src/HOL/Finite_Set.thy
 author haftmann Sun Mar 16 18:09:04 2014 +0100 (2014-03-16) changeset 56166 9a241bc276cd parent 56154 f0a927235162 child 56218 1c3f1f2431f9 permissions -rw-r--r--
normalising simp rules for compound operators
1 (*  Title:      HOL/Finite_Set.thy
2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
3                 with contributions by Jeremy Avigad and Andrei Popescu
4 *)
6 header {* Finite sets *}
8 theory Finite_Set
9 imports Product_Type Sum_Type Nat
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 simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}
21 declare [[simproc del: finite_Collect]]
23 lemma finite_induct [case_names empty insert, induct set: finite]:
24   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
25   assumes "finite F"
26   assumes "P {}"
27     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
28   shows "P F"
29 using `finite F`
30 proof induct
31   show "P {}" by fact
32   fix x F assume F: "finite F" and P: "P F"
33   show "P (insert x F)"
34   proof cases
35     assume "x \<in> F"
36     hence "insert x F = F" by (rule insert_absorb)
37     with P show ?thesis by (simp only:)
38   next
39     assume "x \<notin> F"
40     from F this P show ?thesis by (rule insert)
41   qed
42 qed
44 lemma infinite_finite_induct [case_names infinite empty insert]:
45   assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
46   assumes empty: "P {}"
47   assumes insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
48   shows "P A"
49 proof (cases "finite A")
50   case False with infinite show ?thesis .
51 next
52   case True then show ?thesis by (induct A) (fact empty insert)+
53 qed
56 subsubsection {* Choice principles *}
58 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
59   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
60   shows "\<exists>a::'a. a \<notin> A"
61 proof -
62   from assms have "A \<noteq> UNIV" by blast
63   then show ?thesis by blast
64 qed
66 text {* A finite choice principle. Does not need the SOME choice operator. *}
68 lemma finite_set_choice:
69   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
70 proof (induct rule: finite_induct)
71   case empty then show ?case by simp
72 next
73   case (insert a A)
74   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
75   show ?case (is "EX f. ?P f")
76   proof
77     show "?P(%x. if x = a then b else f x)" using f ab by auto
78   qed
79 qed
82 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
84 lemma finite_imp_nat_seg_image_inj_on:
85   assumes "finite A"
86   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
87 using assms
88 proof induct
89   case empty
90   show ?case
91   proof
92     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
93   qed
94 next
95   case (insert a A)
96   have notinA: "a \<notin> A" by fact
97   from insert.hyps obtain n f
98     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
99   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
100         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
101     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
102   thus ?case by blast
103 qed
105 lemma nat_seg_image_imp_finite:
106   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
107 proof (induct n arbitrary: A)
108   case 0 thus ?case by simp
109 next
110   case (Suc n)
111   let ?B = "f ` {i. i < n}"
112   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
113   show ?case
114   proof cases
115     assume "\<exists>k<n. f n = f k"
116     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
117     thus ?thesis using finB by simp
118   next
119     assume "\<not>(\<exists> k<n. f n = f k)"
120     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
121     thus ?thesis using finB by simp
122   qed
123 qed
125 lemma finite_conv_nat_seg_image:
126   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
127   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
129 lemma finite_imp_inj_to_nat_seg:
130   assumes "finite A"
131   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
132 proof -
133   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
134   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
135     by (auto simp:bij_betw_def)
136   let ?f = "the_inv_into {i. i<n} f"
137   have "inj_on ?f A & ?f ` A = {i. i<n}"
138     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
139   thus ?thesis by blast
140 qed
142 lemma finite_Collect_less_nat [iff]:
143   "finite {n::nat. n < k}"
144   by (fastforce simp: finite_conv_nat_seg_image)
146 lemma finite_Collect_le_nat [iff]:
147   "finite {n::nat. n \<le> k}"
148   by (simp add: le_eq_less_or_eq Collect_disj_eq)
151 subsubsection {* Finiteness and common set operations *}
153 lemma rev_finite_subset:
154   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
155 proof (induct arbitrary: A rule: finite_induct)
156   case empty
157   then show ?case by simp
158 next
159   case (insert x F A)
160   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
161   show "finite A"
162   proof cases
163     assume x: "x \<in> A"
164     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
165     with r have "finite (A - {x})" .
166     hence "finite (insert x (A - {x}))" ..
167     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
168     finally show ?thesis .
169   next
170     show "A \<subseteq> F ==> ?thesis" by fact
171     assume "x \<notin> A"
172     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
173   qed
174 qed
176 lemma finite_subset:
177   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
178   by (rule rev_finite_subset)
180 lemma finite_UnI:
181   assumes "finite F" and "finite G"
182   shows "finite (F \<union> G)"
183   using assms by induct simp_all
185 lemma finite_Un [iff]:
186   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
187   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
189 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
190 proof -
191   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
192   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
193   then show ?thesis by simp
194 qed
196 lemma finite_Int [simp, intro]:
197   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
198   by (blast intro: finite_subset)
200 lemma finite_Collect_conjI [simp, intro]:
201   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
204 lemma finite_Collect_disjI [simp]:
205   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
208 lemma finite_Diff [simp, intro]:
209   "finite A \<Longrightarrow> finite (A - B)"
210   by (rule finite_subset, rule Diff_subset)
212 lemma finite_Diff2 [simp]:
213   assumes "finite B"
214   shows "finite (A - B) \<longleftrightarrow> finite A"
215 proof -
216   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
217   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
218   finally show ?thesis ..
