lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
406 by (blast intro: finite_acyclic_wf wf_acyclic)
409 subsection {* @{typ nat} is well-founded *}
411 lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
412 proof (rule ext, rule ext, rule iffI)
413   fix n m :: nat
414   assume "m < n"
415   then show "(\<lambda>m n. n = Suc m)^++ m n"
416   proof (induct n)
417     case 0 then show ?case by auto
418   next
419     case (Suc n) then show ?case
420       by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
421   qed
422 next
423   fix n m :: nat
424   assume "(\<lambda>m n. n = Suc m)^++ m n"
425   then show "m < n"
426     by (induct n)
427       (simp_all add: less_Suc_eq_le reflexive le_less)
428 qed
430 definition
431   pred_nat :: "(nat * nat) set" where
432   "pred_nat = {(m, n). n = Suc m}"
434 definition
435   less_than :: "(nat * nat) set" where
436   "less_than = pred_nat^+"
438 lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
439   unfolding less_nat_rel pred_nat_def trancl_def by simp
441 lemma pred_nat_trancl_eq_le:
442   "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
443   unfolding less_eq rtrancl_eq_or_trancl by auto
445 lemma wf_pred_nat: "wf pred_nat"
446   apply (unfold wf_def pred_nat_def, clarify)
447   apply (induct_tac x, blast+)
448   done
450 lemma wf_less_than [iff]: "wf less_than"
451   by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
453 lemma trans_less_than [iff]: "trans less_than"
454   by (simp add: less_than_def)
456 lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
457   by (simp add: less_than_def less_eq)
459 lemma wf_less: "wf {(x, y::nat). x < y}"
460   using wf_less_than by (simp add: less_than_def less_eq [symmetric])
463 subsection {* Accessible Part *}
465 text {*
466  Inductive definition of the accessible part @{term "acc r"} of a
468 *}
470 inductive_set
471   acc :: "('a * 'a) set => 'a set"
472   for r :: "('a * 'a) set"
473   where
474     accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
476 abbreviation
477   termip :: "('a => 'a => bool) => 'a => bool" where
478   "termip r \<equiv> accp (r\<inverse>\<inverse>)"
480 abbreviation
481   termi :: "('a * 'a) set => 'a set" where
482   "termi r \<equiv> acc (r\<inverse>)"
484 lemmas accpI = accp.accI
486 text {* Induction rules *}
488 theorem accp_induct:
489   assumes major: "accp r a"
490   assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
491   shows "P a"
492   apply (rule major [THEN accp.induct])
493   apply (rule hyp)
494    apply (rule accp.accI)
495    apply fast
496   apply fast
497   done
499 theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
501 theorem accp_downward: "accp r b ==> r a b ==> accp r a"
502   apply (erule accp.cases)
503   apply fast
504   done
506 lemma not_accp_down:
507   assumes na: "\<not> accp R x"
508   obtains z where "R z x" and "\<not> accp R z"
509 proof -
510   assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
512   show thesis
513   proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
514     case True
515     hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
516     hence "accp R x"
517       by (rule accp.accI)
518     with na show thesis ..
519   next
520     case False then obtain z where "R z x" and "\<not> accp R z"
521       by auto
522     with a show thesis .
523   qed
524 qed
526 lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
527   apply (erule rtranclp_induct)
528    apply blast
529   apply (blast dest: accp_downward)
530   done
532 theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
533   apply (blast dest: accp_downwards_aux)
534   done
536 theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
537   apply (rule wfPUNIVI)
538   apply (rule_tac P=P in accp_induct)
539    apply blast
540   apply blast
541   done
543 theorem accp_wfPD: "wfP r ==> accp r x"
544   apply (erule wfP_induct_rule)
545   apply (rule accp.accI)
546   apply blast
547   done
549 theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
550   apply (blast intro: accp_wfPI dest: accp_wfPD)
551   done
554 text {* Smaller relations have bigger accessible parts: *}
556 lemma accp_subset:
557   assumes sub: "R1 \<le> R2"
558   shows "accp R2 \<le> accp R1"
559 proof (rule predicate1I)
560   fix x assume "accp R2 x"
561   then show "accp R1 x"
562   proof (induct x)
563     fix x
564     assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
565     with sub show "accp R1 x"
566       by (blast intro: accp.accI)
567   qed
568 qed
571 text {* This is a generalized induction theorem that works on
572   subsets of the accessible part. *}
574 lemma accp_subset_induct:
575   assumes subset: "D \<le> accp R"
576     and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
577     and "D x"
578     and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
579   shows "P x"
580 proof -
581   from subset and `D x`
582   have "accp R x" ..
