src/HOL/Finite_Set.thy
 author haftmann Fri May 20 12:35:44 2011 +0200 (2011-05-20) changeset 42875 d1aad0957eb2 parent 42873 da1253ff1764 child 43866 8a50dc70cbff permissions -rw-r--r--
tuned proofs
1 (*  Title:      HOL/Finite_Set.thy
2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
3                 with contributions by Jeremy Avigad
4 *)
6 header {* Finite sets *}
8 theory Finite_Set
9 imports Option Power
10 begin
12 subsection {* Predicate for finite sets *}
14 inductive finite :: "'a set \<Rightarrow> bool"
15   where
16     emptyI [simp, intro!]: "finite {}"
17   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
19 lemma finite_induct [case_names empty insert, induct set: finite]:
20   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
21   assumes "finite F"
22   assumes "P {}"
23     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
24   shows "P F"
25 using `finite F` proof induct
26   show "P {}" by fact
27   fix x F assume F: "finite F" and P: "P F"
28   show "P (insert x F)"
29   proof cases
30     assume "x \<in> F"
31     hence "insert x F = F" by (rule insert_absorb)
32     with P show ?thesis by (simp only:)
33   next
34     assume "x \<notin> F"
35     from F this P show ?thesis by (rule insert)
36   qed
37 qed
40 subsubsection {* Choice principles *}
42 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
43   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
44   shows "\<exists>a::'a. a \<notin> A"
45 proof -
46   from assms have "A \<noteq> UNIV" by blast
47   then show ?thesis by blast
48 qed
50 text {* A finite choice principle. Does not need the SOME choice operator. *}
52 lemma finite_set_choice:
53   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
54 proof (induct rule: finite_induct)
55   case empty then show ?case by simp
56 next
57   case (insert a A)
58   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
59   show ?case (is "EX f. ?P f")
60   proof
61     show "?P(%x. if x = a then b else f x)" using f ab by auto
62   qed
63 qed
66 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
68 lemma finite_imp_nat_seg_image_inj_on:
69   assumes "finite A"
70   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
71 using assms proof induct
72   case empty
73   show ?case
74   proof
75     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
76   qed
77 next
78   case (insert a A)
79   have notinA: "a \<notin> A" by fact
80   from insert.hyps obtain n f
81     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
82   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
83         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
84     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
85   thus ?case by blast
86 qed
88 lemma nat_seg_image_imp_finite:
89   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
90 proof (induct n arbitrary: A)
91   case 0 thus ?case by simp
92 next
93   case (Suc n)
94   let ?B = "f ` {i. i < n}"
95   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
96   show ?case
97   proof cases
98     assume "\<exists>k<n. f n = f k"
99     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
100     thus ?thesis using finB by simp
101   next
102     assume "\<not>(\<exists> k<n. f n = f k)"
103     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
104     thus ?thesis using finB by simp
105   qed
106 qed
108 lemma finite_conv_nat_seg_image:
109   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
110   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
112 lemma finite_imp_inj_to_nat_seg:
113   assumes "finite A"
114   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
115 proof -
116   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
117   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
118     by (auto simp:bij_betw_def)
119   let ?f = "the_inv_into {i. i<n} f"
120   have "inj_on ?f A & ?f ` A = {i. i<n}"
121     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
122   thus ?thesis by blast
123 qed
125 lemma finite_Collect_less_nat [iff]:
126   "finite {n::nat. n < k}"
127   by (fastsimp simp: finite_conv_nat_seg_image)
129 lemma finite_Collect_le_nat [iff]:
130   "finite {n::nat. n \<le> k}"
131   by (simp add: le_eq_less_or_eq Collect_disj_eq)
134 subsubsection {* Finiteness and common set operations *}
136 lemma rev_finite_subset:
137   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
138 proof (induct arbitrary: A rule: finite_induct)
139   case empty
140   then show ?case by simp
141 next
142   case (insert x F A)
143   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
144   show "finite A"
145   proof cases
146     assume x: "x \<in> A"
147     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
148     with r have "finite (A - {x})" .
149     hence "finite (insert x (A - {x}))" ..
150     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
151     finally show ?thesis .
152   next
153     show "A \<subseteq> F ==> ?thesis" by fact
154     assume "x \<notin> A"
155     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
156   qed
157 qed
159 lemma finite_subset:
160   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
161   by (rule rev_finite_subset)
163 lemma finite_UnI:
164   assumes "finite F" and "finite G"
165   shows "finite (F \<union> G)"
166   using assms by induct simp_all
168 lemma finite_Un [iff]:
169   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
170   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
172 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
173 proof -
174   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
175   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
176   then show ?thesis by simp
177 qed
179 lemma finite_Int [simp, intro]:
180   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
181   by (blast intro: finite_subset)
183 lemma finite_Collect_conjI [simp, intro]:
184   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
185   by (simp add: Collect_conj_eq)
187 lemma finite_Collect_disjI [simp]:
188   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
189   by (simp add: Collect_disj_eq)
191 lemma finite_Diff [simp, intro]:
192   "finite A \<Longrightarrow> finite (A - B)"
193   by (rule finite_subset, rule Diff_subset)
195 lemma finite_Diff2 [simp]:
196   assumes "finite B"
197   shows "finite (A - B) \<longleftrightarrow> finite A"
198 proof -
199   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
200   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
201   finally show ?thesis ..
202 qed
204 lemma finite_Diff_insert [iff]:
205   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
206 proof -
207   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
208   moreover have "A - insert a B = A - B - {a}" by auto
209   ultimately show ?thesis by simp
210 qed
212 lemma finite_compl[simp]:
213   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
214   by (simp add: Compl_eq_Diff_UNIV)
216 lemma finite_Collect_not[simp]:
217   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
218   by (simp add: Collect_neg_eq)
220 lemma finite_Union [simp, intro]:
221   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
222   by (induct rule: finite_induct) simp_all
224 lemma finite_UN_I [intro]:
225   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
226   by (induct rule: finite_induct) simp_all
228 lemma finite_UN [simp]:
229   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
230   by (blast intro: finite_subset)
232 lemma finite_Inter [intro]:
233   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
234   by (blast intro: Inter_lower finite_subset)
236 lemma finite_INT [intro]:
237   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
238   by (blast intro: INT_lower finite_subset)
240 lemma finite_imageI [simp, intro]:
241   "finite F \<Longrightarrow> finite (h ` F)"
242   by (induct rule: finite_induct) simp_all
244 lemma finite_image_set [simp]:
245   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
246   by (simp add: image_Collect [symmetric])
248 lemma finite_imageD:
249   assumes "finite (f ` A)" and "inj_on f A"
250   shows "finite A"
251 using assms proof (induct "f ` A" arbitrary: A)
252   case empty then show ?case by simp
253 next
254   case (insert x B)
255   then have B_A: "insert x B = f ` A" by simp
256   then obtain y where "x = f y" and "y \<in> A" by blast
257   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
258   with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
259   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
260   ultimately have "finite (A - {y})" by (rule insert.hyps)
261   then show "finite A" by simp
262 qed
264 lemma finite_surj:
265   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
266   by (erule finite_subset) (rule finite_imageI)
268 lemma finite_range_imageI:
269   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
270   by (drule finite_imageI) (simp add: range_composition)
272 lemma finite_subset_image:
273   assumes "finite B"
274   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
275 using assms proof induct
276   case empty then show ?case by simp
277 next
278   case insert then show ?case
279     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
280        blast
281 qed
283 lemma finite_vimageI:
284   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
285   apply (induct rule: finite_induct)
286    apply simp_all
287   apply (subst vimage_insert)
288   apply (simp add: finite_subset [OF inj_vimage_singleton])
289   done
291 lemma finite_vimageD:
292   assumes fin: "finite (h -` F)" and surj: "surj h"
293   shows "finite F"
294 proof -
295   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
296   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
297   finally show "finite F" .