219 qed
221 lemma finite_Diff_insert [iff]:
222   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
223 proof -
224   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
225   moreover have "A - insert a B = A - B - {a}" by auto
226   ultimately show ?thesis by simp
227 qed
229 lemma finite_compl[simp]:
230   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
233 lemma finite_Collect_not[simp]:
234   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
237 lemma finite_Union [simp, intro]:
238   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
239   by (induct rule: finite_induct) simp_all
241 lemma finite_UN_I [intro]:
242   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
243   by (induct rule: finite_induct) simp_all
245 lemma finite_UN [simp]:
246   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
247   by (blast intro: finite_subset)
249 lemma finite_Inter [intro]:
250   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
251   by (blast intro: Inter_lower finite_subset)
253 lemma finite_INT [intro]:
254   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
255   by (blast intro: INT_lower finite_subset)
257 lemma finite_imageI [simp, intro]:
258   "finite F \<Longrightarrow> finite (h ` F)"
259   by (induct rule: finite_induct) simp_all
261 lemma finite_image_set [simp]:
262   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
263   by (simp add: image_Collect [symmetric])
265 lemma finite_imageD:
266   assumes "finite (f ` A)" and "inj_on f A"
267   shows "finite A"
268 using assms
269 proof (induct "f ` A" arbitrary: A)
270   case empty then show ?case by simp
271 next
272   case (insert x B)
273   then have B_A: "insert x B = f ` A" by simp
274   then obtain y where "x = f y" and "y \<in> A" by blast
275   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
276   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)
277   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
278   ultimately have "finite (A - {y})" by (rule insert.hyps)
279   then show "finite A" by simp
280 qed
282 lemma finite_surj:
283   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
284   by (erule finite_subset) (rule finite_imageI)
286 lemma finite_range_imageI:
287   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
288   by (drule finite_imageI) (simp add: range_composition)
290 lemma finite_subset_image:
291   assumes "finite B"
292   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
293 using assms
294 proof induct
295   case empty then show ?case by simp
296 next
297   case insert then show ?case
298     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
299        blast
300 qed
302 lemma finite_vimage_IntI:
303   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
304   apply (induct rule: finite_induct)
305    apply simp_all
306   apply (subst vimage_insert)
307   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
308   done
310 lemma finite_vimageI:
311   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
312   using finite_vimage_IntI[of F h UNIV] by auto
314 lemma finite_vimageD:
315   assumes fin: "finite (h -` F)" and surj: "surj h"
316   shows "finite F"
317 proof -
318   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
319   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
320   finally show "finite F" .
321 qed
323 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
324   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
326 lemma finite_Collect_bex [simp]:
327   assumes "finite A"
328   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
329 proof -
330   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
331   with assms show ?thesis by simp
332 qed
334 lemma finite_Collect_bounded_ex [simp]:
335   assumes "finite {y. P y}"
336   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
337 proof -
338   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
339   with assms show ?thesis by simp
340 qed
342 lemma finite_Plus:
343   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
346 lemma finite_PlusD:
347   fixes A :: "'a set" and B :: "'b set"
348   assumes fin: "finite (A <+> B)"
349   shows "finite A" "finite B"
350 proof -
351   have "Inl ` A \<subseteq> A <+> B" by auto
352   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
353   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
354 next
355   have "Inr ` B \<subseteq> A <+> B" by auto
356   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
357   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
358 qed
360 lemma finite_Plus_iff [simp]:
361   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
362   by (auto intro: finite_PlusD finite_Plus)
364 lemma finite_Plus_UNIV_iff [simp]:
365   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
366   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
368 lemma finite_SigmaI [simp, intro]:
369   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
370   by (unfold Sigma_def) blast
372 lemma finite_SigmaI2:
373   assumes "finite {x\<in>A. B x \<noteq> {}}"
374   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
375   shows "finite (Sigma A B)"
376 proof -
377   from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
378   also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
379   finally show ?thesis .
380 qed
382 lemma finite_cartesian_product:
383   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
384   by (rule finite_SigmaI)
386 lemma finite_Prod_UNIV:
387   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
388   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
390 lemma finite_cartesian_productD1:
391   assumes "finite (A \<times> B)" and "B \<noteq> {}"
392   shows "finite A"
393 proof -
394   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
395     by (auto simp add: finite_conv_nat_seg_image)
396   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
397   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
399   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
400   then show ?thesis
401     by (auto simp add: finite_conv_nat_seg_image)
402 qed
404 lemma finite_cartesian_productD2:
405   assumes "finite (A \<times> B)" and "A \<noteq> {}"
406   shows "finite B"
407 proof -
408   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
409     by (auto simp add: finite_conv_nat_seg_image)
410   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
411   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
413   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
414   then show ?thesis
415     by (auto simp add: finite_conv_nat_seg_image)
416 qed
418 lemma finite_prod:
419   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
420 by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV
421    dest: finite_cartesian_productD1 finite_cartesian_productD2)
423 lemma finite_Pow_iff [iff]:
424   "finite (Pow A) \<longleftrightarrow> finite A"
425 proof
426   assume "finite (Pow A)"
427   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
428   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
429 next
430   assume "finite A"
431   then show "finite (Pow A)"
432     by induct (simp_all add: Pow_insert)
433 qed
435 corollary finite_Collect_subsets [simp, intro]:
436   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
437   by (simp add: Pow_def [symmetric])
439 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
440 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
442 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
443   by (blast intro: finite_subset [OF subset_Pow_Union])
445 lemma finite_set_of_finite_funs: assumes "finite A" "finite B"
446 shows "finite{f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
447 proof-
448   let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
449   have "?F ` ?S \<subseteq> Pow(A \<times> B)" by auto
450   from finite_subset[OF this] assms have 1: "finite (?F ` ?S)" by simp
451   have 2: "inj_on ?F ?S"
452     by(fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
453   show ?thesis by(rule finite_imageD[OF 1 2])
454 qed
456 subsubsection {* Further induction rules on finite sets *}
458 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
459   assumes "finite F" and "F \<noteq> {}"
460   assumes "\<And>x. P {x}"
461     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
462   shows "P F"
463 using assms
464 proof induct
465   case empty then show ?case by simp
466 next
467   case (insert x F) then show ?case by cases auto
468 qed
470 lemma finite_subset_induct [consumes 2, case_names empty insert]:
471   assumes "finite F" and "F \<subseteq> A"
472   assumes empty: "P {}"
473     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
474   shows "P F"
475 using `finite F` `F \<subseteq> A`
476 proof induct
477   show "P {}" by fact
478 next
479   fix x F
480   assume "finite F" and "x \<notin> F" and
481     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
482   show "P (insert x F)"
483   proof (rule insert)
484     from i show "x \<in> A" by blast
485     from i have "F \<subseteq> A" by blast
486     with P show "P F" .
