src/HOL/Wellfounded_Recursion.thy
 author haftmann Mon Aug 14 13:46:06 2006 +0200 (2006-08-14) changeset 20380 14f9f2a1caa6 parent 20185 183f08468e19 child 20435 d2a30fed7596 permissions -rw-r--r--
simplified code generator setup
1 (*  ID:         \$Id\$
2     Author:     Tobias Nipkow
3     Copyright   1992  University of Cambridge
4 *)
6 header {*Well-founded Recursion*}
8 theory Wellfounded_Recursion
9 imports Transitive_Closure
10 begin
12 consts
13   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => ('a * 'b) set"
15 inductive "wfrec_rel R F"
16 intros
17   wfrecI: "ALL z. (z, x) : R --> (z, g z) : wfrec_rel R F ==>
18             (x, F g x) : wfrec_rel R F"
20 constdefs
21   wf         :: "('a * 'a)set => bool"
22   "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
24   acyclic :: "('a*'a)set => bool"
25   "acyclic r == !x. (x,x) ~: r^+"
27   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
28   "cut f r x == (%y. if (y,x):r then f y else arbitrary)"
30   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
31   "adm_wf R F == ALL f g x.
32      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
34   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
35   "wfrec R F == %x. THE y. (x, y) : wfrec_rel R (%f x. F (cut f R x) x)"
37 axclass wellorder \<subseteq> linorder
38   wf: "wf {(x,y::'a::ord). x<y}"
41 lemma wfUNIVI:
42    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
43 by (unfold wf_def, blast)
45 text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
46     well-founded over their intersection, then @{term "wf r"}*}
47 lemma wfI:
48  "[| r \<subseteq> A <*> B;
49      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
50   ==>  wf r"
51 by (unfold wf_def, blast)
53 lemma wf_induct:
54     "[| wf(r);
55         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
56      |]  ==>  P(a)"
57 by (unfold wf_def, blast)
59 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
61 lemma wf_not_sym [rule_format]: "wf(r) ==> ALL x. (a,x):r --> (x,a)~:r"
62 by (erule_tac a=a in wf_induct, blast)
64 (* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)
65 lemmas wf_asym = wf_not_sym [elim_format]
67 lemma wf_not_refl [simp]: "wf(r) ==> (a,a) ~: r"
68 by (blast elim: wf_asym)
70 (* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)
71 lemmas wf_irrefl = wf_not_refl [elim_format]
73 text{*transitive closure of a well-founded relation is well-founded! *}
74 lemma wf_trancl: "wf(r) ==> wf(r^+)"
75 apply (subst wf_def, clarify)
76 apply (rule allE, assumption)
77   --{*Retains the universal formula for later use!*}
78 apply (erule mp)
79 apply (erule_tac a = x in wf_induct)
80 apply (blast elim: tranclE)
81 done
83 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
84 apply (subst trancl_converse [symmetric])
85 apply (erule wf_trancl)
86 done
89 subsubsection{*Other simple well-foundedness results*}
92 text{*Minimal-element characterization of well-foundedness*}
93 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))"
94 proof (intro iffI strip)
95   fix Q::"'a set" and x
96   assume "wf r" and "x \<in> Q"
97   thus "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
98     by (unfold wf_def,
99         blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"])
100 next
101   assume "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
102   thus "wf r" by (unfold wf_def, force)
103 qed
105 text{*Well-foundedness of subsets*}
106 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
107 apply (simp (no_asm_use) add: wf_eq_minimal)
108 apply fast
109 done
111 text{*Well-foundedness of the empty relation*}
112 lemma wf_empty [iff]: "wf({})"
113 by (simp add: wf_def)
115 lemma wf_Int1: "wf r ==> wf (r Int r')"
116 by (erule wf_subset, rule Int_lower1)
118 lemma wf_Int2: "wf r ==> wf (r' Int r)"
119 by (erule wf_subset, rule Int_lower2)
121 text{*Well-foundedness of insert*}
122 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
123 apply (rule iffI)
124  apply (blast elim: wf_trancl [THEN wf_irrefl]
125               intro: rtrancl_into_trancl1 wf_subset
126                      rtrancl_mono [THEN  rev_subsetD])
127 apply (simp add: wf_eq_minimal, safe)
128 apply (rule allE, assumption, erule impE, blast)
129 apply (erule bexE)
130 apply (rename_tac "a", case_tac "a = x")
131  prefer 2
132 apply blast
133 apply (case_tac "y:Q")
134  prefer 2 apply blast
135 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
136  apply assumption
137 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
138   --{*essential for speed*}
139 txt{*Blast with new substOccur fails*}