298 qed
300 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
301   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
303 lemma finite_Collect_bex [simp]:
304   assumes "finite A"
305   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
306 proof -
307   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
308   with assms show ?thesis by simp
309 qed
311 lemma finite_Collect_bounded_ex [simp]:
312   assumes "finite {y. P y}"
313   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
314 proof -
315   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
316   with assms show ?thesis by simp
317 qed
319 lemma finite_Plus:
320   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
321   by (simp add: Plus_def)
323 lemma finite_PlusD:
324   fixes A :: "'a set" and B :: "'b set"
325   assumes fin: "finite (A <+> B)"
326   shows "finite A" "finite B"
327 proof -
328   have "Inl ` A \<subseteq> A <+> B" by auto
329   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
330   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
331 next
332   have "Inr ` B \<subseteq> A <+> B" by auto
333   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
334   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
335 qed
337 lemma finite_Plus_iff [simp]:
338   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
339   by (auto intro: finite_PlusD finite_Plus)
341 lemma finite_Plus_UNIV_iff [simp]:
342   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
343   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
345 lemma finite_SigmaI [simp, intro]:
346   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
347   by (unfold Sigma_def) blast
349 lemma finite_cartesian_product:
350   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
351   by (rule finite_SigmaI)
353 lemma finite_Prod_UNIV:
354   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
355   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
357 lemma finite_cartesian_productD1:
358   assumes "finite (A \<times> B)" and "B \<noteq> {}"
359   shows "finite A"
360 proof -
361   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
362     by (auto simp add: finite_conv_nat_seg_image)
363   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
364   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
365     by (simp add: image_compose)
366   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
367   then show ?thesis
368     by (auto simp add: finite_conv_nat_seg_image)
369 qed
371 lemma finite_cartesian_productD2:
372   assumes "finite (A \<times> B)" and "A \<noteq> {}"
373   shows "finite B"
374 proof -
375   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
376     by (auto simp add: finite_conv_nat_seg_image)
377   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
378   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
379     by (simp add: image_compose)
380   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
381   then show ?thesis
382     by (auto simp add: finite_conv_nat_seg_image)
383 qed
385 lemma finite_Pow_iff [iff]:
386   "finite (Pow A) \<longleftrightarrow> finite A"
387 proof
388   assume "finite (Pow A)"
389   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
390   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
391 next
392   assume "finite A"
393   then show "finite (Pow A)"
394     by induct (simp_all add: Pow_insert)
395 qed
397 corollary finite_Collect_subsets [simp, intro]:
398   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
399   by (simp add: Pow_def [symmetric])
401 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
402   by (blast intro: finite_subset [OF subset_Pow_Union])
405 subsubsection {* Further induction rules on finite sets *}
407 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
408   assumes "finite F" and "F \<noteq> {}"
409   assumes "\<And>x. P {x}"
410     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
411   shows "P F"
412 using assms proof induct
413   case empty then show ?case by simp
414 next
415   case (insert x F) then show ?case by cases auto
416 qed
418 lemma finite_subset_induct [consumes 2, case_names empty insert]:
419   assumes "finite F" and "F \<subseteq> A"
420   assumes empty: "P {}"
421     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
422   shows "P F"
423 using `finite F` `F \<subseteq> A` proof induct
424   show "P {}" by fact
425 next
426   fix x F
427   assume "finite F" and "x \<notin> F" and
428     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
429   show "P (insert x F)"
430   proof (rule insert)
431     from i show "x \<in> A" by blast
432     from i have "F \<subseteq> A" by blast
433     with P show "P F" .
434     show "finite F" by fact
435     show "x \<notin> F" by fact
436   qed
437 qed
439 lemma finite_empty_induct:
440   assumes "finite A"
441   assumes "P A"
442     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
443   shows "P {}"
444 proof -
445   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
446   proof -
447     fix B :: "'a set"
448     assume "B \<subseteq> A"
449     with `finite A` have "finite B" by (rule rev_finite_subset)
450     from this `B \<subseteq> A` show "P (A - B)"
451     proof induct
452       case empty
453       from `P A` show ?case by simp
454     next
455       case (insert b B)
456       have "P (A - B - {b})"
457       proof (rule remove)
458         from `finite A` show "finite (A - B)" by induct auto
459         from insert show "b \<in> A - B" by simp
460         from insert show "P (A - B)" by simp
461       qed
462       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
463       finally show ?case .
464     qed
465   qed
466   then have "P (A - A)" by blast
467   then show ?thesis by simp
468 qed
471 subsection {* Class @{text finite}  *}
473 class finite =
474   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
475 begin
477 lemma finite [simp]: "finite (A \<Colon> 'a set)"
478   by (rule subset_UNIV finite_UNIV finite_subset)+
480 lemma finite_code [code]: "finite (A \<Colon> 'a set) = True"
481   by simp
483 end
485 lemma UNIV_unit [no_atp]:
486   "UNIV = {()}" by auto
488 instance unit :: finite proof
489 qed (simp add: UNIV_unit)
491 lemma UNIV_bool [no_atp]:
492   "UNIV = {False, True}" by auto
494 instance bool :: finite proof
495 qed (simp add: UNIV_bool)
497 instance prod :: (finite, finite) finite proof
498 qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
500 lemma finite_option_UNIV [simp]:
501   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
502   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
504 instance option :: (finite) finite proof
505 qed (simp add: UNIV_option_conv)
507 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
508   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
510 instance "fun" :: (finite, finite) finite
511 proof
512   show "finite (UNIV :: ('a => 'b) set)"
513   proof (rule finite_imageD)
514     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
515     have "range ?graph \<subseteq> Pow UNIV" by simp
516     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
517       by (simp only: finite_Pow_iff finite)
518     ultimately show "finite (range ?graph)"
519       by (rule finite_subset)
520     show "inj ?graph" by (rule inj_graph)
521   qed
522 qed
524 instance sum :: (finite, finite) finite proof
525 qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
528 subsection {* A basic fold functional for finite sets *}
530 text {* The intended behaviour is
531 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
532 if @{text f} is ``left-commutative'':
533 *}
535 locale comp_fun_commute =
536   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
537   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
538 begin
540 lemma fun_left_comm: "f x (f y z) = f y (f x z)"
541   using comp_fun_commute by (simp add: fun_eq_iff)
543 end
545 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
546 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
547   emptyI [intro]: "fold_graph f z {} z" |
548   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
549       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
551 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
553 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
554   "fold f z A = (THE y. fold_graph f z A y)"
556 text{*A tempting alternative for the definiens is
557 @{term "if finite A then THE y. fold_graph f z A y else e"}.
558 It allows the removal of finiteness assumptions from the theorems
559 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
560 The proofs become ugly. It is not worth the effort. (???) *}
562 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
563 by (induct rule: finite_induct) auto
566 subsubsection{*From @{const fold_graph} to @{term fold}*}
568 context comp_fun_commute
569 begin
571 lemma fold_graph_insertE_aux:
572   "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
573 proof (induct set: fold_graph)
574   case (insertI x A y) show ?case
575   proof (cases "x = a")
576     assume "x = a" with insertI show ?case by auto
577   next
578     assume "x \<noteq> a"
579     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
580       using insertI by auto
581     have "f x y = f a (f x y')"
582       unfolding y by (rule fun_left_comm)
583     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
584       using y' and `x \<noteq> a` and `x \<notin> A`
585       by (simp add: insert_Diff_if fold_graph.insertI)
586     ultimately show ?case by fast
587   qed
588 qed simp
590 lemma fold_graph_insertE:
591   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
592   obtains y where "v = f x y" and "fold_graph f z A y"
593 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
595 lemma fold_graph_determ:
596   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
597 proof (induct arbitrary: y set: fold_graph)
598   case (insertI x A y v)
599   from `fold_graph f z (insert x A) v` and `x \<notin> A`
600   obtain y' where "v = f x y'" and "fold_graph f z A y'"
601     by (rule fold_graph_insertE)
602   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
603   with `v = f x y'` show "v = f x y" by simp
604 qed fast
606 lemma fold_equality:
607   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
608 by (unfold fold_def) (blast intro: fold_graph_determ)
610 lemma fold_graph_fold:
611   assumes "finite A"
612   shows "fold_graph f z A (fold f z A)"
613 proof -
614   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
615   moreover note fold_graph_determ
616   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
617   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
618   then show ?thesis by (unfold fold_def)
619 qed
621 text{* The base case for @{text fold}: *}
623 lemma (in -) fold_empty [simp]: "fold f z {} = z"
624 by (unfold fold_def) blast
626 text{* The various recursion equations for @{const fold}: *}
628 lemma fold_insert [simp]:
629   assumes "finite A" and "x \<notin> A"
630   shows "fold f z (insert x A) = f x (fold f z A)"
631 proof (rule fold_equality)
632   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
633   with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
634 qed
636 lemma fold_fun_comm:
637   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
638 proof (induct rule: finite_induct)
639   case empty then show ?case by simp
640 next
641   case (insert y A) then show ?case
642     by (simp add: fun_left_comm[of x])
643 qed
645 lemma fold_insert2:
646   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
647 by (simp add: fold_fun_comm)
649 lemma fold_rec:
650   assumes "finite A" and "x \<in> A"
651   shows "fold f z A = f x (fold f z (A - {x}))"
652 proof -
653   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
654   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
655   also have "\<dots> = f x (fold f z (A - {x}))"
656     by (rule fold_insert) (simp add: `finite A`)+
657   finally show ?thesis .