487     show "finite F" by fact
488     show "x \<notin> F" by fact
489   qed
490 qed
492 lemma finite_empty_induct:
493   assumes "finite A"
494   assumes "P A"
495     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
496   shows "P {}"
497 proof -
498   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
499   proof -
500     fix B :: "'a set"
501     assume "B \<subseteq> A"
502     with `finite A` have "finite B" by (rule rev_finite_subset)
503     from this `B \<subseteq> A` show "P (A - B)"
504     proof induct
505       case empty
506       from `P A` show ?case by simp
507     next
508       case (insert b B)
509       have "P (A - B - {b})"
510       proof (rule remove)
511         from `finite A` show "finite (A - B)" by induct auto
512         from insert show "b \<in> A - B" by simp
513         from insert show "P (A - B)" by simp
514       qed
515       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
516       finally show ?case .
517     qed
518   qed
519   then have "P (A - A)" by blast
520   then show ?thesis by simp
521 qed
523 subsection {* Class @{text finite}  *}
525 class finite =
526   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
527 begin
529 lemma finite [simp]: "finite (A \<Colon> 'a set)"
530   by (rule subset_UNIV finite_UNIV finite_subset)+
532 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
533   by simp
535 end
537 instance prod :: (finite, finite) finite
538   by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
540 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
541   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
543 instance "fun" :: (finite, finite) finite
544 proof
545   show "finite (UNIV :: ('a => 'b) set)"
546   proof (rule finite_imageD)
547     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
548     have "range ?graph \<subseteq> Pow UNIV" by simp
549     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
550       by (simp only: finite_Pow_iff finite)
551     ultimately show "finite (range ?graph)"
552       by (rule finite_subset)
553     show "inj ?graph" by (rule inj_graph)
554   qed
555 qed
557 instance bool :: finite
558   by default (simp add: UNIV_bool)
560 instance set :: (finite) finite
561   by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
563 instance unit :: finite
564   by default (simp add: UNIV_unit)
566 instance sum :: (finite, finite) finite
567   by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
570 subsection {* A basic fold functional for finite sets *}
572 text {* The intended behaviour is
573 @{text "fold f z {x\<^sub>1, ..., x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)"}
574 if @{text f} is ``left-commutative'':
575 *}
577 locale comp_fun_commute =
578   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
579   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
580 begin
582 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
583   using comp_fun_commute by (simp add: fun_eq_iff)
585 lemma commute_left_comp:
586   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
587   by (simp add: o_assoc comp_fun_commute)
589 end
591 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
592 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
593   emptyI [intro]: "fold_graph f z {} z" |
594   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
595       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
597 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
599 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
600   "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
602 text{*A tempting alternative for the definiens is
603 @{term "if finite A then THE y. fold_graph f z A y else e"}.
604 It allows the removal of finiteness assumptions from the theorems
605 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
606 The proofs become ugly. It is not worth the effort. (???) *}
608 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
609 by (induct rule: finite_induct) auto
612 subsubsection{*From @{const fold_graph} to @{term fold}*}
614 context comp_fun_commute
615 begin
617 lemma fold_graph_finite:
618   assumes "fold_graph f z A y"
619   shows "finite A"
620   using assms by induct simp_all
622 lemma fold_graph_insertE_aux:
623   "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'"
624 proof (induct set: fold_graph)
625   case (insertI x A y) show ?case
626   proof (cases "x = a")
627     assume "x = a" with insertI show ?case by auto
628   next
629     assume "x \<noteq> a"
630     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
631       using insertI by auto
632     have "f x y = f a (f x y')"
633       unfolding y by (rule fun_left_comm)
634     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
635       using y' and `x \<noteq> a` and `x \<notin> A`
636       by (simp add: insert_Diff_if fold_graph.insertI)
637     ultimately show ?case by fast
638   qed
639 qed simp
641 lemma fold_graph_insertE:
642   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
643   obtains y where "v = f x y" and "fold_graph f z A y"
644 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
646 lemma fold_graph_determ:
647   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
648 proof (induct arbitrary: y set: fold_graph)
649   case (insertI x A y v)
650   from `fold_graph f z (insert x A) v` and `x \<notin> A`
651   obtain y' where "v = f x y'" and "fold_graph f z A y'"
652     by (rule fold_graph_insertE)
653   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
654   with `v = f x y'` show "v = f x y" by simp
655 qed fast
657 lemma fold_equality:
658   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
659   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
661 lemma fold_graph_fold:
662   assumes "finite A"
663   shows "fold_graph f z A (fold f z A)"
664 proof -
665   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
666   moreover note fold_graph_determ
667   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
668   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
669   with assms show ?thesis by (simp add: fold_def)
670 qed
672 text {* The base case for @{text fold}: *}
674 lemma (in -) fold_infinite [simp]:
675   assumes "\<not> finite A"
676   shows "fold f z A = z"
677   using assms by (auto simp add: fold_def)
679 lemma (in -) fold_empty [simp]:
680   "fold f z {} = z"
681   by (auto simp add: fold_def)
683 text{* The various recursion equations for @{const fold}: *}
685 lemma fold_insert [simp]:
686   assumes "finite A" and "x \<notin> A"
687   shows "fold f z (insert x A) = f x (fold f z A)"
688 proof (rule fold_equality)
689   fix z
690   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
691   with `x \<notin> A` have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
692   then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
693 qed
695 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
696   -- {* No more proofs involve these. *}
698 lemma fold_fun_left_comm:
699   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
700 proof (induct rule: finite_induct)
701   case empty then show ?case by simp
702 next
703   case (insert y A) then show ?case
704     by (simp add: fun_left_comm [of x])
705 qed
707 lemma fold_insert2:
708   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
711 lemma fold_rec:
712   assumes "finite A" and "x \<in> A"
713   shows "fold f z A = f x (fold f z (A - {x}))"
714 proof -
715   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
716   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
717   also have "\<dots> = f x (fold f z (A - {x}))"
718     by (rule fold_insert) (simp add: `finite A`)+
719   finally show ?thesis .