140 apply (fast intro: converse_rtrancl_into_rtrancl)
141 done
143 text{*Well-foundedness of image*}
144 lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
145 apply (simp only: wf_eq_minimal, clarify)
146 apply (case_tac "EX p. f p : Q")
147 apply (erule_tac x = "{p. f p : Q}" in allE)
148 apply (fast dest: inj_onD, blast)
149 done
152 subsubsection{*Well-Foundedness Results for Unions*}
154 text{*Well-foundedness of indexed union with disjoint domains and ranges*}
156 lemma wf_UN: "[| ALL i:I. wf(r i);
157          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
158       |] ==> wf(UN i:I. r i)"
159 apply (simp only: wf_eq_minimal, clarify)
160 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
161  prefer 2
162  apply force
163 apply clarify
164 apply (drule bspec, assumption)
165 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
166 apply (blast elim!: allE)
167 done
169 lemma wf_Union:
170  "[| ALL r:R. wf r;
171      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}
172   |] ==> wf(Union R)"
173 apply (simp add: Union_def)
174 apply (blast intro: wf_UN)
175 done
177 (*Intuition: we find an (R u S)-min element of a nonempty subset A
178              by case distinction.
179   1. There is a step a -R-> b with a,b : A.
180      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
181      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
182      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
183      have an S-successor and is thus S-min in A as well.
184   2. There is no such step.
185      Pick an S-min element of A. In this case it must be an R-min
186      element of A as well.
188 *)
189 lemma wf_Un:
190      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
191 apply (simp only: wf_eq_minimal, clarify)
192 apply (rename_tac A a)
193 apply (case_tac "EX a:A. EX b:A. (b,a) : r")
194  prefer 2
195  apply simp
196  apply (drule_tac x=A in spec)+
197  apply blast
198 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r) }" in allE)+
199 apply (blast elim!: allE)
200 done
202 subsubsection {*acyclic*}
204 lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
205 by (simp add: acyclic_def)
207 lemma wf_acyclic: "wf r ==> acyclic r"
208 apply (simp add: acyclic_def)
209 apply (blast elim: wf_trancl [THEN wf_irrefl])
210 done
212 lemma acyclic_insert [iff]:
213      "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
214 apply (simp add: acyclic_def trancl_insert)
215 apply (blast intro: rtrancl_trans)
216 done
218 lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
219 by (simp add: acyclic_def trancl_converse)
221 lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
222 apply (simp add: acyclic_def antisym_def)
223 apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
224 done
226 (* Other direction:
227 acyclic = no loops
228 antisym = only self loops
229 Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
230 ==> antisym( r^* ) = acyclic(r - Id)";
231 *)
233 lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
234 apply (simp add: acyclic_def)
235 apply (blast intro: trancl_mono)
236 done
239 subsection{*Well-Founded Recursion*}
241 text{*cut*}
243 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
244 by (simp add: expand_fun_eq cut_def)
246 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
247 by (simp add: cut_def)
249 text{*Inductive characterization of wfrec combinator; for details see:
250 John Harrison, "Inductive definitions: automation and application"*}
252 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. (x, y) : wfrec_rel R F"
254 apply (erule_tac a=x in wf_induct)
255 apply (rule ex1I)
256 apply (rule_tac g = "%x. THE y. (x, y) : wfrec_rel R F" in wfrec_rel.wfrecI)
257 apply (fast dest!: theI')
258 apply (erule wfrec_rel.cases, simp)
259 apply (erule allE, erule allE, erule allE, erule mp)
260 apply (fast intro: the_equality [symmetric])
261 done
263 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
265 apply (intro strip)
266 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
267 apply (rule refl)
268 done
270 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
271 apply (simp add: wfrec_def)
272 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
273 apply (rule wfrec_rel.wfrecI)
274 apply (intro strip)
275 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
276 done
279 text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}
280 lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"
281 apply auto
282 apply (blast intro: wfrec)
283 done
286 subsection {* Code generator setup *}
288 consts_code
289   "wfrec"   ("\<module>wfrec?")