658 qed
660 lemma fold_insert_remove:
661   assumes "finite A"
662   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
663 proof -
664   from `finite A` have "finite (insert x A)" by auto
665   moreover have "x \<in> insert x A" by auto
666   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
667     by (rule fold_rec)
668   then show ?thesis by simp
669 qed
671 end
673 text{* A simplified version for idempotent functions: *}
675 locale comp_fun_idem = comp_fun_commute +
676   assumes comp_fun_idem: "f x o f x = f x"
677 begin
679 lemma fun_left_idem: "f x (f x z) = f x z"
680   using comp_fun_idem by (simp add: fun_eq_iff)
682 lemma fold_insert_idem:
683   assumes fin: "finite A"
684   shows "fold f z (insert x A) = f x (fold f z A)"
685 proof cases
686   assume "x \<in> A"
687   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
688   then show ?thesis using assms by (simp add:fun_left_idem)
689 next
690   assume "x \<notin> A" then show ?thesis using assms by simp
691 qed
693 declare fold_insert[simp del] fold_insert_idem[simp]
695 lemma fold_insert_idem2:
696   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
699 end
702 subsubsection {* Expressing set operations via @{const fold} *}
704 lemma (in comp_fun_commute) comp_comp_fun_commute:
705   "comp_fun_commute (f \<circ> g)"
706 proof
707 qed (simp_all add: comp_fun_commute)
709 lemma (in comp_fun_idem) comp_comp_fun_idem:
710   "comp_fun_idem (f \<circ> g)"
711   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
712     (simp_all add: comp_fun_idem)
714 lemma comp_fun_idem_insert:
715   "comp_fun_idem insert"
716 proof
717 qed auto
719 lemma comp_fun_idem_remove:
720   "comp_fun_idem (\<lambda>x A. A - {x})"
721 proof
722 qed auto
724 lemma (in semilattice_inf) comp_fun_idem_inf:
725   "comp_fun_idem inf"
726 proof
727 qed (auto simp add: inf_left_commute)
729 lemma (in semilattice_sup) comp_fun_idem_sup:
730   "comp_fun_idem sup"
731 proof
732 qed (auto simp add: sup_left_commute)
734 lemma union_fold_insert:
735   assumes "finite A"
736   shows "A \<union> B = fold insert B A"
737 proof -
738   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
739   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
740 qed
742 lemma minus_fold_remove:
743   assumes "finite A"
744   shows "B - A = fold (\<lambda>x A. A - {x}) B A"
745 proof -
746   interpret comp_fun_idem "\<lambda>x A. A - {x}" by (fact comp_fun_idem_remove)
747   from `finite A` show ?thesis by (induct A arbitrary: B) auto
748 qed
750 context complete_lattice
751 begin
753 lemma inf_Inf_fold_inf:
754   assumes "finite A"
755   shows "inf B (Inf A) = fold inf B A"
756 proof -
757   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
758   from `finite A` show ?thesis by (induct A arbitrary: B)
759     (simp_all add: Inf_insert inf_commute fold_fun_comm)
760 qed
762 lemma sup_Sup_fold_sup:
763   assumes "finite A"
764   shows "sup B (Sup A) = fold sup B A"
765 proof -
766   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
767   from `finite A` show ?thesis by (induct A arbitrary: B)
768     (simp_all add: Sup_insert sup_commute fold_fun_comm)
769 qed
771 lemma Inf_fold_inf:
772   assumes "finite A"
773   shows "Inf A = fold inf top A"
774   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
776 lemma Sup_fold_sup:
777   assumes "finite A"
778   shows "Sup A = fold sup bot A"
779   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
781 lemma inf_INFI_fold_inf:
782   assumes "finite A"
783   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
784 proof (rule sym)
785   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
786   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
787   from `finite A` show "?fold = ?inf"
788     by (induct A arbitrary: B)
789       (simp_all add: INFI_def Inf_insert inf_left_commute)
790 qed
792 lemma sup_SUPR_fold_sup:
793   assumes "finite A"
794   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
795 proof (rule sym)
796   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
797   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
798   from `finite A` show "?fold = ?sup"
799     by (induct A arbitrary: B)
800       (simp_all add: SUPR_def Sup_insert sup_left_commute)
801 qed
803 lemma INFI_fold_inf:
804   assumes "finite A"
805   shows "INFI A f = fold (inf \<circ> f) top A"
806   using assms inf_INFI_fold_inf [of A top] by simp
808 lemma SUPR_fold_sup:
809   assumes "finite A"
810   shows "SUPR A f = fold (sup \<circ> f) bot A"
811   using assms sup_SUPR_fold_sup [of A bot] by simp
813 end
816 subsection {* The derived combinator @{text fold_image} *}
818 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
819   where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
821 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
822   by (simp add:fold_image_def)
824 context ab_semigroup_mult
825 begin
827 lemma fold_image_insert[simp]:
828   assumes "finite A" and "a \<notin> A"
829   shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
830 proof -
831   interpret comp_fun_commute "%x y. (g x) * y" proof
832   qed (simp add: fun_eq_iff mult_ac)
833   show ?thesis using assms by (simp add: fold_image_def)
834 qed
836 lemma fold_image_reindex:
837   assumes "finite A"
838   shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
839   using assms by induct auto
841 lemma fold_image_cong:
842   assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
843   shows "fold_image times g z A = fold_image times h z A"
844 proof -
845   from `finite A`
846   have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
847   proof (induct arbitrary: C)
848     case empty then show ?case by simp
849   next
850     case (insert x F) then show ?case apply -
851     apply (simp add: subset_insert_iff, clarify)
852     apply (subgoal_tac "finite C")
853       prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
854     apply (subgoal_tac "C = insert x (C - {x})")
855       prefer 2 apply blast
856     apply (erule ssubst)
857     apply (simp add: Ball_def del: insert_Diff_single)
858     done
859   qed
860   with g_h show ?thesis by simp
861 qed
863 end
865 context comm_monoid_mult
866 begin
868 lemma fold_image_1:
869   "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
870   apply (induct rule: finite_induct)
871   apply simp by auto
873 lemma fold_image_Un_Int:
874   "finite A ==> finite B ==>
875     fold_image times g 1 A * fold_image times g 1 B =
876     fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
877   apply (induct rule: finite_induct)
878 by (induct set: finite)
879    (auto simp add: mult_ac insert_absorb Int_insert_left)
881 lemma fold_image_Un_one:
882   assumes fS: "finite S" and fT: "finite T"
883   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
884   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
885 proof-
886   have "fold_image op * f 1 (S \<inter> T) = 1"
887     apply (rule fold_image_1)
888     using fS fT I0 by auto
889   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
890 qed
892 corollary fold_Un_disjoint:
893   "finite A ==> finite B ==> A Int B = {} ==>
894    fold_image times g 1 (A Un B) =
895    fold_image times g 1 A * fold_image times g 1 B"
896 by (simp add: fold_image_Un_Int)
898 lemma fold_image_UN_disjoint:
899   "\<lbrakk> finite I; ALL i:I. finite (A i);
900      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
901    \<Longrightarrow> fold_image times g 1 (UNION I A) =
902        fold_image times (%i. fold_image times g 1 (A i)) 1 I"
903 apply (induct rule: finite_induct)
904 apply simp
905 apply atomize
906 apply (subgoal_tac "ALL i:F. x \<noteq> i")
907  prefer 2 apply blast
908 apply (subgoal_tac "A x Int UNION F A = {}")
909  prefer 2 apply blast
910 apply (simp add: fold_Un_disjoint)
911 done
913 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
914   fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
915   fold_image times (split g) 1 (SIGMA x:A. B x)"
916 apply (subst Sigma_def)
917 apply (subst fold_image_UN_disjoint, assumption, simp)
918  apply blast
919 apply (erule fold_image_cong)
920 apply (subst fold_image_UN_disjoint, simp, simp)
921  apply blast
922 apply simp
923 done
925 lemma fold_image_distrib: "finite A \<Longrightarrow>
926    fold_image times (%x. g x * h x) 1 A =
927    fold_image times g 1 A *  fold_image times h 1 A"
928 by (erule finite_induct) (simp_all add: mult_ac)
930 lemma fold_image_related:
931   assumes Re: "R e e"
932   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
933   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
934   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
935   using fS by (rule finite_subset_induct) (insert assms, auto)
937 lemma  fold_image_eq_general:
938   assumes fS: "finite S"
939   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
940   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
941   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
942 proof-
943   from h f12 have hS: "h ` S = S'" by auto
944   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
945     from f12 h H  have "x = y" by auto }
946   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
947   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
948   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
949   also have "\<dots> = fold_image (op *) (f2 o h) e S"
950     using fold_image_reindex[OF fS hinj, of f2 e] .
