1 (* Title: HOLCF/UpperPD.thy
6 header {* Upper powerdomain *}
12 subsection {* Basis preorder *}
15 upper_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<sharp>" 50) where
16 "upper_le = (\<lambda>u v. \<forall>y\<in>Rep_pd_basis v. \<exists>x\<in>Rep_pd_basis u. compact_le x y)"
18 lemma upper_le_refl [simp]: "t \<le>\<sharp> t"
19 unfolding upper_le_def by (fast intro: compact_le_refl)
21 lemma upper_le_trans: "\<lbrakk>t \<le>\<sharp> u; u \<le>\<sharp> v\<rbrakk> \<Longrightarrow> t \<le>\<sharp> v"
22 unfolding upper_le_def
24 apply (drule (1) bspec, erule bexE)
25 apply (drule (1) bspec, erule bexE)
26 apply (erule rev_bexI)
27 apply (erule (1) compact_le_trans)
30 interpretation upper_le: preorder [upper_le]
31 by (rule preorder.intro, rule upper_le_refl, rule upper_le_trans)
33 lemma upper_le_minimal [simp]: "PDUnit compact_bot \<le>\<sharp> t"
34 unfolding upper_le_def Rep_PDUnit by simp
36 lemma PDUnit_upper_mono: "compact_le x y \<Longrightarrow> PDUnit x \<le>\<sharp> PDUnit y"
37 unfolding upper_le_def Rep_PDUnit by simp
39 lemma PDPlus_upper_mono: "\<lbrakk>s \<le>\<sharp> t; u \<le>\<sharp> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<sharp> PDPlus t v"
40 unfolding upper_le_def Rep_PDPlus by fast
42 lemma PDPlus_upper_less: "PDPlus t u \<le>\<sharp> t"
43 unfolding upper_le_def Rep_PDPlus by (fast intro: compact_le_refl)
45 lemma upper_le_PDUnit_PDUnit_iff [simp]:
46 "(PDUnit a \<le>\<sharp> PDUnit b) = compact_le a b"
47 unfolding upper_le_def Rep_PDUnit by fast
49 lemma upper_le_PDPlus_PDUnit_iff:
50 "(PDPlus t u \<le>\<sharp> PDUnit a) = (t \<le>\<sharp> PDUnit a \<or> u \<le>\<sharp> PDUnit a)"
51 unfolding upper_le_def Rep_PDPlus Rep_PDUnit by fast
53 lemma upper_le_PDPlus_iff: "(t \<le>\<sharp> PDPlus u v) = (t \<le>\<sharp> u \<and> t \<le>\<sharp> v)"
54 unfolding upper_le_def Rep_PDPlus by fast
56 lemma upper_le_induct [induct set: upper_le]:
57 assumes le: "t \<le>\<sharp> u"
58 assumes 1: "\<And>a b. compact_le a b \<Longrightarrow> P (PDUnit a) (PDUnit b)"
59 assumes 2: "\<And>t u a. P t (PDUnit a) \<Longrightarrow> P (PDPlus t u) (PDUnit a)"
60 assumes 3: "\<And>t u v. \<lbrakk>P t u; P t v\<rbrakk> \<Longrightarrow> P t (PDPlus u v)"
62 using le apply (induct u arbitrary: t rule: pd_basis_induct)
64 apply (induct_tac t rule: pd_basis_induct)
66 apply (simp add: upper_le_PDPlus_PDUnit_iff)
68 apply (subst PDPlus_commute)
70 apply (simp add: upper_le_PDPlus_iff 3)
73 lemma approx_pd_upper_mono1:
74 "i \<le> j \<Longrightarrow> approx_pd i t \<le>\<sharp> approx_pd j t"
75 apply (induct t rule: pd_basis_induct)
76 apply (simp add: compact_approx_mono1)