290 attach {*
291 fun wfrec f x = f (wfrec f) x;
292 *}
294 setup {*
295   CodegenPackage.add_appconst ("wfrec", CodegenPackage.appgen_wfrec)
296 *}
298 code_constapp
299   wfrec
300     ml (target_atom "(let fun wfrec f x = f (wfrec f) x in wfrec end)")
301     haskell (target_atom "(wfrec where wfrec f x = f (wfrec f) x)")
303 subsection{*Variants for TFL: the Recdef Package*}
305 lemma tfl_wf_induct: "ALL R. wf R -->
306        (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
307 apply clarify
308 apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
309 done
311 lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
312 apply clarify
313 apply (rule cut_apply, assumption)
314 done
316 lemma tfl_wfrec:
317      "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
318 apply clarify
319 apply (erule wfrec)
320 done
322 subsection {*LEAST and wellorderings*}
324 text{* See also @{text wf_linord_ex_has_least} and its consequences in
325  @{text Wellfounded_Relations.ML}*}
327 lemma wellorder_Least_lemma [rule_format]:
328      "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k"
329 apply (rule_tac a = k in wf [THEN wf_induct])
330 apply (rule impI)
331 apply (rule classical)
332 apply (rule_tac s = x in Least_equality [THEN ssubst], auto)
333 apply (auto simp add: linorder_not_less [symmetric])
334 done
336 lemmas LeastI   = wellorder_Least_lemma [THEN conjunct1, standard]
337 lemmas Least_le = wellorder_Least_lemma [THEN conjunct2, standard]
339 -- "The following 3 lemmas are due to Brian Huffman"
340 lemma LeastI_ex: "EX x::'a::wellorder. P x ==> P (Least P)"
341 apply (erule exE)
342 apply (erule LeastI)
343 done
345 lemma LeastI2:
346   "[| P (a::'a::wellorder); !!x. P x ==> Q x |] ==> Q (Least P)"
347 by (blast intro: LeastI)
349 lemma LeastI2_ex:
350   "[| EX a::'a::wellorder. P a; !!x. P x ==> Q x |] ==> Q (Least P)"
351 by (blast intro: LeastI_ex)
353 lemma not_less_Least: "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)"
354 apply (simp (no_asm_use) add: linorder_not_le [symmetric])
355 apply (erule contrapos_nn)
356 apply (erule Least_le)
357 done
359 ML
360 {*
361 val wf_def = thm "wf_def";
362 val wfUNIVI = thm "wfUNIVI";
363 val wfI = thm "wfI";
364 val wf_induct = thm "wf_induct";
365 val wf_not_sym = thm "wf_not_sym";
366 val wf_asym = thm "wf_asym";
367 val wf_not_refl = thm "wf_not_refl";
368 val wf_irrefl = thm "wf_irrefl";
369 val wf_trancl = thm "wf_trancl";
370 val wf_converse_trancl = thm "wf_converse_trancl";
371 val wf_eq_minimal = thm "wf_eq_minimal";
372 val wf_subset = thm "wf_subset";
373 val wf_empty = thm "wf_empty";
374 val wf_insert = thm "wf_insert";
375 val wf_UN = thm "wf_UN";
376 val wf_Union = thm "wf_Union";
377 val wf_Un = thm "wf_Un";
378 val wf_prod_fun_image = thm "wf_prod_fun_image";
379 val acyclicI = thm "acyclicI";
380 val wf_acyclic = thm "wf_acyclic";
381 val acyclic_insert = thm "acyclic_insert";
382 val acyclic_converse = thm "acyclic_converse";
383 val acyclic_impl_antisym_rtrancl = thm "acyclic_impl_antisym_rtrancl";
384 val acyclic_subset = thm "acyclic_subset";
385 val cuts_eq = thm "cuts_eq";
386 val cut_apply = thm "cut_apply";
387 val wfrec_unique = thm "wfrec_unique";
388 val wfrec = thm "wfrec";
389 val def_wfrec = thm "def_wfrec";
390 val tfl_wf_induct = thm "tfl_wf_induct";
391 val tfl_cut_apply = thm "tfl_cut_apply";
392 val tfl_wfrec = thm "tfl_wfrec";
393 val LeastI = thm "LeastI";
394 val Least_le = thm "Least_le";
395 val not_less_Least = thm "not_less_Least";
396 *}
398 end