951   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
952     by blast
953   finally show ?thesis ..
954 qed
956 lemma fold_image_eq_general_inverses:
957   assumes fS: "finite S"
958   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
959   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
960   shows "fold_image (op *) f e S = fold_image (op *) g e T"
961   (* metis solves it, but not yet available here *)
962   apply (rule fold_image_eq_general[OF fS, of T h g f e])
963   apply (rule ballI)
964   apply (frule kh)
965   apply (rule ex1I[])
966   apply blast
967   apply clarsimp
968   apply (drule hk) apply simp
969   apply (rule sym)
970   apply (erule conjunct1[OF conjunct2[OF hk]])
971   apply (rule ballI)
972   apply (drule  hk)
973   apply blast
974   done
976 end
979 subsection {* A fold functional for non-empty sets *}
981 text{* Does not require start value. *}
983 inductive
984   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
985   for f :: "'a => 'a => 'a"
986 where
987   fold1Set_insertI [intro]:
988    "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
990 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
991   "fold1 f A == THE x. fold1Set f A x"
993 lemma fold1Set_nonempty:
994   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
995 by(erule fold1Set.cases, simp_all)
997 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
999 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
1002 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
1003 by (blast elim: fold_graph.cases)
1005 lemma fold1_singleton [simp]: "fold1 f {a} = a"
1006 by (unfold fold1_def) blast
1008 lemma finite_nonempty_imp_fold1Set:
1009   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
1010 apply (induct A rule: finite_induct)
1011 apply (auto dest: finite_imp_fold_graph [of _ f])
1012 done
1014 text{*First, some lemmas about @{const fold_graph}.*}
1016 context ab_semigroup_mult
1017 begin
1019 lemma comp_fun_commute: "comp_fun_commute (op *)" proof
1020 qed (simp add: fun_eq_iff mult_ac)
1022 lemma fold_graph_insert_swap:
1023 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
1024 shows "fold_graph times z (insert b A) (z * y)"
1025 proof -
1026   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1027 from assms show ?thesis
1028 proof (induct rule: fold_graph.induct)
1029   case emptyI show ?case by (subst mult_commute [of z b], fast)
1030 next
1031   case (insertI x A y)
1032     have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
1033       using insertI by force  --{*how does @{term id} get unfolded?*}
1034     thus ?case by (simp add: insert_commute mult_ac)
1035 qed
1036 qed
1038 lemma fold_graph_permute_diff:
1039 assumes fold: "fold_graph times b A x"
1040 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
1041 using fold
1042 proof (induct rule: fold_graph.induct)
1043   case emptyI thus ?case by simp
1044 next
1045   case (insertI x A y)
1046   have "a = x \<or> a \<in> A" using insertI by simp
1047   thus ?case
1048   proof
1049     assume "a = x"
1050     with insertI show ?thesis
1051       by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
1052   next
1053     assume ainA: "a \<in> A"
1054     hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
1055       using insertI by force
1056     moreover
1057     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
1058       using ainA insertI by blast
1059     ultimately show ?thesis by simp
1060   qed
1061 qed
1063 lemma fold1_eq_fold:
1064 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
1065 proof -
1066   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1067   from assms show ?thesis
1068 apply (simp add: fold1_def fold_def)
1069 apply (rule the_equality)
1070 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
1071 apply (rule sym, clarify)
1072 apply (case_tac "Aa=A")
1073  apply (best intro: fold_graph_determ)
1074 apply (subgoal_tac "fold_graph times a A x")
1075  apply (best intro: fold_graph_determ)
1076 apply (subgoal_tac "insert aa (Aa - {a}) = A")
1077  prefer 2 apply (blast elim: equalityE)
1078 apply (auto dest: fold_graph_permute_diff [where a=a])
1079 done
1080 qed
1082 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
1083 apply safe
1084  apply simp
1085  apply (drule_tac x=x in spec)
1086  apply (drule_tac x="A-{x}" in spec, auto)
1087 done
1089 lemma fold1_insert:
1090   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
1091   shows "fold1 times (insert x A) = x * fold1 times A"
1092 proof -
1093   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1094   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
1095     by (auto simp add: nonempty_iff)
1096   with A show ?thesis
1097     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
1098 qed
1100 end
1102 context ab_semigroup_idem_mult
1103 begin
1105 lemma comp_fun_idem: "comp_fun_idem (op *)" proof
1106 qed (simp_all add: fun_eq_iff mult_left_commute)
1108 lemma fold1_insert_idem [simp]:
1109   assumes nonempty: "A \<noteq> {}" and A: "finite A"
1110   shows "fold1 times (insert x A) = x * fold1 times A"
1111 proof -
1112   interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
1113     by (rule comp_fun_idem)
1114   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
1115     by (auto simp add: nonempty_iff)
1116   show ?thesis
1117   proof cases
1118     assume a: "a = x"
1119     show ?thesis
1120     proof cases
1121       assume "A' = {}"
1122       with A' a show ?thesis by simp
1123     next
1124       assume "A' \<noteq> {}"
1125       with A A' a show ?thesis
1126         by (simp add: fold1_insert mult_assoc [symmetric])
1127     qed
1128   next
1129     assume "a \<noteq> x"
1130     with A A' show ?thesis
1131       by (simp add: insert_commute fold1_eq_fold)
1132   qed
1133 qed
1135 lemma hom_fold1_commute:
1136 assumes hom: "!!x y. h (x * y) = h x * h y"
1137 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
1138 using N proof (induct rule: finite_ne_induct)
1139   case singleton thus ?case by simp
1140 next
1141   case (insert n N)
1142   then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
1143   also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
1144   also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
1145   also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
1146     using insert by(simp)
1147   also have "insert (h n) (h ` N) = h ` insert n N" by simp
1148   finally show ?case .