720 qed
722 lemma fold_insert_remove:
723   assumes "finite A"
724   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
725 proof -
726   from `finite A` have "finite (insert x A)" by auto
727   moreover have "x \<in> insert x A" by auto
728   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
729     by (rule fold_rec)
730   then show ?thesis by simp
731 qed
733 end
735 text{* Other properties of @{const fold}: *}
737 lemma fold_image:
738   assumes "inj_on g A"
739   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
740 proof (cases "finite A")
741   case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def)
742 next
743   case True
744   have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
745   proof
746     fix w
747     show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
748     proof
749       assume ?P then show ?Q using assms
750       proof (induct "g ` A" w arbitrary: A)
751         case emptyI then show ?case by (auto intro: fold_graph.emptyI)
752       next
753         case (insertI x A r B)
754         from `inj_on g B` `x \<notin> A` `insert x A = image g B` obtain x' A' where
755           "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
756           by (rule inj_img_insertE)
757         from insertI.prems have "fold_graph (f o g) z A' r"
758           by (auto intro: insertI.hyps)
759         with `x' \<notin> A'` have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
760           by (rule fold_graph.insertI)
761         then show ?case by simp
762       qed
763     next
764       assume ?Q then show ?P using assms
765       proof induct
766         case emptyI thus ?case by (auto intro: fold_graph.emptyI)
767       next
768         case (insertI x A r)
769         from `x \<notin> A` insertI.prems have "g x \<notin> g ` A" by auto
770         moreover from insertI have "fold_graph f z (g ` A) r" by simp
771         ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
772           by (rule fold_graph.insertI)
773         then show ?case by simp
774       qed
775     qed
776   qed
777   with True assms show ?thesis by (auto simp add: fold_def)
778 qed
780 lemma fold_cong:
781   assumes "comp_fun_commute f" "comp_fun_commute g"
782   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
783     and "s = t" and "A = B"
784   shows "fold f s A = fold g t B"
785 proof -
786   have "fold f s A = fold g s A"
787   using `finite A` cong proof (induct A)
788     case empty then show ?case by simp
789   next
790     case (insert x A)
791     interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
792     interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
793     from insert show ?case by simp
794   qed
795   with assms show ?thesis by simp
796 qed
799 text {* A simplified version for idempotent functions: *}
801 locale comp_fun_idem = comp_fun_commute +
802   assumes comp_fun_idem: "f x \<circ> f x = f x"
803 begin
805 lemma fun_left_idem: "f x (f x z) = f x z"
806   using comp_fun_idem by (simp add: fun_eq_iff)
808 lemma fold_insert_idem:
809   assumes fin: "finite A"
810   shows "fold f z (insert x A)  = f x (fold f z A)"
811 proof cases
812   assume "x \<in> A"
813   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
814   then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
815 next
816   assume "x \<notin> A" then show ?thesis using assms by simp
817 qed
819 declare fold_insert [simp del] fold_insert_idem [simp]
821 lemma fold_insert_idem2:
822   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
825 end
828 subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
830 lemma (in comp_fun_commute) comp_comp_fun_commute:
831   "comp_fun_commute (f \<circ> g)"
832 proof
835 lemma (in comp_fun_idem) comp_comp_fun_idem:
836   "comp_fun_idem (f \<circ> g)"
837   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
840 lemma (in comp_fun_commute) comp_fun_commute_funpow:
841   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
842 proof
843   fix y x
844   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
845   proof (cases "x = y")
846     case True then show ?thesis by simp
847   next
848     case False show ?thesis
849     proof (induct "g x" arbitrary: g)
850       case 0 then show ?case by simp
851     next
852       case (Suc n g)
853       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
854       proof (induct "g y" arbitrary: g)
855         case 0 then show ?case by simp
856       next
857         case (Suc n g)
858         def h \<equiv> "\<lambda>z. g z - 1"
859         with Suc have "n = h y" by simp
860         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
861           by auto
862         from Suc h_def have "g y = Suc (h y)" by simp
863         then show ?case by (simp add: comp_assoc hyp)
865       qed
866       def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
867       with Suc have "n = h x" by simp
868       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
869         by auto
870       with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
871       from Suc h_def have "g x = Suc (h x)" by simp
872       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
874     qed
875   qed
876 qed
879 subsubsection {* Expressing set operations via @{const fold} *}
881 lemma comp_fun_commute_const:
882   "comp_fun_commute (\<lambda>_. f)"
883 proof
884 qed rule
886 lemma comp_fun_idem_insert:
887   "comp_fun_idem insert"
888 proof
889 qed auto
891 lemma comp_fun_idem_remove:
892   "comp_fun_idem Set.remove"
893 proof
894 qed auto
896 lemma (in semilattice_inf) comp_fun_idem_inf:
897   "comp_fun_idem inf"
898 proof
899 qed (auto simp add: inf_left_commute)
901 lemma (in semilattice_sup) comp_fun_idem_sup:
902   "comp_fun_idem sup"
903 proof
904 qed (auto simp add: sup_left_commute)
906 lemma union_fold_insert:
907   assumes "finite A"
908   shows "A \<union> B = fold insert B A"
909 proof -
910   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
911   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
912 qed
914 lemma minus_fold_remove:
915   assumes "finite A"
916   shows "B - A = fold Set.remove B A"
917 proof -
918   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
919   from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
920   then show ?thesis ..