77 apply (simp add: PDPlus_upper_mono)
80 lemma approx_pd_upper_le: "approx_pd i t \<le>\<sharp> t"
81 apply (induct t rule: pd_basis_induct)
82 apply (simp add: compact_approx_le)
83 apply (simp add: PDPlus_upper_mono)
86 lemma approx_pd_upper_mono:
87 "t \<le>\<sharp> u \<Longrightarrow> approx_pd n t \<le>\<sharp> approx_pd n u"
88 apply (erule upper_le_induct)
89 apply (simp add: compact_approx_mono)
90 apply (simp add: upper_le_PDPlus_PDUnit_iff)
91 apply (simp add: upper_le_PDPlus_iff)
95 subsection {* Type definition *}
97 cpodef (open) 'a upper_pd =
98 "{S::'a::bifinite pd_basis set. upper_le.ideal S}"
99 apply (simp add: upper_le.adm_ideal)
100 apply (fast intro: upper_le.ideal_principal)
103 lemma ideal_Rep_upper_pd: "upper_le.ideal (Rep_upper_pd x)"
104 by (rule Rep_upper_pd [simplified])
107 upper_principal :: "'a pd_basis \<Rightarrow> 'a upper_pd" where
108 "upper_principal t = Abs_upper_pd {u. u \<le>\<sharp> t}"
110 lemma Rep_upper_principal:
111 "Rep_upper_pd (upper_principal t) = {u. u \<le>\<sharp> t}"
112 unfolding upper_principal_def
113 apply (rule Abs_upper_pd_inverse [simplified])
114 apply (rule upper_le.ideal_principal)
117 interpretation upper_pd:
118 bifinite_basis [upper_le upper_principal Rep_upper_pd approx_pd]
120 apply (rule ideal_Rep_upper_pd)
121 apply (rule cont_Rep_upper_pd)
122 apply (rule Rep_upper_principal)
123 apply (simp only: less_upper_pd_def less_set_def)
124 apply (rule approx_pd_upper_le)
125 apply (rule approx_pd_idem)
126 apply (erule approx_pd_upper_mono)
127 apply (rule approx_pd_upper_mono1, simp)
128 apply (rule finite_range_approx_pd)
129 apply (rule ex_approx_pd_eq)
132 lemma upper_principal_less_iff [simp]:
133 "(upper_principal t \<sqsubseteq> upper_principal u) = (t \<le>\<sharp> u)"
134 unfolding less_upper_pd_def Rep_upper_principal less_set_def
135 by (fast intro: upper_le_refl elim: upper_le_trans)
137 lemma upper_principal_mono:
138 "t \<le>\<sharp> u \<Longrightarrow> upper_principal t \<sqsubseteq> upper_principal u"
139 by (rule upper_pd.principal_mono)
141 lemma compact_upper_principal: "compact (upper_principal t)"
142 by (rule upper_pd.compact_principal)
144 lemma upper_pd_minimal: "upper_principal (PDUnit compact_bot) \<sqsubseteq> ys"
145 by (induct ys rule: upper_pd.principal_induct, simp, simp)
147 instance upper_pd :: (bifinite) pcpo
148 by (intro_classes, fast intro: upper_pd_minimal)
150 lemma inst_upper_pd_pcpo: "\<bottom> = upper_principal (PDUnit compact_bot)"
151 by (rule upper_pd_minimal [THEN UU_I, symmetric])
154 subsection {* Approximation *}
156 instance upper_pd :: (bifinite) approx ..