1149 qed
1151 lemma fold1_eq_fold_idem:
1152   assumes "finite A"
1153   shows "fold1 times (insert a A) = fold times a A"
1154 proof (cases "a \<in> A")
1155   case False
1156   with assms show ?thesis by (simp add: fold1_eq_fold)
1157 next
1158   interpret comp_fun_idem times by (fact comp_fun_idem)
1159   case True then obtain b B
1160     where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
1161   with assms have "finite B" by auto
1162   then have "fold times a (insert a B) = fold times (a * a) B"
1163     using `a \<notin> B` by (rule fold_insert2)
1164   then show ?thesis
1165     using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
1166 qed
1168 end
1171 text{* Now the recursion rules for definitions: *}
1173 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
1174 by simp
1176 lemma (in ab_semigroup_mult) fold1_insert_def:
1177   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1178 by (simp add:fold1_insert)
1180 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
1181   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1182 by simp
1184 subsubsection{* Determinacy for @{term fold1Set} *}
1186 (*Not actually used!!*)
1187 (*
1188 context ab_semigroup_mult
1189 begin
1191 lemma fold_graph_permute:
1192   "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
1193    ==> fold_graph times id a (insert b A) x"
1194 apply (cases "a=b")
1195 apply (auto dest: fold_graph_permute_diff)
1196 done
1198 lemma fold1Set_determ:
1199   "fold1Set times A x ==> fold1Set times A y ==> y = x"
1200 proof (clarify elim!: fold1Set.cases)
1201   fix A x B y a b
1202   assume Ax: "fold_graph times id a A x"
1203   assume By: "fold_graph times id b B y"
1204   assume anotA:  "a \<notin> A"
1205   assume bnotB:  "b \<notin> B"
1206   assume eq: "insert a A = insert b B"
1207   show "y=x"
1208   proof cases
1209     assume same: "a=b"
1210     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
1211     thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
1212   next
1213     assume diff: "a\<noteq>b"
1214     let ?D = "B - {a}"
1215     have B: "B = insert a ?D" and A: "A = insert b ?D"
1216      and aB: "a \<in> B" and bA: "b \<in> A"
1217       using eq anotA bnotB diff by (blast elim!:equalityE)+
1218     with aB bnotB By
1219     have "fold_graph times id a (insert b ?D) y"
1220       by (auto intro: fold_graph_permute simp add: insert_absorb)
1221     moreover
1222     have "fold_graph times id a (insert b ?D) x"
1223       by (simp add: A [symmetric] Ax)
1224     ultimately show ?thesis by (blast intro: fold_graph_determ)
1225   qed
1226 qed
1228 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
1229   by (unfold fold1_def) (blast intro: fold1Set_determ)
1231 end
1232 *)
1234 declare
1235   empty_fold_graphE [rule del]  fold_graph.intros [rule del]
1236   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
1237   -- {* No more proofs involve these relations. *}
1239 subsubsection {* Lemmas about @{text fold1} *}
1241 context ab_semigroup_mult
1242 begin
1244 lemma fold1_Un:
1245 assumes A: "finite A" "A \<noteq> {}"
1246 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
1247        fold1 times (A Un B) = fold1 times A * fold1 times B"
1248 using A by (induct rule: finite_ne_induct)
1249   (simp_all add: fold1_insert mult_assoc)
1251 lemma fold1_in:
1252   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
1253   shows "fold1 times A \<in> A"
1254 using A
1255 proof (induct rule:finite_ne_induct)
1256   case singleton thus ?case by simp
1257 next
1258   case insert thus ?case using elem by (force simp add:fold1_insert)
1259 qed
1261 end
1263 lemma (in ab_semigroup_idem_mult) fold1_Un2:
1264 assumes A: "finite A" "A \<noteq> {}"
1265 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
1266        fold1 times (A Un B) = fold1 times A * fold1 times B"
1267 using A
1268 proof(induct rule:finite_ne_induct)
1269   case singleton thus ?case by simp
1270 next
1271   case insert thus ?case by (simp add: mult_assoc)
1272 qed
1275 subsection {* Locales as mini-packages for fold operations *}
1277 subsubsection {* The natural case *}
1279 locale folding =
1280   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
1281   fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
1282   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
1283   assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
1284 begin
1286 lemma empty [simp]:
1287   "F {} = id"
1288   by (simp add: eq_fold fun_eq_iff)
1290 lemma insert [simp]:
1291   assumes "finite A" and "x \<notin> A"
1292   shows "F (insert x A) = F A \<circ> f x"
1293 proof -
1294   interpret comp_fun_commute f proof
1295   qed (insert comp_fun_commute, simp add: fun_eq_iff)
1296   from fold_insert2 assms
1297   have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
1298   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
1299 qed
1301 lemma remove:
1302   assumes "finite A" and "x \<in> A"
1303   shows "F A = F (A - {x}) \<circ> f x"
1304 proof -
1305   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1306     by (auto dest: mk_disjoint_insert)
1307   moreover from `finite A` this have "finite B" by simp
1308   ultimately show ?thesis by simp
1309 qed
1311 lemma insert_remove:
1312   assumes "finite A"
1313   shows "F (insert x A) = F (A - {x}) \<circ> f x"
1314   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1316 lemma commute_left_comp:
1317   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
1318   by (simp add: o_assoc comp_fun_commute)
1320 lemma comp_fun_commute':
1321   assumes "finite A"
1322   shows "f x \<circ> F A = F A \<circ> f x"
1323   using assms by (induct A)
1324     (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] comp_fun_commute)
1326 lemma commute_left_comp':
1327   assumes "finite A"
1328   shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
1329   using assms by (simp add: o_assoc comp_fun_commute')
1331 lemma comp_fun_commute'':
1332   assumes "finite A" and "finite B"
1333   shows "F B \<circ> F A = F A \<circ> F B"
1334   using assms by (induct A)
1335     (simp_all add: o_assoc, simp add: o_assoc [symmetric] comp_fun_commute')
1337 lemma commute_left_comp'':
1338   assumes "finite A" and "finite B"
1339   shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
1340   using assms by (simp add: o_assoc comp_fun_commute'')
1342 lemmas comp_fun_commutes = o_assoc [symmetric] comp_fun_commute commute_left_comp
1343   comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
1345 lemma union_inter:
1346   assumes "finite A" and "finite B"
1347   shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
1348   using assms by (induct A)
1349     (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
1350       simp add: o_assoc)
1352 lemma union:
1353   assumes "finite A" and "finite B"
1354   and "A \<inter> B = {}"
1355   shows "F (A \<union> B) = F A \<circ> F B"
1356 proof -
1357   from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
1358   with `A \<inter> B = {}` show ?thesis by simp
1359 qed
1361 end
1364 subsubsection {* The natural case with idempotency *}
1366 locale folding_idem = folding +
1367   assumes idem_comp: "f x \<circ> f x = f x"
1368 begin
1370 lemma idem_left_comp:
1371   "f x \<circ> (f x \<circ> g) = f x \<circ> g"
1372   by (simp add: o_assoc idem_comp)
1374 lemma in_comp_idem:
1375   assumes "finite A" and "x \<in> A"
1376   shows "F A \<circ> f x = F A"
1377 using assms by (induct A)
1378   (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
1380 lemma subset_comp_idem:
1381   assumes "finite A" and "B \<subseteq> A"
1382   shows "F A \<circ> F B = F A"
1383 proof -
1384   from assms have "finite B" by (blast dest: finite_subset)
1385   then show ?thesis using `B \<subseteq> A` by (induct B)
1386     (simp_all add: o_assoc in_comp_idem `finite A`)
1387 qed
1389 declare insert [simp del]
1391 lemma insert_idem [simp]:
1392   assumes "finite A"
1393   shows "F (insert x A) = F A \<circ> f x"
1394   using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
1396 lemma union_idem:
1397   assumes "finite A" and "finite B"
1398   shows "F (A \<union> B) = F A \<circ> F B"
1399 proof -
1400   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1401   then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
1402   with assms show ?thesis by (simp add: union_inter)
1403 qed
1405 end
1408 subsubsection {* The image case with fixed function *}
1410 no_notation times (infixl "*" 70)
1411 no_notation Groups.one ("1")
1413 locale folding_image_simple = comm_monoid +
1414   fixes g :: "('b \<Rightarrow> 'a)"
1415   fixes F :: "'b set \<Rightarrow> 'a"
1416   assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
1417 begin
1419 lemma empty [simp]:
1420   "F {} = 1"
1421   by (simp add: eq_fold_g)
1423 lemma insert [simp]:
1424   assumes "finite A" and "x \<notin> A"
1425   shows "F (insert x A) = g x * F A"
1426 proof -
1427   interpret comp_fun_commute "%x y. (g x) * y" proof
1428   qed (simp add: ac_simps fun_eq_iff)
1429   with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
1430     by (simp add: fold_image_def)
1431   with `finite A` show ?thesis by (simp add: eq_fold_g)
1432 qed
1434 lemma remove:
1435   assumes "finite A" and "x \<in> A"
1436   shows "F A = g x * F (A - {x})"
1437 proof -
1438   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1439     by (auto dest: mk_disjoint_insert)
1440   moreover from `finite A` this have "finite B" by simp
1441   ultimately show ?thesis by simp
1442 qed
1444 lemma insert_remove:
1445   assumes "finite A"
1446   shows "F (insert x A) = g x * F (A - {x})"
1447   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1449 lemma neutral:
1450   assumes "finite A" and "\<forall>x\<in>A. g x = 1"
1451   shows "F A = 1"
1452   using assms by (induct A) simp_all
1454 lemma union_inter:
1455   assumes "finite A" and "finite B"
1456   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1457 using assms proof (induct A)
1458   case empty then show ?case by simp
1459 next
1460   case (insert x A) then show ?case
1461     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
1462 qed
1464 corollary union_inter_neutral:
1465   assumes "finite A" and "finite B"
1466   and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
1467   shows "F (A \<union> B) = F A * F B"
1468   using assms by (simp add: union_inter [symmetric] neutral)
1470 corollary union_disjoint:
1471   assumes "finite A" and "finite B"
1472   assumes "A \<inter> B = {}"
1473   shows "F (A \<union> B) = F A * F B"
1474   using assms by (simp add: union_inter_neutral)
1476 end
1479 subsubsection {* The image case with flexible function *}
1481 locale folding_image = comm_monoid +