921 qed
923 lemma comp_fun_commute_filter_fold:
924   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
925 proof -
926   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
927   show ?thesis by default (auto simp: fun_eq_iff)
928 qed
930 lemma Set_filter_fold:
931   assumes "finite A"
932   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
933 using assms
934 by (induct A)
935   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
937 lemma inter_Set_filter:
938   assumes "finite B"
939   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
940 using assms
941 by (induct B) (auto simp: Set.filter_def)
943 lemma image_fold_insert:
944   assumes "finite A"
945   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
946 using assms
947 proof -
948   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
949   show ?thesis using assms by (induct A) auto
950 qed
952 lemma Ball_fold:
953   assumes "finite A"
954   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
955 using assms
956 proof -
957   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
958   show ?thesis using assms by (induct A) auto
959 qed
961 lemma Bex_fold:
962   assumes "finite A"
963   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
964 using assms
965 proof -
966   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
967   show ?thesis using assms by (induct A) auto
968 qed
970 lemma comp_fun_commute_Pow_fold:
971   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
972   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
974 lemma Pow_fold:
975   assumes "finite A"
976   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
977 using assms
978 proof -
979   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
980   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
981 qed
983 lemma fold_union_pair:
984   assumes "finite B"
985   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
986 proof -
987   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
988   show ?thesis using assms  by (induct B arbitrary: A) simp_all
989 qed
991 lemma comp_fun_commute_product_fold:
992   assumes "finite B"
993   shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
994 by default (auto simp: fold_union_pair[symmetric] assms)
996 lemma product_fold:
997   assumes "finite A"
998   assumes "finite B"
999   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
1000 using assms unfolding Sigma_def
1001 by (induct A)
1002   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
1005 context complete_lattice
1006 begin
1008 lemma inf_Inf_fold_inf:
1009   assumes "finite A"
1010   shows "inf (Inf A) B = fold inf B A"
1011 proof -
1012   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
1013   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
1015 qed
1017 lemma sup_Sup_fold_sup:
1018   assumes "finite A"
1019   shows "sup (Sup A) B = fold sup B A"
1020 proof -
1021   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
1022   from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
1024 qed
1026 lemma Inf_fold_inf:
1027   assumes "finite A"
1028   shows "Inf A = fold inf top A"
1029   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
1031 lemma Sup_fold_sup:
1032   assumes "finite A"
1033   shows "Sup A = fold sup bot A"
1034   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
1036 lemma inf_INF_fold_inf:
1037   assumes "finite A"
1038   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
1039 proof (rule sym)
1040   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
1041   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
1042   from `finite A` show "?fold = ?inf"
1043     by (induct A arbitrary: B)
1045 qed
1047 lemma sup_SUP_fold_sup:
1048   assumes "finite A"
1049   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
1050 proof (rule sym)
1051   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
1052   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
1053   from `finite A` show "?fold = ?sup"
1054     by (induct A arbitrary: B)
1056 qed
1058 lemma INF_fold_inf:
1059   assumes "finite A"
1060   shows "INFI A f = fold (inf \<circ> f) top A"
1061   using assms inf_INF_fold_inf [of A top] by simp
1063 lemma SUP_fold_sup:
1064   assumes "finite A"
1065   shows "SUPR A f = fold (sup \<circ> f) bot A"
1066   using assms sup_SUP_fold_sup [of A bot] by simp
1068 end
1071 subsection {* Locales as mini-packages for fold operations *}
1073 subsubsection {* The natural case *}
1075 locale folding =
1076   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
1077   fixes z :: "'b"
1078   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
1079 begin
1081 interpretation fold?: comp_fun_commute f
1082   by default (insert comp_fun_commute, simp add: fun_eq_iff)
1084 definition F :: "'a set \<Rightarrow> 'b"
1085 where
1086   eq_fold: "F A = fold f z A"
1088 lemma empty [simp]:
1089   "F {} = z"
1092 lemma infinite [simp]:
1093   "\<not> finite A \<Longrightarrow> F A = z"
1096 lemma insert [simp]:
1097   assumes "finite A" and "x \<notin> A"
1098   shows "F (insert x A) = f x (F A)"
1099 proof -
1100   from fold_insert assms
1101   have "fold f z (insert x A) = f x (fold f z A)" by simp
1102   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
1103 qed
1105 lemma remove:
1106   assumes "finite A" and "x \<in> A"
1107   shows "F A = f x (F (A - {x}))"
1108 proof -
1109   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1110     by (auto dest: mk_disjoint_insert)
1111   moreover from `finite A` A have "finite B" by simp
1112   ultimately show ?thesis by simp
1113 qed
1115 lemma insert_remove:
1116   assumes "finite A"
1117   shows "F (insert x A) = f x (F (A - {x}))"
1118   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1120 end
1123 subsubsection {* With idempotency *}
1125 locale folding_idem = folding +
1126   assumes comp_fun_idem: "f x \<circ> f x = f x"
1127 begin
1129 declare insert [simp del]
1131 interpretation fold?: comp_fun_idem f
1132   by default (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
1134 lemma insert_idem [simp]:
1135   assumes "finite A"
1136   shows "F (insert x A) = f x (F A)"
1137 proof -
1138   from fold_insert_idem assms
1139   have "fold f z (insert x A) = f x (fold f z A)" by simp
1140   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
1141 qed
1143 end
1146 subsection {* Finite cardinality *}
1148 text {*
1150   @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
1151   is ugly to work with.