160 "approx \<equiv> (\<lambda>n. upper_pd.basis_fun (\<lambda>t. upper_principal (approx_pd n t)))"
162 lemma approx_upper_principal [simp]:
163 "approx n\<cdot>(upper_principal t) = upper_principal (approx_pd n t)"
164 unfolding approx_upper_pd_def
165 apply (rule upper_pd.basis_fun_principal)
166 apply (erule upper_principal_mono [OF approx_pd_upper_mono])
169 lemma chain_approx_upper_pd:
170 "chain (approx :: nat \<Rightarrow> 'a upper_pd \<rightarrow> 'a upper_pd)"
171 unfolding approx_upper_pd_def
172 by (rule upper_pd.chain_basis_fun_take)
174 lemma lub_approx_upper_pd:
175 "(\<Squnion>i. approx i\<cdot>xs) = (xs::'a upper_pd)"
176 unfolding approx_upper_pd_def
177 by (rule upper_pd.lub_basis_fun_take)
179 lemma approx_upper_pd_idem:
180 "approx n\<cdot>(approx n\<cdot>xs) = approx n\<cdot>(xs::'a upper_pd)"
181 apply (induct xs rule: upper_pd.principal_induct, simp)
182 apply (simp add: approx_pd_idem)
185 lemma approx_eq_upper_principal:
186 "\<exists>t\<in>Rep_upper_pd xs. approx n\<cdot>xs = upper_principal (approx_pd n t)"
187 unfolding approx_upper_pd_def
188 by (rule upper_pd.basis_fun_take_eq_principal)
190 lemma finite_fixes_approx_upper_pd:
191 "finite {xs::'a upper_pd. approx n\<cdot>xs = xs}"
192 unfolding approx_upper_pd_def
193 by (rule upper_pd.finite_fixes_basis_fun_take)
195 instance upper_pd :: (bifinite) bifinite
197 apply (simp add: chain_approx_upper_pd)
198 apply (rule lub_approx_upper_pd)
199 apply (rule approx_upper_pd_idem)
200 apply (rule finite_fixes_approx_upper_pd)
203 lemma compact_imp_upper_principal:
204 "compact xs \<Longrightarrow> \<exists>t. xs = upper_principal t"
205 apply (drule bifinite_compact_eq_approx)
208 apply (cut_tac n=i and xs=xs in approx_eq_upper_principal)
212 lemma upper_principal_induct:
213 "\<lbrakk>adm P; \<And>t. P (upper_principal t)\<rbrakk> \<Longrightarrow> P xs"
214 apply (erule approx_induct, rename_tac xs)
215 apply (cut_tac n=n and xs=xs in approx_eq_upper_principal)
216 apply (clarify, simp)
219 lemma upper_principal_induct2:
220 "\<lbrakk>\<And>ys. adm (\<lambda>xs. P xs ys); \<And>xs. adm (\<lambda>ys. P xs ys);
221 \<And>t u. P (upper_principal t) (upper_principal u)\<rbrakk> \<Longrightarrow> P xs ys"
222 apply (rule_tac x=ys in spec)
223 apply (rule_tac xs=xs in upper_principal_induct, simp)
224 apply (rule allI, rename_tac ys)
225 apply (rule_tac xs=ys in upper_principal_induct, simp)
230 subsection {* Monadic unit *}
233 upper_unit :: "'a \<rightarrow> 'a upper_pd" where
234 "upper_unit = compact_basis.basis_fun (\<lambda>a. upper_principal (PDUnit a))"
236 lemma upper_unit_Rep_compact_basis [simp]:
237 "upper_unit\<cdot>(Rep_compact_basis a) = upper_principal (PDUnit a)"
238 unfolding upper_unit_def
239 apply (rule compact_basis.basis_fun_principal)
240 apply (rule upper_principal_mono)
241 apply (erule PDUnit_upper_mono)
244 lemma upper_unit_strict [simp]: "upper_unit\<cdot>\<bottom> = \<bottom>"
245 unfolding inst_upper_pd_pcpo Rep_compact_bot [symmetric] by simp
247 lemma approx_upper_unit [simp]:
248 "approx n\<cdot>(upper_unit\<cdot>x) = upper_unit\<cdot>(approx n\<cdot>x)"
249 apply (induct x rule: compact_basis_induct, simp)
250 apply (simp add: approx_Rep_compact_basis)
253 lemma upper_unit_less_iff [simp]:
254 "(upper_unit\<cdot>x \<sqsubseteq> upper_unit\<cdot>y) = (x \<sqsubseteq> y)"
256 apply (rule bifinite_less_ext)
257 apply (drule_tac f="approx i" in monofun_cfun_arg, simp)
258 apply (cut_tac x="approx i\<cdot>x" in compact_imp_Rep_compact_basis, simp)
259 apply (cut_tac x="approx i\<cdot>y" in compact_imp_Rep_compact_basis, simp)
260 apply (clarify, simp add: compact_le_def)
261 apply (erule monofun_cfun_arg)
264 lemma upper_unit_eq_iff [simp]:
265 "(upper_unit\<cdot>x = upper_unit\<cdot>y) = (x = y)"
266 unfolding po_eq_conv by simp
268 lemma upper_unit_strict_iff [simp]: "(upper_unit\<cdot>x = \<bottom>) = (x = \<bottom>)"
269 unfolding upper_unit_strict [symmetric] by (rule upper_unit_eq_iff)
271 lemma compact_upper_unit_iff [simp]:
272 "compact (upper_unit\<cdot>x) = compact x"
273 unfolding bifinite_compact_iff by simp
276 subsection {* Monadic plus *}
279 upper_plus :: "'a upper_pd \<rightarrow> 'a upper_pd \<rightarrow> 'a upper_pd" where
280 "upper_plus = upper_pd.basis_fun (\<lambda>t. upper_pd.basis_fun (\<lambda>u.
