src/HOL/Library/Order_Continuity.thy
 author wenzelm Fri Jul 22 11:00:43 2016 +0200 (2016-07-22) changeset 63540 f8652d0534fa parent 62374 cb27a55d868a child 63979 95c3ae4baba8 permissions -rw-r--r--
tuned proofs -- avoid unstructured calculation;
1 (*  Title:      HOL/Library/Order_Continuity.thy
2     Author:     David von Oheimb, TU München
3     Author:     Johannes Hölzl, TU München
4 *)
6 section \<open>Continuity and iterations\<close>
8 theory Order_Continuity
9 imports Complex_Main Countable_Complete_Lattices
10 begin
12 (* TODO: Generalize theory to chain-complete partial orders *)
14 lemma SUP_nat_binary:
15   "(SUP n::nat. if n = 0 then A else B) = (sup A B::'a::countable_complete_lattice)"
16   apply (auto intro!: antisym ccSUP_least)
17   apply (rule ccSUP_upper2[where i=0])
18   apply simp_all
19   apply (rule ccSUP_upper2[where i=1])
20   apply simp_all
21   done
23 lemma INF_nat_binary:
24   "(INF n::nat. if n = 0 then A else B) = (inf A B::'a::countable_complete_lattice)"
25   apply (auto intro!: antisym ccINF_greatest)
26   apply (rule ccINF_lower2[where i=0])
27   apply simp_all
28   apply (rule ccINF_lower2[where i=1])
29   apply simp_all
30   done
32 text \<open>
33   The name \<open>continuous\<close> is already taken in \<open>Complex_Main\<close>, so we use
34   \<open>sup_continuous\<close> and \<open>inf_continuous\<close>. These names appear sometimes in literature
35   and have the advantage that these names are duals.
36 \<close>
38 named_theorems order_continuous_intros
40 subsection \<open>Continuity for complete lattices\<close>
42 definition
43   sup_continuous :: "('a::countable_complete_lattice \<Rightarrow> 'b::countable_complete_lattice) \<Rightarrow> bool"
44 where
45   "sup_continuous F \<longleftrightarrow> (\<forall>M::nat \<Rightarrow> 'a. mono M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
47 lemma sup_continuousD: "sup_continuous F \<Longrightarrow> mono M \<Longrightarrow> F (SUP i::nat. M i) = (SUP i. F (M i))"
48   by (auto simp: sup_continuous_def)
50 lemma sup_continuous_mono:
51   assumes [simp]: "sup_continuous F" shows "mono F"
52 proof
53   fix A B :: "'a" assume [simp]: "A \<le> B"
54   have "F B = F (SUP n::nat. if n = 0 then A else B)"
55     by (simp add: sup_absorb2 SUP_nat_binary)
56   also have "\<dots> = (SUP n::nat. if n = 0 then F A else F B)"
57     by (auto simp: sup_continuousD mono_def intro!: SUP_cong)
58   finally show "F A \<le> F B"
59     by (simp add: SUP_nat_binary le_iff_sup)
60 qed
62 lemma [order_continuous_intros]:
63   shows sup_continuous_const: "sup_continuous (\<lambda>x. c)"
64     and sup_continuous_id: "sup_continuous (\<lambda>x. x)"
65     and sup_continuous_apply: "sup_continuous (\<lambda>f. f x)"
66     and sup_continuous_fun: "(\<And>s. sup_continuous (\<lambda>x. P x s)) \<Longrightarrow> sup_continuous P"
67     and sup_continuous_If: "sup_continuous F \<Longrightarrow> sup_continuous G \<Longrightarrow> sup_continuous (\<lambda>f. if C then F f else G f)"
68   by (auto simp: sup_continuous_def)
70 lemma sup_continuous_compose:
71   assumes f: "sup_continuous f" and g: "sup_continuous g"
72   shows "sup_continuous (\<lambda>x. f (g x))"
73   unfolding sup_continuous_def
74 proof safe
75   fix M :: "nat \<Rightarrow> 'c"
76   assume M: "mono M"
77   then have "mono (\<lambda>i. g (M i))"
78     using sup_continuous_mono[OF g] by (auto simp: mono_def)