1482   fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
1483   assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
1485 sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
1486 qed (fact eq_fold)
1488 context folding_image
1489 begin
1491 lemma reindex: (* FIXME polymorhism *)
1492   assumes "finite A" and "inj_on h A"
1493   shows "F g (h ` A) = F (g \<circ> h) A"
1494   using assms by (induct A) auto
1496 lemma cong:
1497   assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
1498   shows "F g A = F h A"
1499 proof -
1500   from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
1501   apply - apply (erule finite_induct) apply simp
1502   apply (simp add: subset_insert_iff, clarify)
1503   apply (subgoal_tac "finite C")
1504   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
1505   apply (subgoal_tac "C = insert x (C - {x})")
1506   prefer 2 apply blast
1507   apply (erule ssubst)
1508   apply (drule spec)
1509   apply (erule (1) notE impE)
1510   apply (simp add: Ball_def del: insert_Diff_single)
1511   done
1512   with assms show ?thesis by simp
1513 qed
1515 lemma UNION_disjoint:
1516   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
1517   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
1518   shows "F g (UNION I A) = F (F g \<circ> A) I"
1519 apply (insert assms)
1520 apply (induct rule: finite_induct)
1521 apply simp
1522 apply atomize
1523 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
1524  prefer 2 apply blast
1525 apply (subgoal_tac "A x Int UNION Fa A = {}")
1526  prefer 2 apply blast
1527 apply (simp add: union_disjoint)
1528 done
1530 lemma distrib:
1531   assumes "finite A"
1532   shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
1533   using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
1535 lemma related:
1536   assumes Re: "R 1 1"
1537   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
1538   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
1539   shows "R (F h S) (F g S)"
1540   using fS by (rule finite_subset_induct) (insert assms, auto)
1542 lemma eq_general:
1543   assumes fS: "finite S"
1544   and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
1545   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
1546   shows "F f1 S = F f2 S'"
1547 proof-
1548   from h f12 have hS: "h ` S = S'" by blast
1549   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1550     from f12 h H  have "x = y" by auto }
1551   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1552   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1553   from hS have "F f2 S' = F f2 (h ` S)" by simp
1554   also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
1555   also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
1556     by blast
1557   finally show ?thesis ..
1558 qed
1560 lemma eq_general_inverses:
1561   assumes fS: "finite S"
1562   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1563   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
1564   shows "F j S = F g T"
1565   (* metis solves it, but not yet available here *)
1566   apply (rule eq_general [OF fS, of T h g j])
1567   apply (rule ballI)
1568   apply (frule kh)
1569   apply (rule ex1I[])
1570   apply blast
1571   apply clarsimp
1572   apply (drule hk) apply simp
1573   apply (rule sym)
1574   apply (erule conjunct1[OF conjunct2[OF hk]])
1575   apply (rule ballI)
1576   apply (drule hk)
1577   apply blast
1578   done
1580 end
1583 subsubsection {* The image case with fixed function and idempotency *}
1585 locale folding_image_simple_idem = folding_image_simple +
1586   assumes idem: "x * x = x"
1588 sublocale folding_image_simple_idem < semilattice proof
1589 qed (fact idem)
1591 context folding_image_simple_idem
1592 begin
1594 lemma in_idem:
1595   assumes "finite A" and "x \<in> A"
1596   shows "g x * F A = F A"
1597   using assms by (induct A) (auto simp add: left_commute)
1599 lemma subset_idem:
1600   assumes "finite A" and "B \<subseteq> A"
1601   shows "F B * F A = F A"
1602 proof -
1603   from assms have "finite B" by (blast dest: finite_subset)
1604   then show ?thesis using `B \<subseteq> A` by (induct B)
1605     (auto simp add: assoc in_idem `finite A`)
1606 qed
1608 declare insert [simp del]
1610 lemma insert_idem [simp]:
1611   assumes "finite A"
1612   shows "F (insert x A) = g x * F A"
1613   using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
1615 lemma union_idem:
1616   assumes "finite A" and "finite B"
1617   shows "F (A \<union> B) = F A * F B"
1618 proof -
1619   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1620   then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
1621   with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1622 qed
1624 end
1627 subsubsection {* The image case with flexible function and idempotency *}
1629 locale folding_image_idem = folding_image +
1630   assumes idem: "x * x = x"
1632 sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
1633 qed (fact idem)
1636 subsubsection {* The neutral-less case *}
1638 locale folding_one = abel_semigroup +
1639   fixes F :: "'a set \<Rightarrow> 'a"
1640   assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
1641 begin
1643 lemma singleton [simp]:
1644   "F {x} = x"
1645   by (simp add: eq_fold)
1647 lemma eq_fold':
1648   assumes "finite A" and "x \<notin> A"
1649   shows "F (insert x A) = fold (op *) x A"
1650 proof -
1651   interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
1652   with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
1653 qed
1655 lemma insert [simp]:
1656   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
1657   shows "F (insert x A) = x * F A"
1658 proof -
1659   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
1660   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
1661   with `finite A` have "finite B" by simp
1662   interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
1663   qed (simp_all add: fun_eq_iff ac_simps)
1664   thm fold.comp_fun_commute' [of B b, simplified fun_eq_iff, simplified]
1665   from `finite B` fold.comp_fun_commute' [of B x]
1666     have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
1667   then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
1668   from `finite B` * fold.insert [of B b]
1669     have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
1670   then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
1671   from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
1672 qed
1674 lemma remove:
1675   assumes "finite A" and "x \<in> A"
1676   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
1677 proof -
1678   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
1679   with assms show ?thesis by simp
1680 qed
1682 lemma insert_remove:
1683   assumes "finite A"
1684   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
1685   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
1687 lemma union_disjoint:
1688   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
1689   shows "F (A \<union> B) = F A * F B"
1690   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
1692 lemma union_inter:
1693   assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
1694   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1695 proof -
1696   from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
1697   from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
1698     case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
1699   next
1700     case (insert x A) show ?case proof (cases "x \<in> B")
1701       case True then have "B \<noteq> {}" by auto
1702       with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
1703         (simp_all add: insert_absorb ac_simps union_disjoint)
1704     next
1705       case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
1706       moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
1707         by auto
1708       ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
1709     qed
1710   qed
1711 qed
1713 lemma closed:
1714   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
1715   shows "F A \<in> A"
1716 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
1717   case singleton then show ?case by simp
1718 next
1719   case insert with elem show ?case by force
1720 qed
1722 end
1725 subsubsection {* The neutral-less case with idempotency *}
1727 locale folding_one_idem = folding_one +
1728   assumes idem: "x * x = x"
1730 sublocale folding_one_idem < semilattice proof
1731 qed (fact idem)
1733 context folding_one_idem
1734 begin
1736 lemma in_idem:
1737   assumes "finite A" and "x \<in> A"
1738   shows "x * F A = F A"
1739 proof -
1740   from assms have "A \<noteq> {}" by auto
1741   with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
1742 qed
1744 lemma subset_idem:
1745   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
1746   shows "F B * F A = F A"
1747 proof -
1748   from assms have "finite B" by (blast dest: finite_subset)
1749   then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
1750     (simp_all add: assoc in_idem `finite A`)
1751 qed
1753 lemma eq_fold_idem':
1754   assumes "finite A"
1755   shows "F (insert a A) = fold (op *) a A"
1756 proof -
1757   interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
1758   with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
1759 qed
1761 lemma insert_idem [simp]:
1762   assumes "finite A" and "A \<noteq> {}"
1763   shows "F (insert x A) = x * F A"
1764 proof (cases "x \<in> A")
1765   case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
1766 next
1767   case True
1768   from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
1769 qed
1771 lemma union_idem:
1772   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
1773   shows "F (A \<union> B) = F A * F B"
1774 proof (cases "A \<inter> B = {}")
1775   case True with assms show ?thesis by (simp add: union_disjoint)
1776 next
1777   case False
1778   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1779   with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
1780   with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1781 qed
1783 lemma hom_commute:
1784   assumes hom: "\<And>x y. h (x * y) = h x * h y"
1785   and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
1786 using N proof (induct rule: finite_ne_induct)
1787   case singleton thus ?case by simp
1788 next
1789   case (insert n N)
1790   then have "h (F (insert n N)) = h (n * F N)" by simp
1791   also have "\<dots> = h n * h (F N)" by (rule hom)
1792   also have "h (F N) = F (h ` N)" by(rule insert)
1793   also have "h n * \<dots> = F (insert (h n) (h ` N))"
1794     using insert by(simp)
1795   also have "insert (h n) (h ` N) = h ` insert n N" by simp
1796   finally show ?case .