583   then show "P x" using `D x`
584   proof (induct x)
585     fix x
586     assume "D x"
587       and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
588     with dcl and istep show "P x" by blast
589   qed
590 qed
593 text {* Set versions of the above theorems *}
595 lemmas acc_induct = accp_induct [to_set]
597 lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
599 lemmas acc_downward = accp_downward [to_set]
601 lemmas not_acc_down = not_accp_down [to_set]
603 lemmas acc_downwards_aux = accp_downwards_aux [to_set]
605 lemmas acc_downwards = accp_downwards [to_set]
607 lemmas acc_wfI = accp_wfPI [to_set]
609 lemmas acc_wfD = accp_wfPD [to_set]
611 lemmas wf_acc_iff = wfP_accp_iff [to_set]
613 lemmas acc_subset = accp_subset [to_set]
615 lemmas acc_subset_induct = accp_subset_induct [to_set]
618 subsection {* Tools for building wellfounded relations *}
620 text {* Inverse Image *}
622 lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
623 apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
624 apply clarify
625 apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
626 prefer 2 apply (blast del: allE)
627 apply (erule allE)
628 apply (erule (1) notE impE)
629 apply blast
630 done
632 text {* Measure functions into @{typ nat} *}
634 definition measure :: "('a => nat) => ('a * 'a)set"
635 where "measure = inv_image less_than"
637 lemma in_measure[simp, code_unfold]: "((x,y) : measure f) = (f x < f y)"
638   by (simp add:measure_def)
640 lemma wf_measure [iff]: "wf (measure f)"
641 apply (unfold measure_def)
642 apply (rule wf_less_than [THEN wf_inv_image])
643 done
645 lemma wf_if_measure: fixes f :: "'a \<Rightarrow> nat"
646 shows "(!!x. P x \<Longrightarrow> f(g x) < f x) \<Longrightarrow> wf {(y,x). P x \<and> y = g x}"
647 apply(insert wf_measure[of f])
648 apply(simp only: measure_def inv_image_def less_than_def less_eq)
649 apply(erule wf_subset)
650 apply auto
651 done
654 text{* Lexicographic combinations *}
656 definition lex_prod :: "('a \<times>'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set" (infixr "<*lex*>" 80) where
657   "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
659 lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
660 apply (unfold wf_def lex_prod_def)
661 apply (rule allI, rule impI)
662 apply (simp (no_asm_use) only: split_paired_All)
663 apply (drule spec, erule mp)
664 apply (rule allI, rule impI)
665 apply (drule spec, erule mp, blast)
666 done
668 lemma in_lex_prod[simp]:
669   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
670   by (auto simp:lex_prod_def)
672 text{* @{term "op <*lex*>"} preserves transitivity *}
674 lemma trans_lex_prod [intro!]:
675     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
676 by (unfold trans_def lex_prod_def, blast)
678 text {* lexicographic combinations with measure functions *}
680 definition
681   mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
682 where
683   "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
685 lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
686 unfolding mlex_prod_def
687 by auto
689 lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
690 unfolding mlex_prod_def by simp
692 lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
693 unfolding mlex_prod_def by auto
695 text {* proper subset relation on finite sets *}
697 definition finite_psubset  :: "('a set * 'a set) set"
698 where "finite_psubset = {(A,B). A < B & finite B}"
700 lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
701 apply (unfold finite_psubset_def)
702 apply (rule wf_measure [THEN wf_subset])
703 apply (simp add: measure_def inv_image_def less_than_def less_eq)
704 apply (fast elim!: psubset_card_mono)
705 done
707 lemma trans_finite_psubset: "trans finite_psubset"
708 by (simp add: finite_psubset_def less_le trans_def, blast)
710 lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
711 unfolding finite_psubset_def by auto
713 text {* max- and min-extension of order to finite sets *}
715 inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
716 for R :: "('a \<times> 'a) set"
717 where
718   max_extI[intro]: "finite X \<Longrightarrow> finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> R) \<Longrightarrow> (X, Y) \<in> max_ext R"
720 lemma max_ext_wf:
721   assumes wf: "wf r"
722   shows "wf (max_ext r)"
723 proof (rule acc_wfI, intro allI)
724   fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
725   proof cases
726     assume "finite M"
727     thus ?thesis
728     proof (induct M)