79   with M show "f (g (SUPREMUM UNIV M)) = (SUP i. f (g (M i)))"
80     by (auto simp: sup_continuous_def g[THEN sup_continuousD] f[THEN sup_continuousD])
81 qed
83 lemma sup_continuous_sup[order_continuous_intros]:
84   "sup_continuous f \<Longrightarrow> sup_continuous g \<Longrightarrow> sup_continuous (\<lambda>x. sup (f x) (g x))"
85   by (simp add: sup_continuous_def ccSUP_sup_distrib)
87 lemma sup_continuous_inf[order_continuous_intros]:
88   fixes P Q :: "'a :: countable_complete_lattice \<Rightarrow> 'b :: countable_complete_distrib_lattice"
89   assumes P: "sup_continuous P" and Q: "sup_continuous Q"
90   shows "sup_continuous (\<lambda>x. inf (P x) (Q x))"
91   unfolding sup_continuous_def
92 proof (safe intro!: antisym)
93   fix M :: "nat \<Rightarrow> 'a" assume M: "incseq M"
94   have "inf (P (SUP i. M i)) (Q (SUP i. M i)) \<le> (SUP j i. inf (P (M i)) (Q (M j)))"
95     by (simp add: sup_continuousD[OF P M] sup_continuousD[OF Q M] inf_ccSUP ccSUP_inf)
96   also have "\<dots> \<le> (SUP i. inf (P (M i)) (Q (M i)))"
97   proof (intro ccSUP_least)
98     fix i j from M assms[THEN sup_continuous_mono] show "inf (P (M i)) (Q (M j)) \<le> (SUP i. inf (P (M i)) (Q (M i)))"
99       by (intro ccSUP_upper2[of _ "sup i j"] inf_mono) (auto simp: mono_def)
100   qed auto
101   finally show "inf (P (SUP i. M i)) (Q (SUP i. M i)) \<le> (SUP i. inf (P (M i)) (Q (M i)))" .
103   show "(SUP i. inf (P (M i)) (Q (M i))) \<le> inf (P (SUP i. M i)) (Q (SUP i. M i))"
104     unfolding sup_continuousD[OF P M] sup_continuousD[OF Q M] by (intro ccSUP_least inf_mono ccSUP_upper) auto
105 qed
107 lemma sup_continuous_and[order_continuous_intros]:
108   "sup_continuous P \<Longrightarrow> sup_continuous Q \<Longrightarrow> sup_continuous (\<lambda>x. P x \<and> Q x)"
109   using sup_continuous_inf[of P Q] by simp
111 lemma sup_continuous_or[order_continuous_intros]:
112   "sup_continuous P \<Longrightarrow> sup_continuous Q \<Longrightarrow> sup_continuous (\<lambda>x. P x \<or> Q x)"
113   by (auto simp: sup_continuous_def)
115 lemma sup_continuous_lfp:
116   assumes "sup_continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)" (is "lfp F = ?U")
117 proof (rule antisym)
118   note mono = sup_continuous_mono[OF \<open>sup_continuous F\<close>]
119   show "?U \<le> lfp F"
120   proof (rule SUP_least)
121     fix i show "(F ^^ i) bot \<le> lfp F"
122     proof (induct i)
123       case (Suc i)
124       have "(F ^^ Suc i) bot = F ((F ^^ i) bot)" by simp
125       also have "\<dots> \<le> F (lfp F)" by (rule monoD[OF mono Suc])
126       also have "\<dots> = lfp F" by (simp add: lfp_unfold[OF mono, symmetric])
127       finally show ?case .
128     qed simp
129   qed
130   show "lfp F \<le> ?U"
131   proof (rule lfp_lowerbound)
132     have "mono (\<lambda>i::nat. (F ^^ i) bot)"
133     proof -
134       { fix i::nat have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot"
135         proof (induct i)
136           case 0 show ?case by simp
137         next
138           case Suc thus ?case using monoD[OF mono Suc] by auto
139         qed }
140       thus ?thesis by (auto simp add: mono_iff_le_Suc)
141     qed
142     hence "F ?U = (SUP i. (F ^^ Suc i) bot)"
143       using \<open>sup_continuous F\<close> by (simp add: sup_continuous_def)
144     also have "\<dots> \<le> ?U"
145       by (fast intro: SUP_least SUP_upper)
146     finally show "F ?U \<le> ?U" .