281 upper_principal (PDPlus t u)))"
284 upper_add :: "'a upper_pd \<Rightarrow> 'a upper_pd \<Rightarrow> 'a upper_pd"
285 (infixl "+\<sharp>" 65) where
286 "xs +\<sharp> ys == upper_plus\<cdot>xs\<cdot>ys"
288 lemma upper_plus_principal [simp]:
289 "upper_plus\<cdot>(upper_principal t)\<cdot>(upper_principal u) =
290 upper_principal (PDPlus t u)"
291 unfolding upper_plus_def
292 by (simp add: upper_pd.basis_fun_principal
293 upper_pd.basis_fun_mono PDPlus_upper_mono)
295 lemma approx_upper_plus [simp]:
296 "approx n\<cdot>(upper_plus\<cdot>xs\<cdot>ys) = upper_plus\<cdot>(approx n\<cdot>xs)\<cdot>(approx n\<cdot>ys)"
297 by (induct xs ys rule: upper_principal_induct2, simp, simp, simp)
299 lemma upper_plus_commute: "upper_plus\<cdot>xs\<cdot>ys = upper_plus\<cdot>ys\<cdot>xs"
300 apply (induct xs ys rule: upper_principal_induct2, simp, simp)
301 apply (simp add: PDPlus_commute)
304 lemma upper_plus_assoc:
305 "upper_plus\<cdot>(upper_plus\<cdot>xs\<cdot>ys)\<cdot>zs = upper_plus\<cdot>xs\<cdot>(upper_plus\<cdot>ys\<cdot>zs)"
306 apply (induct xs ys arbitrary: zs rule: upper_principal_induct2, simp, simp)
307 apply (rule_tac xs=zs in upper_principal_induct, simp)
308 apply (simp add: PDPlus_assoc)
311 lemma upper_plus_absorb: "upper_plus\<cdot>xs\<cdot>xs = xs"
312 apply (induct xs rule: upper_principal_induct, simp)
313 apply (simp add: PDPlus_absorb)
316 lemma upper_plus_less1: "upper_plus\<cdot>xs\<cdot>ys \<sqsubseteq> xs"
317 apply (induct xs ys rule: upper_principal_induct2, simp, simp)
318 apply (simp add: PDPlus_upper_less)
321 lemma upper_plus_less2: "upper_plus\<cdot>xs\<cdot>ys \<sqsubseteq> ys"
322 by (subst upper_plus_commute, rule upper_plus_less1)
324 lemma upper_plus_greatest: "\<lbrakk>xs \<sqsubseteq> ys; xs \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs \<sqsubseteq> upper_plus\<cdot>ys\<cdot>zs"
325 apply (subst upper_plus_absorb [of xs, symmetric])
326 apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
329 lemma upper_less_plus_iff:
330 "(xs \<sqsubseteq> upper_plus\<cdot>ys\<cdot>zs) = (xs \<sqsubseteq> ys \<and> xs \<sqsubseteq> zs)"
332 apply (erule trans_less [OF _ upper_plus_less1])
333 apply (erule trans_less [OF _ upper_plus_less2])
334 apply (erule (1) upper_plus_greatest)
337 lemma upper_plus_strict1 [simp]: "upper_plus\<cdot>\<bottom>\<cdot>ys = \<bottom>"
338 by (rule UU_I, rule upper_plus_less1)
340 lemma upper_plus_strict2 [simp]: "upper_plus\<cdot>xs\<cdot>\<bottom> = \<bottom>"
341 by (rule UU_I, rule upper_plus_less2)
343 lemma upper_plus_less_unit_iff:
344 "(upper_plus\<cdot>xs\<cdot>ys \<sqsubseteq> upper_unit\<cdot>z) =
345 (xs \<sqsubseteq> upper_unit\<cdot>z \<or> ys \<sqsubseteq> upper_unit\<cdot>z)"
348 "adm (\<lambda>f. f\<cdot>xs \<sqsubseteq> f\<cdot>(upper_unit\<cdot>z) \<or> f\<cdot>ys \<sqsubseteq> f\<cdot>(upper_unit\<cdot>z))")
349 apply (drule admD, rule chain_approx)
350 apply (drule_tac f="approx i" in monofun_cfun_arg)