729       show "{} \<in> ?W"
730         by (rule accI) (auto elim: max_ext.cases)
731     next
732       fix M a assume "M \<in> ?W" "finite M"
733       with wf show "insert a M \<in> ?W"
734       proof (induct arbitrary: M)
735         fix M a
736         assume "M \<in> ?W"  and  [intro]: "finite M"
737         assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
738         {
739           fix N M :: "'a set"
740           assume "finite N" "finite M"
741           then
742           have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
743             by (induct N arbitrary: M) (auto simp: hyp)
744         }
745         note add_less = this
747         show "insert a M \<in> ?W"
748         proof (rule accI)
749           fix N assume Nless: "(N, insert a M) \<in> max_ext r"
750           hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
751             by (auto elim!: max_ext.cases)
753           let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
754           let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
755           have N: "?N1 \<union> ?N2 = N" by (rule set_eqI) auto
756           from Nless have "finite N" by (auto elim: max_ext.cases)
757           then have finites: "finite ?N1" "finite ?N2" by auto
759           have "?N2 \<in> ?W"
760           proof cases
761             assume [simp]: "M = {}"
762             have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
764             from asm1 have "?N2 = {}" by auto
765             with Mw show "?N2 \<in> ?W" by (simp only:)
766           next
767             assume "M \<noteq> {}"
768             have N2: "(?N2, M) \<in> max_ext r"
769               by (rule max_extI[OF _ _ `M \<noteq> {}`]) (insert asm1, auto intro: finites)
771             with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
772           qed
773           with finites have "?N1 \<union> ?N2 \<in> ?W"
774             by (rule add_less) simp
775           then show "N \<in> ?W" by (simp only: N)
776         qed
777       qed
778     qed
779   next
780     assume [simp]: "\<not> finite M"
781     show ?thesis
782       by (rule accI) (auto elim: max_ext.cases)
783   qed
784 qed
787  "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
788   (A \<union> C, B \<union> D) \<in> max_ext R"
789 by (force elim!: max_ext.cases)
792 definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"  where
793   "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
795 lemma min_ext_wf:
796   assumes "wf r"
797   shows "wf (min_ext r)"
798 proof (rule wfI_min)
799   fix Q :: "'a set set"
800   fix x
801   assume nonempty: "x \<in> Q"
802   show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
803   proof cases
804     assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
805   next
806     assume "Q \<noteq> {{}}"
807     with nonempty
808     obtain e x where "x \<in> Q" "e \<in> x" by force
809     then have eU: "e \<in> \<Union>Q" by auto
810     with `wf r`
811     obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q"
812       by (erule wfE_min)
813     from z obtain m where "m \<in> Q" "z \<in> m" by auto
814     from `m \<in> Q`
815     show ?thesis
816     proof (rule, intro bexI allI impI)
817       fix n
818       assume smaller: "(n, m) \<in> min_ext r"
819       with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
820       then show "n \<notin> Q" using z(2) by auto
821     qed
822   qed
823 qed
825 text{* Bounded increase must terminate: *}
827 lemma wf_bounded_measure:
828 fixes ub :: "'a \<Rightarrow> nat" and f :: "'a \<Rightarrow> nat"
829 assumes "!!a b. (b,a) : r \<Longrightarrow> ub b \<le> ub a & ub a \<ge> f b & f b > f a"
830 shows "wf r"
831 apply(rule wf_subset[OF wf_measure[of "%a. ub a - f a"]])
832 apply (auto dest: assms)
833 done
835 lemma wf_bounded_set:
836 fixes ub :: "'a \<Rightarrow> 'b set" and f :: "'a \<Rightarrow> 'b set"
837 assumes "!!a b. (b,a) : r \<Longrightarrow>
838   finite(ub a) & ub b \<subseteq> ub a & ub a \<supseteq> f b & f b \<supset> f a"
839 shows "wf r"
840 apply(rule wf_bounded_measure[of r "%a. card(ub a)" "%a. card(f a)"])
841 apply(drule assms)
842 apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])
843 done
846 subsection {* size of a datatype value *}
848 use "Tools/Function/size.ML"
850 setup Size.setup
852 lemma size_bool [code]:
853   "size (b\<Colon>bool) = 0" by (cases b) auto
855 lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
856   by (induct n) simp_all
858 declare "prod.size" [no_atp]
860 end