1 (* Title: HOL/Finite_Set.thy
2 Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
3 with contributions by Jeremy Avigad
6 header {* Finite sets *}
12 subsection {* Predicate for finite sets *}
14 inductive finite :: "'a set \<Rightarrow> bool"
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. *}
23 and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
25 using `finite F` proof induct
27 fix x F assume F: "finite F" and P: "P F"
31 hence "insert x F = F" by (rule insert_absorb)
32 with P show ?thesis by (simp only:)
35 from F this P show ?thesis by (rule insert)
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"
46 from assms have "A \<noteq> UNIV" by blast
47 then show ?thesis by blast
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
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")
61 show "?P(%x. if x = a then b else f x)" using f ab by auto
66 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
68 lemma finite_imp_nat_seg_image_inj_on:
70 shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
71 using assms proof induct
75 show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
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)
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
94 let ?B = "f ` {i. i < n}"
95 have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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
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
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:
114 shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
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
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)
140 then show ?case by simp
143 have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
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 .
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)
160 "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
161 by (rule rev_finite_subset)
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"
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
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]:
197 shows "finite (A - B) \<longleftrightarrow> finite A"
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 ..
204 lemma finite_Diff_insert [iff]:
205 "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
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
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])
249 assumes "finite (f ` A)" and "inj_on f A"
251 using assms proof (induct "f ` A" arbitrary: A)
252 case empty then show ?case by simp
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
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:
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
278 case insert then show ?case
279 by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
283 lemma finite_vimage_IntI:
284 "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
285 apply (induct rule: finite_induct)
287 apply (subst vimage_insert)
288 apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
291 lemma finite_vimageI:
292 "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
293 using finite_vimage_IntI[of F h UNIV] by auto
295 lemma finite_vimageD:
296 assumes fin: "finite (h -` F)" and surj: "surj h"
299 have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
300 also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
301 finally show "finite F" .
304 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
305 unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
307 lemma finite_Collect_bex [simp]:
309 shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
311 have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
312 with assms show ?thesis by simp
315 lemma finite_Collect_bounded_ex [simp]:
316 assumes "finite {y. P y}"
317 shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
319 have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
320 with assms show ?thesis by simp
324 "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
325 by (simp add: Plus_def)
328 fixes A :: "'a set" and B :: "'b set"
329 assumes fin: "finite (A <+> B)"
330 shows "finite A" "finite B"
332 have "Inl ` A \<subseteq> A <+> B" by auto
333 then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
334 then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
336 have "Inr ` B \<subseteq> A <+> B" by auto
337 then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
338 then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
341 lemma finite_Plus_iff [simp]:
342 "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
343 by (auto intro: finite_PlusD finite_Plus)
345 lemma finite_Plus_UNIV_iff [simp]:
346 "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
347 by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
349 lemma finite_SigmaI [simp, intro]:
350 "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
351 by (unfold Sigma_def) blast
353 lemma finite_cartesian_product:
354 "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
355 by (rule finite_SigmaI)
357 lemma finite_Prod_UNIV:
358 "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
359 by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
361 lemma finite_cartesian_productD1:
362 assumes "finite (A \<times> B)" and "B \<noteq> {}"
365 from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
366 by (auto simp add: finite_conv_nat_seg_image)
367 then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
368 with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
369 by (simp add: image_compose)
370 then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
372 by (auto simp add: finite_conv_nat_seg_image)
375 lemma finite_cartesian_productD2:
376 assumes "finite (A \<times> B)" and "A \<noteq> {}"
379 from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
380 by (auto simp add: finite_conv_nat_seg_image)
381 then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
382 with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
383 by (simp add: image_compose)
384 then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
386 by (auto simp add: finite_conv_nat_seg_image)
389 lemma finite_Pow_iff [iff]:
390 "finite (Pow A) \<longleftrightarrow> finite A"
392 assume "finite (Pow A)"
393 then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
394 then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
397 then show "finite (Pow A)"
398 by induct (simp_all add: Pow_insert)
401 corollary finite_Collect_subsets [simp, intro]:
402 "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
403 by (simp add: Pow_def [symmetric])
405 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
406 by (blast intro: finite_subset [OF subset_Pow_Union])
409 subsubsection {* Further induction rules on finite sets *}
411 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
412 assumes "finite F" and "F \<noteq> {}"
413 assumes "\<And>x. P {x}"
414 and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