351 apply (cut_tac xs="approx i\<cdot>xs" in compact_imp_upper_principal, simp)
352 apply (cut_tac xs="approx i\<cdot>ys" in compact_imp_upper_principal, simp)
353 apply (cut_tac x="approx i\<cdot>z" in compact_imp_Rep_compact_basis, simp)
354 apply (clarify, simp add: upper_le_PDPlus_PDUnit_iff)
358 apply (erule trans_less [OF upper_plus_less1])
359 apply (erule trans_less [OF upper_plus_less2])
362 lemmas upper_pd_less_simps =
365 upper_plus_less_unit_iff
368 subsection {* Induction rules *}
370 lemma upper_pd_induct1:
372 assumes unit: "\<And>x. P (upper_unit\<cdot>x)"
374 "\<And>x ys. \<lbrakk>P (upper_unit\<cdot>x); P ys\<rbrakk> \<Longrightarrow> P (upper_plus\<cdot>(upper_unit\<cdot>x)\<cdot>ys)"
375 shows "P (xs::'a upper_pd)"
376 apply (induct xs rule: upper_principal_induct, rule P)
377 apply (induct_tac t rule: pd_basis_induct1)
378 apply (simp only: upper_unit_Rep_compact_basis [symmetric])
380 apply (simp only: upper_unit_Rep_compact_basis [symmetric]
381 upper_plus_principal [symmetric])
382 apply (erule insert [OF unit])
385 lemma upper_pd_induct:
387 assumes unit: "\<And>x. P (upper_unit\<cdot>x)"
388 assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (upper_plus\<cdot>xs\<cdot>ys)"
389 shows "P (xs::'a upper_pd)"
390 apply (induct xs rule: upper_principal_induct, rule P)
391 apply (induct_tac t rule: pd_basis_induct)
392 apply (simp only: upper_unit_Rep_compact_basis [symmetric] unit)
393 apply (simp only: upper_plus_principal [symmetric] plus)
397 subsection {* Monadic bind *}
401 "'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b upper_pd) \<rightarrow> 'b upper_pd" where
402 "upper_bind_basis = fold_pd
403 (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
404 (\<lambda>x y. \<Lambda> f. upper_plus\<cdot>(x\<cdot>f)\<cdot>(y\<cdot>f))"
406 lemma ACI_upper_bind: "ACIf (\<lambda>x y. \<Lambda> f. upper_plus\<cdot>(x\<cdot>f)\<cdot>(y\<cdot>f))"
408 apply (simp add: upper_plus_commute)
409 apply (simp add: upper_plus_assoc)
410 apply (simp add: upper_plus_absorb eta_cfun)
413 lemma upper_bind_basis_simps [simp]:
414 "upper_bind_basis (PDUnit a) =
415 (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
416 "upper_bind_basis (PDPlus t u) =
417 (\<Lambda> f. upper_plus\<cdot>(upper_bind_basis t\<cdot>f)\<cdot>(upper_bind_basis u\<cdot>f))"
418 unfolding upper_bind_basis_def
420 apply (rule ACIf.fold_pd_PDUnit [OF ACI_upper_bind])
421 apply (rule ACIf.fold_pd_PDPlus [OF ACI_upper_bind])
424 lemma upper_bind_basis_mono:
425 "t \<le>\<sharp> u \<Longrightarrow> upper_bind_basis t \<sqsubseteq> upper_bind_basis u"
426 unfolding expand_cfun_less
427 apply (erule upper_le_induct, safe)
428 apply (simp add: compact_le_def monofun_cfun)
429 apply (simp add: trans_less [OF upper_plus_less1])
430 apply (simp add: upper_less_plus_iff)
434 upper_bind :: "'a upper_pd \<rightarrow> ('a \<rightarrow> 'b upper_pd) \<rightarrow> 'b upper_pd" where