1152   But now that we have @{const fold} things are easy:
1153 *}
1155 definition card :: "'a set \<Rightarrow> nat" where
1156   "card = folding.F (\<lambda>_. Suc) 0"
1158 interpretation card!: folding "\<lambda>_. Suc" 0
1159 where
1160   "folding.F (\<lambda>_. Suc) 0 = card"
1161 proof -
1162   show "folding (\<lambda>_. Suc)" by default rule
1163   then interpret card!: folding "\<lambda>_. Suc" 0 .
1164   from card_def show "folding.F (\<lambda>_. Suc) 0 = card" by rule
1165 qed
1167 lemma card_infinite:
1168   "\<not> finite A \<Longrightarrow> card A = 0"
1169   by (fact card.infinite)
1171 lemma card_empty:
1172   "card {} = 0"
1173   by (fact card.empty)
1175 lemma card_insert_disjoint:
1176   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
1177   by (fact card.insert)
1179 lemma card_insert_if:
1180   "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
1181   by auto (simp add: card.insert_remove card.remove)
1183 lemma card_ge_0_finite:
1184   "card A > 0 \<Longrightarrow> finite A"
1185   by (rule ccontr) simp
1187 lemma card_0_eq [simp]:
1188   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
1189   by (auto dest: mk_disjoint_insert)
1191 lemma finite_UNIV_card_ge_0:
1192   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
1193   by (rule ccontr) simp
1195 lemma card_eq_0_iff:
1196   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
1197   by auto
1199 lemma card_gt_0_iff:
1200   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
1201   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
1203 lemma card_Suc_Diff1:
1204   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
1205 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
1206 apply(simp del:insert_Diff_single)
1207 done
1209 lemma card_Diff_singleton:
1210   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
1211   by (simp add: card_Suc_Diff1 [symmetric])
1213 lemma card_Diff_singleton_if:
1214   "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
1217 lemma card_Diff_insert[simp]:
1218   assumes "finite A" and "a \<in> A" and "a \<notin> B"
1219   shows "card (A - insert a B) = card (A - B) - 1"
1220 proof -
1221   have "A - insert a B = (A - B) - {a}" using assms by blast
1222   then show ?thesis using assms by(simp add: card_Diff_singleton)
1223 qed
1225 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
1226   by (fact card.insert_remove)
1228 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
1231 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
1232 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
1234 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
1235 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
1237 lemma card_mono:
1238   assumes "finite B" and "A \<subseteq> B"
1239   shows "card A \<le> card B"
1240 proof -
1241   from assms have "finite A" by (auto intro: finite_subset)
1242   then show ?thesis using assms proof (induct A arbitrary: B)
1243     case empty then show ?case by simp
1244   next
1245     case (insert x A)
1246     then have "x \<in> B" by simp
1247     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
1248     with insert.hyps have "card A \<le> card (B - {x})" by auto
1249     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
1250   qed
1251 qed
1253 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
1254 apply (induct rule: finite_induct)
1255 apply simp
1256 apply clarify
1257 apply (subgoal_tac "finite A & A - {x} <= F")
1258  prefer 2 apply (blast intro: finite_subset, atomize)
1259 apply (drule_tac x = "A - {x}" in spec)
1261 apply (case_tac "card A", auto)
1262 done
1264 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
1265 apply (simp add: psubset_eq linorder_not_le [symmetric])
1266 apply (blast dest: card_seteq)
1267 done
1269 lemma card_Un_Int:
1270   assumes "finite A" and "finite B"
1271   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
1272 using assms proof (induct A)
1273   case empty then show ?case by simp
1274 next
1275  case (insert x A) then show ?case
1276     by (auto simp add: insert_absorb Int_insert_left)
1277 qed
1279 lemma card_Un_disjoint:
1280   assumes "finite A" and "finite B"
1281   assumes "A \<inter> B = {}"
1282   shows "card (A \<union> B) = card A + card B"
1283 using assms card_Un_Int [of A B] by simp
1285 lemma card_Diff_subset:
1286   assumes "finite B" and "B \<subseteq> A"
1287   shows "card (A - B) = card A - card B"
1288 proof (cases "finite A")
1289   case False with assms show ?thesis by simp
1290 next
1291   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
1292 qed
1294 lemma card_Diff_subset_Int:
1295   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
1296 proof -
1297   have "A - B = A - A \<inter> B" by auto
1298   thus ?thesis
1299     by (simp add: card_Diff_subset AB)
1300 qed
1302 lemma diff_card_le_card_Diff:
1303 assumes "finite B" shows "card A - card B \<le> card(A - B)"
1304 proof-
1305   have "card A - card B \<le> card A - card (A \<inter> B)"
1306     using card_mono[OF assms Int_lower2, of A] by arith
1307   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
1308   finally show ?thesis .