1797 qed
1799 end
1801 notation times (infixl "*" 70)
1802 notation Groups.one ("1")
1805 subsection {* Finite cardinality *}
1807 text {* This definition, although traditional, is ugly to work with:
1808 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
1809 But now that we have @{text fold_image} things are easy:
1810 *}
1812 definition card :: "'a set \<Rightarrow> nat" where
1813   "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
1815 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
1816 qed (simp add: card_def)
1818 lemma card_infinite [simp]:
1819   "\<not> finite A \<Longrightarrow> card A = 0"
1820   by (simp add: card_def)
1822 lemma card_empty:
1823   "card {} = 0"
1824   by (fact card.empty)
1826 lemma card_insert_disjoint:
1827   "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
1828   by simp
1830 lemma card_insert_if:
1831   "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
1832   by auto (simp add: card.insert_remove card.remove)
1834 lemma card_ge_0_finite:
1835   "card A > 0 \<Longrightarrow> finite A"
1836   by (rule ccontr) simp
1838 lemma card_0_eq [simp, no_atp]:
1839   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
1840   by (auto dest: mk_disjoint_insert)
1842 lemma finite_UNIV_card_ge_0:
1843   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
1844   by (rule ccontr) simp
1846 lemma card_eq_0_iff:
1847   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
1848   by auto
1850 lemma card_gt_0_iff:
1851   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
1852   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
1854 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
1855 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
1856 apply(simp del:insert_Diff_single)
1857 done
1859 lemma card_Diff_singleton:
1860   "finite A ==> x: A ==> card (A - {x}) = card A - 1"
1861 by (simp add: card_Suc_Diff1 [symmetric])
1863 lemma card_Diff_singleton_if:
1864   "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
1865 by (simp add: card_Diff_singleton)
1867 lemma card_Diff_insert[simp]:
1868 assumes "finite A" and "a:A" and "a ~: B"
1869 shows "card(A - insert a B) = card(A - B) - 1"
1870 proof -
1871   have "A - insert a B = (A - B) - {a}" using assms by blast
1872   then show ?thesis using assms by(simp add:card_Diff_singleton)
1873 qed
1875 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
1876 by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
1878 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
1879 by (simp add: card_insert_if)
1881 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
1882 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
1884 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
1885 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
1887 lemma card_mono:
1888   assumes "finite B" and "A \<subseteq> B"
1889   shows "card A \<le> card B"
1890 proof -
1891   from assms have "finite A" by (auto intro: finite_subset)
1892   then show ?thesis using assms proof (induct A arbitrary: B)
1893     case empty then show ?case by simp
1894   next
1895     case (insert x A)
1896     then have "x \<in> B" by simp
1897     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
1898     with insert.hyps have "card A \<le> card (B - {x})" by auto
1899     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
1900   qed
1901 qed
1903 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
1904 apply (induct rule: finite_induct)
1905 apply simp
1906 apply clarify
1907 apply (subgoal_tac "finite A & A - {x} <= F")
1908  prefer 2 apply (blast intro: finite_subset, atomize)
1909 apply (drule_tac x = "A - {x}" in spec)
1910 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
1911 apply (case_tac "card A", auto)
1912 done
1914 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
1915 apply (simp add: psubset_eq linorder_not_le [symmetric])
1916 apply (blast dest: card_seteq)
1917 done
1919 lemma card_Un_Int: "finite A ==> finite B
1920     ==> card A + card B = card (A Un B) + card (A Int B)"
1921   by (fact card.union_inter [symmetric])
1923 lemma card_Un_disjoint: "finite A ==> finite B
1924     ==> A Int B = {} ==> card (A Un B) = card A + card B"
1925   by (fact card.union_disjoint)
1927 lemma card_Diff_subset:
1928   assumes "finite B" and "B \<subseteq> A"
1929   shows "card (A - B) = card A - card B"
1930 proof (cases "finite A")
1931   case False with assms show ?thesis by simp
1932 next
1933   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
1934 qed
1936 lemma card_Diff_subset_Int:
1937   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
1938 proof -
1939   have "A - B = A - A \<inter> B" by auto
1940   thus ?thesis
1941     by (simp add: card_Diff_subset AB)
1942 qed
1944 lemma diff_card_le_card_Diff:
1945 assumes "finite B" shows "card A - card B \<le> card(A - B)"
1946 proof-
1947   have "card A - card B \<le> card A - card (A \<inter> B)"
1948     using card_mono[OF assms Int_lower2, of A] by arith
1949   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
1950   finally show ?thesis .