147   qed
148 qed
150 lemma lfp_transfer_bounded:
151   assumes P: "P bot" "\<And>x. P x \<Longrightarrow> P (f x)" "\<And>M. (\<And>i. P (M i)) \<Longrightarrow> P (SUP i::nat. M i)"
152   assumes \<alpha>: "\<And>M. mono M \<Longrightarrow> (\<And>i::nat. P (M i)) \<Longrightarrow> \<alpha> (SUP i. M i) = (SUP i. \<alpha> (M i))"
153   assumes f: "sup_continuous f" and g: "sup_continuous g"
154   assumes [simp]: "\<And>x. P x \<Longrightarrow> x \<le> lfp f \<Longrightarrow> \<alpha> (f x) = g (\<alpha> x)"
155   assumes g_bound: "\<And>x. \<alpha> bot \<le> g x"
156   shows "\<alpha> (lfp f) = lfp g"
157 proof (rule antisym)
158   note mono_g = sup_continuous_mono[OF g]
159   note mono_f = sup_continuous_mono[OF f]
160   have lfp_bound: "\<alpha> bot \<le> lfp g"
161     by (subst lfp_unfold[OF mono_g]) (rule g_bound)
163   have P_pow: "P ((f ^^ i) bot)" for i
164     by (induction i) (auto intro!: P)
165   have incseq_pow: "mono (\<lambda>i. (f ^^ i) bot)"
166     unfolding mono_iff_le_Suc
167   proof
168     fix i show "(f ^^ i) bot \<le> (f ^^ (Suc i)) bot"
169     proof (induct i)
170       case Suc thus ?case using monoD[OF sup_continuous_mono[OF f] Suc] by auto
171     qed (simp add: le_fun_def)
172   qed
173   have P_lfp: "P (lfp f)"
174     using P_pow unfolding sup_continuous_lfp[OF f] by (auto intro!: P)
176   have iter_le_lfp: "(f ^^ n) bot \<le> lfp f" for n
177     apply (induction n)
178     apply simp
179     apply (subst lfp_unfold[OF mono_f])
180     apply (auto intro!: monoD[OF mono_f])
181     done
183   have "\<alpha> (lfp f) = (SUP i. \<alpha> ((f^^i) bot))"
184     unfolding sup_continuous_lfp[OF f] using incseq_pow P_pow by (rule \<alpha>)
185   also have "\<dots> \<le> lfp g"
186   proof (rule SUP_least)
187     fix i show "\<alpha> ((f^^i) bot) \<le> lfp g"
188     proof (induction i)
189       case (Suc n) then show ?case
190         by (subst lfp_unfold[OF mono_g]) (simp add: monoD[OF mono_g] P_pow iter_le_lfp)
191     qed (simp add: lfp_bound)
192   qed
193   finally show "\<alpha> (lfp f) \<le> lfp g" .
195   show "lfp g \<le> \<alpha> (lfp f)"
196   proof (induction rule: lfp_ordinal_induct[OF mono_g])
197     case (1 S) then show ?case
198       by (subst lfp_unfold[OF sup_continuous_mono[OF f]])
199          (simp add: monoD[OF mono_g] P_lfp)
200   qed (auto intro: Sup_least)
201 qed
203 lemma lfp_transfer:
204   "sup_continuous \<alpha> \<Longrightarrow> sup_continuous f \<Longrightarrow> sup_continuous g \<Longrightarrow>
205     (\<And>x. \<alpha> bot \<le> g x) \<Longrightarrow> (\<And>x. x \<le> lfp f \<Longrightarrow> \<alpha> (f x) = g (\<alpha> x)) \<Longrightarrow> \<alpha> (lfp f) = lfp g"
206   by (rule lfp_transfer_bounded[where P=top]) (auto dest: sup_continuousD)
208 definition
209   inf_continuous :: "('a::countable_complete_lattice \<Rightarrow> 'b::countable_complete_lattice) \<Rightarrow> bool"