1309 qed
1311 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
1312 apply (rule Suc_less_SucD)
1313 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
1314 done
1316 lemma card_Diff2_less:
1317   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
1318 apply (case_tac "x = y")
1319  apply (simp add: card_Diff1_less del:card_Diff_insert)
1320 apply (rule less_trans)
1321  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
1322 done
1324 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
1325 apply (case_tac "x : A")
1326  apply (simp_all add: card_Diff1_less less_imp_le)
1327 done
1329 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
1330 by (erule psubsetI, blast)
1332 lemma card_le_inj:
1333   assumes fA: "finite A"
1334     and fB: "finite B"
1335     and c: "card A \<le> card B"
1336   shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
1337   using fA fB c
1338 proof (induct arbitrary: B rule: finite_induct)
1339   case empty
1340   then show ?case by simp
1341 next
1342   case (insert x s t)
1343   then show ?case
1344   proof (induct rule: finite_induct[OF "insert.prems"(1)])
1345     case 1
1346     then show ?case by simp
1347   next
1348     case (2 y t)
1349     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
1350       by simp
1351     from "2.prems"(3) [OF "2.hyps"(1) cst]
1352     obtain f where "f ` s \<subseteq> t" "inj_on f s"
1353       by blast
1354     with "2.prems"(2) "2.hyps"(2) show ?case
1355       apply -
1356       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
1357       apply (auto simp add: inj_on_def)
1358       done
1359   qed
1360 qed
1362 lemma card_subset_eq:
1363   assumes fB: "finite B"
1364     and AB: "A \<subseteq> B"
1365     and c: "card A = card B"
1366   shows "A = B"
1367 proof -
1368   from fB AB have fA: "finite A"
1369     by (auto intro: finite_subset)
1370   from fA fB have fBA: "finite (B - A)"
1371     by auto
1372   have e: "A \<inter> (B - A) = {}"
1373     by blast
1374   have eq: "A \<union> (B - A) = B"
1375     using AB by blast
1376   from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
1377     by arith
1378   then have "B - A = {}"
1379     unfolding card_eq_0_iff using fA fB by simp
1380   with AB show "A = B"
1381     by blast
1382 qed
1384 lemma insert_partition:
1385   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
1386   \<Longrightarrow> x \<inter> \<Union> F = {}"
1387 by auto
1389 lemma finite_psubset_induct[consumes 1, case_names psubset]:
1390   assumes fin: "finite A"
1391   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
1392   shows "P A"
1393 using fin
1394 proof (induct A taking: card rule: measure_induct_rule)
1395   case (less A)
1396   have fin: "finite A" by fact
1397   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
1398   { fix B
1399     assume asm: "B \<subset> A"
1400     from asm have "card B < card A" using psubset_card_mono fin by blast
1401     moreover
1402     from asm have "B \<subseteq> A" by auto
1403     then have "finite B" using fin finite_subset by blast
1404     ultimately
1405     have "P B" using ih by simp
1406   }
1407   with fin show "P A" using major by blast
1408 qed
1410 lemma finite_induct_select[consumes 1, case_names empty select]:
1411   assumes "finite S"
1412   assumes "P {}"
1413   assumes select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
1414   shows "P S"
1415 proof -
1416   have "0 \<le> card S" by simp
1417   then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
1418   proof (induct rule: dec_induct)
1419     case base with `P {}` show ?case
1420       by (intro exI[of _ "{}"]) auto
1421   next
1422     case (step n)
1423     then obtain T where T: "T \<subseteq> S" "card T = n" "P T"
1424       by auto
1425     with `n < card S` have "T \<subset> S" "P T"
1426       by auto
1427     with select[of T] obtain s where "s \<in> S" "s \<notin> T" "P (insert s T)"
1428       by auto
1429     with step(2) T `finite S` show ?case
1430       by (intro exI[of _ "insert s T"]) (auto dest: finite_subset)
1431   qed
1432   with `finite S` show "P S"
1433     by (auto dest: card_subset_eq)
1434 qed
1436 text{* main cardinality theorem *}
1437 lemma card_partition [rule_format]:
1438   "finite C ==>
1439      finite (\<Union> C) -->
1440      (\<forall>c\<in>C. card c = k) -->
1441      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
1442      k * card(C) = card (\<Union> C)"
1443 apply (erule finite_induct, simp)
1444 apply (simp add: card_Un_disjoint insert_partition
1445        finite_subset [of _ "\<Union> (insert x F)"])
1446 done
1448 lemma card_eq_UNIV_imp_eq_UNIV:
1449   assumes fin: "finite (UNIV :: 'a set)"
1450   and card: "card A = card (UNIV :: 'a set)"
1451   shows "A = (UNIV :: 'a set)"
1452 proof
1453   show "A \<subseteq> UNIV" by simp
1454   show "UNIV \<subseteq> A"
1455   proof
1456     fix x
1457     show "x \<in> A"
1458     proof (rule ccontr)
1459       assume "x \<notin> A"
1460       then have "A \<subset> UNIV" by auto
1461       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
1462       with card show False by simp
1463     qed
1464   qed
1465 qed
1467 text{*The form of a finite set of given cardinality*}
1469 lemma card_eq_SucD:
1470 assumes "card A = Suc k"
1471 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
1472 proof -
1473   have fin: "finite A" using assms by (auto intro: ccontr)
1474   moreover have "card A \<noteq> 0" using assms by auto
1475   ultimately obtain b where b: "b \<in> A" by auto
1476   show ?thesis
1477   proof (intro exI conjI)
1478     show "A = insert b (A-{b})" using b by blast
1479     show "b \<notin> A - {b}" by blast
1480     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
1481       using assms b fin by(fastforce dest:mk_disjoint_insert)+
1482   qed
1483 qed
1485 lemma card_Suc_eq:
1486   "(card A = Suc k) =
1487    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
1488  apply(auto elim!: card_eq_SucD)
1489  apply(subst card.insert)
1491  done
1493 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
1494   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
1495 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
1496   dest: subset_singletonD split: nat.splits if_splits)
1498 lemma finite_fun_UNIVD2:
1499   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
1500   shows "finite (UNIV :: 'b set)"
1501 proof -
1502   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