416 using assms proof induct
417 case empty then show ?case by simp
419 case (insert x F) then show ?case by cases auto
422 lemma finite_subset_induct [consumes 2, case_names empty insert]:
423 assumes "finite F" and "F \<subseteq> A"
424 assumes empty: "P {}"
425 and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
427 using `finite F` `F \<subseteq> A` proof induct
431 assume "finite F" and "x \<notin> F" and
432 P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
433 show "P (insert x F)"
435 from i show "x \<in> A" by blast
436 from i have "F \<subseteq> A" by blast
438 show "finite F" by fact
439 show "x \<notin> F" by fact
443 lemma finite_empty_induct:
446 and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
449 have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
452 assume "B \<subseteq> A"
453 with `finite A` have "finite B" by (rule rev_finite_subset)
454 from this `B \<subseteq> A` show "P (A - B)"
457 from `P A` show ?case by simp
460 have "P (A - B - {b})"
462 from `finite A` show "finite (A - B)" by induct auto
463 from insert show "b \<in> A - B" by simp
464 from insert show "P (A - B)" by simp
466 also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
470 then have "P (A - A)" by blast
471 then show ?thesis by simp
475 subsection {* Class @{text finite} *}
478 assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
481 lemma finite [simp]: "finite (A \<Colon> 'a set)"
482 by (rule subset_UNIV finite_UNIV finite_subset)+
484 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
489 instance prod :: (finite, finite) finite proof
490 qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
492 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
493 by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
495 instance "fun" :: (finite, finite) finite
497 show "finite (UNIV :: ('a => 'b) set)"
498 proof (rule finite_imageD)
499 let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
500 have "range ?graph \<subseteq> Pow UNIV" by simp
501 moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
502 by (simp only: finite_Pow_iff finite)
503 ultimately show "finite (range ?graph)"
504 by (rule finite_subset)
505 show "inj ?graph" by (rule inj_graph)
509 instance bool :: finite proof
510 qed (simp add: UNIV_bool)
512 instance unit :: finite proof
513 qed (simp add: UNIV_unit)
515 instance sum :: (finite, finite) finite proof
516 qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
518 lemma finite_option_UNIV [simp]:
519 "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
520 by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
522 instance option :: (finite) finite proof
523 qed (simp add: UNIV_option_conv)
526 subsection {* A basic fold functional for finite sets *}
528 text {* The intended behaviour is
529 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
530 if @{text f} is ``left-commutative'':
533 locale comp_fun_commute =
534 fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
535 assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
538 lemma fun_left_comm: "f x (f y z) = f y (f x z)"
539 using comp_fun_commute by (simp add: fun_eq_iff)
543 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
544 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
545 emptyI [intro]: "fold_graph f z {} z" |
546 insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
547 \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
549 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
551 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
552 "fold f z A = (THE y. fold_graph f z A y)"
554 text{*A tempting alternative for the definiens is
555 @{term "if finite A then THE y. fold_graph f z A y else e"}.
556 It allows the removal of finiteness assumptions from the theorems
557 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
558 The proofs become ugly. It is not worth the effort. (???) *}
560 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
561 by (induct rule: finite_induct) auto
564 subsubsection{*From @{const fold_graph} to @{term fold}*}
566 context comp_fun_commute
569 lemma fold_graph_insertE_aux:
570 "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'"
571 proof (induct set: fold_graph)
572 case (insertI x A y) show ?case
573 proof (cases "x = a")
574 assume "x = a" with insertI show ?case by auto
576 assume "x \<noteq> a"
577 then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
578 using insertI by auto
579 have "f x y = f a (f x y')"
580 unfolding y by (rule fun_left_comm)
581 moreover have "fold_graph f z (insert x A - {a}) (f x y')"
582 using y' and `x \<noteq> a` and `x \<notin> A`
583 by (simp add: insert_Diff_if fold_graph.insertI)
584 ultimately show ?case by fast
588 lemma fold_graph_insertE:
589 assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
590 obtains y where "v = f x y" and "fold_graph f z A y"
591 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
593 lemma fold_graph_determ:
594 "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
595 proof (induct arbitrary: y set: fold_graph)
596 case (insertI x A y v)
597 from `fold_graph f z (insert x A) v` and `x \<notin> A`
598 obtain y' where "v = f x y'" and "fold_graph f z A y'"
599 by (rule fold_graph_insertE)
600 from `fold_graph f z A y'` have "y' = y" by (rule insertI)
601 with `v = f x y'` show "v = f x y" by simp
605 "fold_graph f z A y \<Longrightarrow> fold f z A = y"
606 by (unfold fold_def) (blast intro: fold_graph_determ)
608 lemma fold_graph_fold:
610 shows "fold_graph f z A (fold f z A)"
612 from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
613 moreover note fold_graph_determ
614 ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
615 then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
616 then show ?thesis by (unfold fold_def)
619 text{* The base case for @{text fold}: *}
621 lemma (in -) fold_empty [simp]: "fold f z {} = z"
622 by (unfold fold_def) blast
624 text{* The various recursion equations for @{const fold}: *}
626 lemma fold_insert [simp]:
627 assumes "finite A" and "x \<notin> A"
628 shows "fold f z (insert x A) = f x (fold f z A)"
629 proof (rule fold_equality)
630 from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
631 with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
635 "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
636 proof (induct rule: finite_induct)
637 case empty then show ?case by simp
639 case (insert y A) then show ?case
640 by (simp add: fun_left_comm[of x])
644 "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
645 by (simp add: fold_fun_comm)
648 assumes "finite A" and "x \<in> A"
649 shows "fold f z A = f x (fold f z (A - {x}))"
651 have A: "A = insert x (A - {x})" using `x \<in> A` by blast
652 then have "fold f z A = fold f z (insert x (A - {x}))" by simp
653 also have "\<dots> = f x (fold f z (A - {x}))"
654 by (rule fold_insert) (simp add: `finite A`)+
655 finally show ?thesis .