210 where
211   "inf_continuous F \<longleftrightarrow> (\<forall>M::nat \<Rightarrow> 'a. antimono M \<longrightarrow> F (INF i. M i) = (INF i. F (M i)))"
213 lemma inf_continuousD: "inf_continuous F \<Longrightarrow> antimono M \<Longrightarrow> F (INF i::nat. M i) = (INF i. F (M i))"
214   by (auto simp: inf_continuous_def)
216 lemma inf_continuous_mono:
217   assumes [simp]: "inf_continuous F" shows "mono F"
218 proof
219   fix A B :: "'a" assume [simp]: "A \<le> B"
220   have "F A = F (INF n::nat. if n = 0 then B else A)"
221     by (simp add: inf_absorb2 INF_nat_binary)
222   also have "\<dots> = (INF n::nat. if n = 0 then F B else F A)"
223     by (auto simp: inf_continuousD antimono_def intro!: INF_cong)
224   finally show "F A \<le> F B"
225     by (simp add: INF_nat_binary le_iff_inf inf_commute)
226 qed
228 lemma [order_continuous_intros]:
229   shows inf_continuous_const: "inf_continuous (\<lambda>x. c)"
230     and inf_continuous_id: "inf_continuous (\<lambda>x. x)"
231     and inf_continuous_apply: "inf_continuous (\<lambda>f. f x)"
232     and inf_continuous_fun: "(\<And>s. inf_continuous (\<lambda>x. P x s)) \<Longrightarrow> inf_continuous P"
233     and inf_continuous_If: "inf_continuous F \<Longrightarrow> inf_continuous G \<Longrightarrow> inf_continuous (\<lambda>f. if C then F f else G f)"
234   by (auto simp: inf_continuous_def)
236 lemma inf_continuous_inf[order_continuous_intros]:
237   "inf_continuous f \<Longrightarrow> inf_continuous g \<Longrightarrow> inf_continuous (\<lambda>x. inf (f x) (g x))"
238   by (simp add: inf_continuous_def ccINF_inf_distrib)
240 lemma inf_continuous_sup[order_continuous_intros]:
241   fixes P Q :: "'a :: countable_complete_lattice \<Rightarrow> 'b :: countable_complete_distrib_lattice"
242   assumes P: "inf_continuous P" and Q: "inf_continuous Q"
243   shows "inf_continuous (\<lambda>x. sup (P x) (Q x))"
244   unfolding inf_continuous_def
245 proof (safe intro!: antisym)
246   fix M :: "nat \<Rightarrow> 'a" assume M: "decseq M"
247   show "sup (P (INF i. M i)) (Q (INF i. M i)) \<le> (INF i. sup (P (M i)) (Q (M i)))"
248     unfolding inf_continuousD[OF P M] inf_continuousD[OF Q M] by (intro ccINF_greatest sup_mono ccINF_lower) auto
250   have "(INF i. sup (P (M i)) (Q (M i))) \<le> (INF j i. sup (P (M i)) (Q (M j)))"
251   proof (intro ccINF_greatest)
252     fix i j from M assms[THEN inf_continuous_mono] show "sup (P (M i)) (Q (M j)) \<ge> (INF i. sup (P (M i)) (Q (M i)))"
253       by (intro ccINF_lower2[of _ "sup i j"] sup_mono) (auto simp: mono_def antimono_def)
254   qed auto
255   also have "\<dots> \<le> sup (P (INF i. M i)) (Q (INF i. M i))"
256     by (simp add: inf_continuousD[OF P M] inf_continuousD[OF Q M] ccINF_sup sup_ccINF)
257   finally show "sup (P (INF i. M i)) (Q (INF i. M i)) \<ge> (INF i. sup (P (M i)) (Q (M i)))" .