1503     by (rule finite_imageI)
1504   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
1505     by (rule UNIV_eq_I) auto
1506   ultimately show "finite (UNIV :: 'b set)" by simp
1507 qed
1509 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
1510   unfolding UNIV_unit by simp
1513 subsubsection {* Cardinality of image *}
1515 lemma card_image_le: "finite A ==> card (f ` A) \<le> card A"
1516   by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)
1518 lemma card_image:
1519   assumes "inj_on f A"
1520   shows "card (f ` A) = card A"
1521 proof (cases "finite A")
1522   case True then show ?thesis using assms by (induct A) simp_all
1523 next
1524   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
1525   with False show ?thesis by simp
1526 qed
1528 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
1529 by(auto simp: card_image bij_betw_def)
1531 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
1532 by (simp add: card_seteq card_image)
1534 lemma eq_card_imp_inj_on:
1535   assumes "finite A" "card(f ` A) = card A" shows "inj_on f A"
1536 using assms
1537 proof (induct rule:finite_induct)
1538   case empty show ?case by simp
1539 next
1540   case (insert x A)
1541   then show ?case using card_image_le [of A f]
1542     by (simp add: card_insert_if split: if_splits)
1543 qed
1545 lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card(f ` A) = card A"
1546   by (blast intro: card_image eq_card_imp_inj_on)
1548 lemma card_inj_on_le:
1549   assumes "inj_on f A" "f ` A \<subseteq> B" "finite B" shows "card A \<le> card B"
1550 proof -
1551   have "finite A" using assms
1552     by (blast intro: finite_imageD dest: finite_subset)
1553   then show ?thesis using assms
1554    by (force intro: card_mono simp: card_image [symmetric])
1555 qed
1557 lemma card_bij_eq:
1558   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
1559      finite A; finite B |] ==> card A = card B"
1560 by (auto intro: le_antisym card_inj_on_le)
1562 lemma bij_betw_finite:
1563   assumes "bij_betw f A B"
1564   shows "finite A \<longleftrightarrow> finite B"
1565 using assms unfolding bij_betw_def
1566 using finite_imageD[of f A] by auto
1568 lemma inj_on_finite:
1569 assumes "inj_on f A" "f ` A \<le> B" "finite B"
1570 shows "finite A"
1571 using assms finite_imageD finite_subset by blast
1574 subsubsection {* Pigeonhole Principles *}
1576 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
1577 by (auto dest: card_image less_irrefl_nat)
1579 lemma pigeonhole_infinite:
1580 assumes  "~ finite A" and "finite(f`A)"
1581 shows "EX a0:A. ~finite{a:A. f a = f a0}"
1582 proof -
1583   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
1584   proof(induct "f`A" arbitrary: A rule: finite_induct)
1585     case empty thus ?case by simp
1586   next
1587     case (insert b F)
1588     show ?case
1589     proof cases
1590       assume "finite{a:A. f a = b}"
1591       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
1592       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
1593       finally have "~ finite({a:A. f a \<noteq> b})" .
1594       from insert(3)[OF _ this]
1595       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
1596     next
1597       assume 1: "~finite{a:A. f a = b}"
1598       hence "{a \<in> A. f a = b} \<noteq> {}" by force
1599       thus ?thesis using 1 by blast
1600     qed
1601   qed
1602   from this[OF assms(2,1)] show ?thesis .
1603 qed
1605 lemma pigeonhole_infinite_rel:
1606 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
1607 shows "EX b:B. ~finite{a:A. R a b}"
1608 proof -
1609    let ?F = "%a. {b:B. R a b}"
1610    from finite_Pow_iff[THEN iffD2, OF `finite B`]
1611    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
1612    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
1613    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
1614    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
1615    { assume "finite{a:A. R a b0}"
1616      then have "finite {a\<in>A. ?F a = ?F a0}"
1617        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
1618    }
1619    with 1 `b0 : B` show ?thesis by blast
1620 qed
1623 subsubsection {* Cardinality of sums *}
1625 lemma card_Plus:
1626   assumes "finite A" and "finite B"
1627   shows "card (A <+> B) = card A + card B"
1628 proof -
1629   have "Inl`A \<inter> Inr`B = {}" by fast
1630   with assms show ?thesis
1631     unfolding Plus_def
1632     by (simp add: card_Un_disjoint card_image)
1633 qed
1635 lemma card_Plus_conv_if:
1636   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
1637   by (auto simp add: card_Plus)
1639 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
1641 lemma dvd_partition:
1642   assumes f: "finite (\<Union>C)" and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
1643     shows "k dvd card (\<Union>C)"
1644 proof -
1645   have "finite C"
1646     by (rule finite_UnionD [OF f])
1647   then show ?thesis using assms
1648   proof (induct rule: finite_induct)
1649     case empty show ?case by simp
1650   next
1651     case (insert c C)
1652     then show ?case
1653       apply simp
1654       apply (subst card_Un_disjoint)
1655       apply (auto simp add: disjoint_eq_subset_Compl)
1656       done
1657   qed
1658 qed
1660 subsubsection {* Relating injectivity and surjectivity *}
1662 lemma finite_surj_inj: assumes "finite A" "A \<subseteq> f ` A" shows "inj_on f A"
1663 proof -
1664   have "f ` A = A"
1665     by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
1666   then show ?thesis using assms
1668 qed
1670 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
1671 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
1672 by (blast intro: finite_surj_inj subset_UNIV)
1674 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
1675 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
1676 by(fastforce simp:surj_def dest!: endo_inj_surj)
1678 corollary infinite_UNIV_nat [iff]:
1679   "\<not> finite (UNIV :: nat set)"
1680 proof
1681   assume "finite (UNIV :: nat set)"
1682   with finite_UNIV_inj_surj [of Suc]
1683   show False by simp (blast dest: Suc_neq_Zero surjD)
1684 qed
1686 lemma infinite_UNIV_char_0:
1687   "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
1688 proof
1689   assume "finite (UNIV :: 'a set)"
1690   with subset_UNIV have "finite (range of_nat :: 'a set)"
1691     by (rule finite_subset)
1692   moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"