435 "upper_bind = upper_pd.basis_fun upper_bind_basis"
437 lemma upper_bind_principal [simp]:
438 "upper_bind\<cdot>(upper_principal t) = upper_bind_basis t"
439 unfolding upper_bind_def
440 apply (rule upper_pd.basis_fun_principal)
441 apply (erule upper_bind_basis_mono)
444 lemma upper_bind_unit [simp]:
445 "upper_bind\<cdot>(upper_unit\<cdot>x)\<cdot>f = f\<cdot>x"
446 by (induct x rule: compact_basis_induct, simp, simp)
448 lemma upper_bind_plus [simp]:
449 "upper_bind\<cdot>(upper_plus\<cdot>xs\<cdot>ys)\<cdot>f =
450 upper_plus\<cdot>(upper_bind\<cdot>xs\<cdot>f)\<cdot>(upper_bind\<cdot>ys\<cdot>f)"
451 by (induct xs ys rule: upper_principal_induct2, simp, simp, simp)
453 lemma upper_bind_strict [simp]: "upper_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
454 unfolding upper_unit_strict [symmetric] by (rule upper_bind_unit)
457 subsection {* Map and join *}
460 upper_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a upper_pd \<rightarrow> 'b upper_pd" where
461 "upper_map = (\<Lambda> f xs. upper_bind\<cdot>xs\<cdot>(\<Lambda> x. upper_unit\<cdot>(f\<cdot>x)))"
464 upper_join :: "'a upper_pd upper_pd \<rightarrow> 'a upper_pd" where
465 "upper_join = (\<Lambda> xss. upper_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"
467 lemma upper_map_unit [simp]:
468 "upper_map\<cdot>f\<cdot>(upper_unit\<cdot>x) = upper_unit\<cdot>(f\<cdot>x)"
469 unfolding upper_map_def by simp
471 lemma upper_map_plus [simp]:
472 "upper_map\<cdot>f\<cdot>(upper_plus\<cdot>xs\<cdot>ys) =
473 upper_plus\<cdot>(upper_map\<cdot>f\<cdot>xs)\<cdot>(upper_map\<cdot>f\<cdot>ys)"
474 unfolding upper_map_def by simp
476 lemma upper_join_unit [simp]:
477 "upper_join\<cdot>(upper_unit\<cdot>xs) = xs"
478 unfolding upper_join_def by simp
480 lemma upper_join_plus [simp]:
481 "upper_join\<cdot>(upper_plus\<cdot>xss\<cdot>yss) =
482 upper_plus\<cdot>(upper_join\<cdot>xss)\<cdot>(upper_join\<cdot>yss)"
483 unfolding upper_join_def by simp
485 lemma upper_map_ident: "upper_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
486 by (induct xs rule: upper_pd_induct, simp_all)
489 "upper_map\<cdot>f\<cdot>(upper_map\<cdot>g\<cdot>xs) = upper_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
490 by (induct xs rule: upper_pd_induct, simp_all)
492 lemma upper_join_map_unit:
493 "upper_join\<cdot>(upper_map\<cdot>upper_unit\<cdot>xs) = xs"
494 by (induct xs rule: upper_pd_induct, simp_all)
496 lemma upper_join_map_join:
497 "upper_join\<cdot>(upper_map\<cdot>upper_join\<cdot>xsss) = upper_join\<cdot>(upper_join\<cdot>xsss)"
498 by (induct xsss rule: upper_pd_induct, simp_all)
500 lemma upper_join_map_map:
501 "upper_join\<cdot>(upper_map\<cdot>(upper_map\<cdot>f)\<cdot>xss) =
502 upper_map\<cdot>f\<cdot>(upper_join\<cdot>xss)"
503 by (induct xss rule: upper_pd_induct, simp_all)
505 lemma upper_map_approx: "upper_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs"
506 by (induct xs rule: upper_pd_induct, simp_all)