258 qed
260 lemma inf_continuous_and[order_continuous_intros]:
261   "inf_continuous P \<Longrightarrow> inf_continuous Q \<Longrightarrow> inf_continuous (\<lambda>x. P x \<and> Q x)"
262   using inf_continuous_inf[of P Q] by simp
264 lemma inf_continuous_or[order_continuous_intros]:
265   "inf_continuous P \<Longrightarrow> inf_continuous Q \<Longrightarrow> inf_continuous (\<lambda>x. P x \<or> Q x)"
266   using inf_continuous_sup[of P Q] by simp
268 lemma inf_continuous_compose:
269   assumes f: "inf_continuous f" and g: "inf_continuous g"
270   shows "inf_continuous (\<lambda>x. f (g x))"
271   unfolding inf_continuous_def
272 proof safe
273   fix M :: "nat \<Rightarrow> 'c"
274   assume M: "antimono M"
275   then have "antimono (\<lambda>i. g (M i))"
276     using inf_continuous_mono[OF g] by (auto simp: mono_def antimono_def)
277   with M show "f (g (INFIMUM UNIV M)) = (INF i. f (g (M i)))"
278     by (auto simp: inf_continuous_def g[THEN inf_continuousD] f[THEN inf_continuousD])
279 qed
281 lemma inf_continuous_gfp:
282   assumes "inf_continuous F" shows "gfp F = (INF i. (F ^^ i) top)" (is "gfp F = ?U")
283 proof (rule antisym)
284   note mono = inf_continuous_mono[OF \<open>inf_continuous F\<close>]
285   show "gfp F \<le> ?U"
286   proof (rule INF_greatest)
287     fix i show "gfp F \<le> (F ^^ i) top"
288     proof (induct i)
289       case (Suc i)
290       have "gfp F = F (gfp F)" by (simp add: gfp_unfold[OF mono, symmetric])
291       also have "\<dots> \<le> F ((F ^^ i) top)" by (rule monoD[OF mono Suc])
292       also have "\<dots> = (F ^^ Suc i) top" by simp
293       finally show ?case .
294     qed simp
295   qed
296   show "?U \<le> gfp F"
297   proof (rule gfp_upperbound)
298     have *: "antimono (\<lambda>i::nat. (F ^^ i) top)"
299     proof -
300       { fix i::nat have "(F ^^ Suc i) top \<le> (F ^^ i) top"
301         proof (induct i)
302           case 0 show ?case by simp
303         next
304           case Suc thus ?case using monoD[OF mono Suc] by auto
305         qed }
306       thus ?thesis by (auto simp add: antimono_iff_le_Suc)
307     qed
308     have "?U \<le> (INF i. (F ^^ Suc i) top)"
309       by (fast intro: INF_greatest INF_lower)
310     also have "\<dots> \<le> F ?U"
311       by (simp add: inf_continuousD \<open>inf_continuous F\<close> *)
312     finally show "?U \<le> F ?U" .
313   qed
314 qed
316 lemma gfp_transfer:
317   assumes \<alpha>: "inf_continuous \<alpha>" and f: "inf_continuous f" and g: "inf_continuous g"
318   assumes [simp]: "\<alpha> top = top" "\<And>x. \<alpha> (f x) = g (\<alpha> x)"
319   shows "\<alpha> (gfp f) = gfp g"
320 proof -
321   have "\<alpha> (gfp f) = (INF i. \<alpha> ((f^^i) top))"
322     unfolding inf_continuous_gfp[OF f] by (intro f \<alpha> inf_continuousD antimono_funpow inf_continuous_mono)
323   moreover have "\<alpha> ((f^^i) top) = (g^^i) top" for i
324     by (induction i; simp)
325   ultimately show ?thesis
326     unfolding inf_continuous_gfp[OF g] by simp
327 qed
329 lemma gfp_transfer_bounded:
330   assumes P: "P (f top)" "\<And>x. P x \<Longrightarrow> P (f x)" "\<And>M. antimono M \<Longrightarrow> (\<And>i. P (M i)) \<Longrightarrow> P (INF i::nat. M i)"
331   assumes \<alpha>: "\<And>M. antimono M \<Longrightarrow> (\<And>i::nat. P (M i)) \<Longrightarrow> \<alpha> (INF i. M i) = (INF i. \<alpha> (M i))"
332   assumes f: "inf_continuous f" and g: "inf_continuous g"
333   assumes [simp]: "\<And>x. P x \<Longrightarrow> \<alpha> (f x) = g (\<alpha> x)"
334   assumes g_bound: "\<And>x. g x \<le> \<alpha> (f top)"
335   shows "\<alpha> (gfp f) = gfp g"
336 proof (rule antisym)
337   note mono_g = inf_continuous_mono[OF g]
339   have P_pow: "P ((f ^^ i) (f top))" for i