1951 qed
1953 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
1954 apply (rule Suc_less_SucD)
1955 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
1956 done
1958 lemma card_Diff2_less:
1959   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
1960 apply (case_tac "x = y")
1961  apply (simp add: card_Diff1_less del:card_Diff_insert)
1962 apply (rule less_trans)
1963  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
1964 done
1966 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
1967 apply (case_tac "x : A")
1968  apply (simp_all add: card_Diff1_less less_imp_le)
1969 done
1971 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
1972 by (erule psubsetI, blast)
1974 lemma insert_partition:
1975   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
1976   \<Longrightarrow> x \<inter> \<Union> F = {}"
1977 by auto
1979 lemma finite_psubset_induct[consumes 1, case_names psubset]:
1980   assumes fin: "finite A"
1981   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
1982   shows "P A"
1983 using fin
1984 proof (induct A taking: card rule: measure_induct_rule)
1985   case (less A)
1986   have fin: "finite A" by fact
1987   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
1988   { fix B
1989     assume asm: "B \<subset> A"
1990     from asm have "card B < card A" using psubset_card_mono fin by blast
1991     moreover
1992     from asm have "B \<subseteq> A" by auto
1993     then have "finite B" using fin finite_subset by blast
1994     ultimately
1995     have "P B" using ih by simp
1996   }
1997   with fin show "P A" using major by blast
1998 qed
2000 text{* main cardinality theorem *}
2001 lemma card_partition [rule_format]:
2002   "finite C ==>
2003      finite (\<Union> C) -->
2004      (\<forall>c\<in>C. card c = k) -->
2005      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
2006      k * card(C) = card (\<Union> C)"
2007 apply (erule finite_induct, simp)
2008 apply (simp add: card_Un_disjoint insert_partition
2009        finite_subset [of _ "\<Union> (insert x F)"])
2010 done
2012 lemma card_eq_UNIV_imp_eq_UNIV:
2013   assumes fin: "finite (UNIV :: 'a set)"
2014   and card: "card A = card (UNIV :: 'a set)"
2015   shows "A = (UNIV :: 'a set)"
2016 proof
2017   show "A \<subseteq> UNIV" by simp
2018   show "UNIV \<subseteq> A"
2019   proof
2020     fix x
2021     show "x \<in> A"
2022     proof (rule ccontr)
2023       assume "x \<notin> A"
2024       then have "A \<subset> UNIV" by auto
2025       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
2026       with card show False by simp
2027     qed
2028   qed
2029 qed
2031 text{*The form of a finite set of given cardinality*}
2033 lemma card_eq_SucD:
2034 assumes "card A = Suc k"
2035 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
2036 proof -
2037   have fin: "finite A" using assms by (auto intro: ccontr)
2038   moreover have "card A \<noteq> 0" using assms by auto
2039   ultimately obtain b where b: "b \<in> A" by auto
2040   show ?thesis
2041   proof (intro exI conjI)
2042     show "A = insert b (A-{b})" using b by blast
2043     show "b \<notin> A - {b}" by blast
2044     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
2045       using assms b fin by(fastsimp dest:mk_disjoint_insert)+
2046   qed
2047 qed
2049 lemma card_Suc_eq:
2050   "(card A = Suc k) =
2051    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
2052 apply(rule iffI)
2053  apply(erule card_eq_SucD)
2054 apply(auto)
2055 apply(subst card_insert)
2056  apply(auto intro:ccontr)
2057 done
2059 lemma finite_fun_UNIVD2:
2060   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
2061   shows "finite (UNIV :: 'b set)"
2062 proof -
2063   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
2064     by(rule finite_imageI)
2065   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
2066     by(rule UNIV_eq_I) auto
2067   ultimately show "finite (UNIV :: 'b set)" by simp
2068 qed
2070 lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
2071   unfolding UNIV_unit by simp
2074 subsubsection {* Cardinality of image *}
2076 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
2077 apply (induct rule: finite_induct)
2078  apply simp
2079 apply (simp add: le_SucI card_insert_if)
2080 done
2082 lemma card_image:
2083   assumes "inj_on f A"
2084   shows "card (f ` A) = card A"
2085 proof (cases "finite A")
2086   case True then show ?thesis using assms by (induct A) simp_all
2087 next
2088   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
2089   with False show ?thesis by simp
2090 qed
2092 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
2093 by(auto simp: card_image bij_betw_def)
2095 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
2096 by (simp add: card_seteq card_image)
2098 lemma eq_card_imp_inj_on:
2099   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
2100 apply (induct rule:finite_induct)
2101 apply simp
2102 apply(frule card_image_le[where f = f])
2103 apply(simp add:card_insert_if split:if_splits)
2104 done
2106 lemma inj_on_iff_eq_card:
2107   "finite A ==> inj_on f A = (card(f ` A) = card A)"
2108 by(blast intro: card_image eq_card_imp_inj_on)
2111 lemma card_inj_on_le:
2112   "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
2113 apply (subgoal_tac "finite A")
2114  apply (force intro: card_mono simp add: card_image [symmetric])
2115 apply (blast intro: finite_imageD dest: finite_subset)
2116 done
2118 lemma card_bij_eq:
2119   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
2120      finite A; finite B |] ==> card A = card B"
2121 by (auto intro: le_antisym card_inj_on_le)
2123 lemma bij_betw_finite:
2124   assumes "bij_betw f A B"
2125   shows "finite A \<longleftrightarrow> finite B"
2126 using assms unfolding bij_betw_def
2127 using finite_imageD[of f A] by auto
2130 subsubsection {* Pigeonhole Principles *}
2132 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
2133 by (auto dest: card_image less_irrefl_nat)
2135 lemma pigeonhole_infinite:
2136 assumes  "~ finite A" and "finite(f`A)"
2137 shows "EX a0:A. ~finite{a:A. f a = f a0}"
2138 proof -
2139   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
2140   proof(induct "f`A" arbitrary: A rule: finite_induct)
2141     case empty thus ?case by simp
2142   next
2143     case (insert b F)
2144     show ?case
2145     proof cases
2146       assume "finite{a:A. f a = b}"
2147       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
2148       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
2149       finally have "~ finite({a:A. f a \<noteq> b})" .
2150       from insert(3)[OF _ this]
2151       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
2152     next
2153       assume 1: "~finite{a:A. f a = b}"
2154       hence "{a \<in> A. f a = b} \<noteq> {}" by force
2155       thus ?thesis using 1 by blast
2156     qed
2157   qed
2158   from this[OF assms(2,1)] show ?thesis .
2159 qed
2161 lemma pigeonhole_infinite_rel:
2162 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
2163 shows "EX b:B. ~finite{a:A. R a b}"
2164 proof -
2165    let ?F = "%a. {b:B. R a b}"
2166    from finite_Pow_iff[THEN iffD2, OF `finite B`]
2167    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
2168    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
2169    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
2170    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
2171    { assume "finite{a:A. R a b0}"
2172      then have "finite {a\<in>A. ?F a = ?F a0}"
2173        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
2174    }
2175    with 1 `b0 : B` show ?thesis by blast
2176 qed
2179 subsubsection {* Cardinality of sums *}
2181 lemma card_Plus:
2182   assumes "finite A" and "finite B"
2183   shows "card (A <+> B) = card A + card B"
2184 proof -
2185   have "Inl`A \<inter> Inr`B = {}" by fast
2186   with assms show ?thesis
2187     unfolding Plus_def
2188     by (simp add: card_Un_disjoint card_image)
2189 qed
2191 lemma card_Plus_conv_if:
2192   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
2193   by (auto simp add: card_Plus)
2196 subsubsection {* Cardinality of the Powerset *}
2198 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
2199 apply (induct rule: finite_induct)
2200  apply (simp_all add: Pow_insert)
2201 apply (subst card_Un_disjoint, blast)
2202   apply (blast, blast)
2203 apply (subgoal_tac "inj_on (insert x) (Pow F)")
2204  apply (simp add: card_image Pow_insert)
2205 apply (unfold inj_on_def)
2206 apply (blast elim!: equalityE)
2207 done
2209 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
2211 lemma dvd_partition:
2212   "finite (Union C) ==>
2213     ALL c : C. k dvd card c ==>
2214     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
2215   k dvd card (Union C)"
2216 apply (frule finite_UnionD)
2217 apply (rotate_tac -1)
2218 apply (induct rule: finite_induct)
2219 apply simp_all
2220 apply clarify
2221 apply (subst card_Un_disjoint)
2222    apply (auto simp add: disjoint_eq_subset_Compl)
2223 done
2226 subsubsection {* Relating injectivity and surjectivity *}
2228 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
2229 apply(rule eq_card_imp_inj_on, assumption)
2230 apply(frule finite_imageI)
2231 apply(drule (1) card_seteq)
2232  apply(erule card_image_le)
2233 apply simp
2234 done
2236 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
2237 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
2238 by (blast intro: finite_surj_inj subset_UNIV)
2240 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
2241 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
2242 by(fastsimp simp:surj_def dest!: endo_inj_surj)
2244 corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
2245 proof
2246   assume "finite(UNIV::nat set)"
2247   with finite_UNIV_inj_surj[of Suc]
2248   show False by simp (blast dest: Suc_neq_Zero surjD)
2249 qed
2251 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
2252 lemma infinite_UNIV_char_0[no_atp]:
2253   "\<not> finite (UNIV::'a::semiring_char_0 set)"
2254 proof
2255   assume "finite (UNIV::'a set)"
2256   with subset_UNIV have "finite (range of_nat::'a set)"
2257     by (rule finite_subset)
2258   moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
2259     by (simp add: inj_on_def)
2260   ultimately have "finite (UNIV::nat set)"
2261     by (rule finite_imageD)
2262   then show "False"
2263     by simp
2264 qed
2266 end