340     by (induction i) (auto intro!: P)
342   have antimono_pow: "antimono (\<lambda>i. (f ^^ i) top)"
343     unfolding antimono_iff_le_Suc
344   proof
345     fix i show "(f ^^ Suc i) top \<le> (f ^^ i) top"
346     proof (induct i)
347       case Suc thus ?case using monoD[OF inf_continuous_mono[OF f] Suc] by auto
348     qed (simp add: le_fun_def)
349   qed
350   have antimono_pow2: "antimono (\<lambda>i. (f ^^ i) (f top))"
351   proof
352     show "x \<le> y \<Longrightarrow> (f ^^ y) (f top) \<le> (f ^^ x) (f top)" for x y
353       using antimono_pow[THEN antimonoD, of "Suc x" "Suc y"]
354       unfolding funpow_Suc_right by simp
355   qed
357   have gfp_f: "gfp f = (INF i. (f ^^ i) (f top))"
358     unfolding inf_continuous_gfp[OF f]
359   proof (rule INF_eq)
360     show "\<exists>j\<in>UNIV. (f ^^ j) (f top) \<le> (f ^^ i) top" for i
361       by (intro bexI[of _ "i - 1"]) (auto simp: diff_Suc funpow_Suc_right simp del: funpow.simps(2) split: nat.split)
362     show "\<exists>j\<in>UNIV. (f ^^ j) top \<le> (f ^^ i) (f top)" for i
363       by (intro bexI[of _ "Suc i"]) (auto simp: funpow_Suc_right simp del: funpow.simps(2))
364   qed
366   have P_lfp: "P (gfp f)"
367     unfolding gfp_f by (auto intro!: P P_pow antimono_pow2)
369   have "\<alpha> (gfp f) = (INF i. \<alpha> ((f^^i) (f top)))"
370     unfolding gfp_f by (rule \<alpha>) (auto intro!: P_pow antimono_pow2)
371   also have "\<dots> \<ge> gfp g"
372   proof (rule INF_greatest)
373     fix i show "gfp g \<le> \<alpha> ((f^^i) (f top))"
374     proof (induction i)
375       case (Suc n) then show ?case
376         by (subst gfp_unfold[OF mono_g]) (simp add: monoD[OF mono_g] P_pow)
377     next
378       case 0
379       have "gfp g \<le> \<alpha> (f top)"
380         by (subst gfp_unfold[OF mono_g]) (rule g_bound)
381       then show ?case
382         by simp
383     qed
384   qed
385   finally show "gfp g \<le> \<alpha> (gfp f)" .
387   show "\<alpha> (gfp f) \<le> gfp g"
388   proof (induction rule: gfp_ordinal_induct[OF mono_g])
389     case (1 S) then show ?case
390       by (subst gfp_unfold[OF inf_continuous_mono[OF f]])
391          (simp add: monoD[OF mono_g] P_lfp)
392   qed (auto intro: Inf_greatest)
393 qed
395 subsubsection \<open>Least fixed points in countable complete lattices\<close>
397 definition (in countable_complete_lattice) cclfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
398   where "cclfp f = (SUP i. (f ^^ i) bot)"
400 lemma cclfp_unfold:
401   assumes "sup_continuous F" shows "cclfp F = F (cclfp F)"
402 proof -
403   have "cclfp F = (SUP i. F ((F ^^ i) bot))"
404     unfolding cclfp_def by (subst UNIV_nat_eq) auto
405   also have "\<dots> = F (cclfp F)"
406     unfolding cclfp_def
407     by (intro sup_continuousD[symmetric] assms mono_funpow sup_continuous_mono)
408   finally show ?thesis .
409 qed
411 lemma cclfp_lowerbound: assumes f: "mono f" and A: "f A \<le> A" shows "cclfp f \<le> A"
412   unfolding cclfp_def
413 proof (intro ccSUP_least)
414   fix i show "(f ^^ i) bot \<le> A"
415   proof (induction i)
416     case (Suc i) from monoD[OF f this] A show ?case
417       by auto
418   qed simp
419 qed simp
421 lemma cclfp_transfer:
422   assumes "sup_continuous \<alpha>" "mono f"
423   assumes "\<alpha> bot = bot" "\<And>x. \<alpha> (f x) = g (\<alpha> x)"
424   shows "\<alpha> (cclfp f) = cclfp g"
425 proof -
426   have "\<alpha> (cclfp f) = (SUP i. \<alpha> ((f ^^ i) bot))"
427     unfolding cclfp_def by (intro sup_continuousD assms mono_funpow sup_continuous_mono)
428   moreover have "\<alpha> ((f ^^ i) bot) = (g ^^ i) bot" for i
429     by (induction i) (simp_all add: assms)
430   ultimately show ?thesis
431     by (simp add: cclfp_def)
432 qed
434 end