658 lemma fold_insert_remove:
660 shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
662 from `finite A` have "finite (insert x A)" by auto
663 moreover have "x \<in> insert x A" by auto
664 ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
666 then show ?thesis by simp
671 text{* A simplified version for idempotent functions: *}
673 locale comp_fun_idem = comp_fun_commute +
674 assumes comp_fun_idem: "f x o f x = f x"
677 lemma fun_left_idem: "f x (f x z) = f x z"
678 using comp_fun_idem by (simp add: fun_eq_iff)
680 lemma fold_insert_idem:
681 assumes fin: "finite A"
682 shows "fold f z (insert x A) = f x (fold f z A)"
685 then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
686 then show ?thesis using assms by (simp add:fun_left_idem)
688 assume "x \<notin> A" then show ?thesis using assms by simp
691 declare fold_insert[simp del] fold_insert_idem[simp]
693 lemma fold_insert_idem2:
694 "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
695 by(simp add:fold_fun_comm)
700 subsubsection {* Expressing set operations via @{const fold} *}
702 lemma (in comp_fun_commute) comp_comp_fun_commute:
703 "comp_fun_commute (f \<circ> g)"
705 qed (simp_all add: comp_fun_commute)
707 lemma (in comp_fun_idem) comp_comp_fun_idem:
708 "comp_fun_idem (f \<circ> g)"
709 by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
710 (simp_all add: comp_fun_idem)
712 lemma comp_fun_idem_insert:
713 "comp_fun_idem insert"
717 lemma comp_fun_idem_remove:
718 "comp_fun_idem (\<lambda>x A. A - {x})"
722 lemma (in semilattice_inf) comp_fun_idem_inf:
725 qed (auto simp add: inf_left_commute)
727 lemma (in semilattice_sup) comp_fun_idem_sup:
730 qed (auto simp add: sup_left_commute)
732 lemma union_fold_insert:
734 shows "A \<union> B = fold insert B A"
736 interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
737 from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
740 lemma minus_fold_remove:
742 shows "B - A = fold (\<lambda>x A. A - {x}) B A"
744 interpret comp_fun_idem "\<lambda>x A. A - {x}" by (fact comp_fun_idem_remove)
745 from `finite A` show ?thesis by (induct A arbitrary: B) auto
748 context complete_lattice
751 lemma inf_Inf_fold_inf:
753 shows "inf B (Inf A) = fold inf B A"
755 interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
756 from `finite A` show ?thesis by (induct A arbitrary: B)
757 (simp_all add: Inf_insert inf_commute fold_fun_comm)
760 lemma sup_Sup_fold_sup:
762 shows "sup B (Sup A) = fold sup B A"
764 interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
765 from `finite A` show ?thesis by (induct A arbitrary: B)
766 (simp_all add: Sup_insert sup_commute fold_fun_comm)
771 shows "Inf A = fold inf top A"
772 using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
776 shows "Sup A = fold sup bot A"
777 using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
779 lemma inf_INFI_fold_inf:
781 shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
783 interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
784 interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
785 from `finite A` show "?fold = ?inf"
786 by (induct A arbitrary: B)
787 (simp_all add: INFI_def Inf_insert inf_left_commute)
790 lemma sup_SUPR_fold_sup:
792 shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
794 interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
795 interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
796 from `finite A` show "?fold = ?sup"
797 by (induct A arbitrary: B)
798 (simp_all add: SUPR_def Sup_insert sup_left_commute)
803 shows "INFI A f = fold (inf \<circ> f) top A"
804 using assms inf_INFI_fold_inf [of A top] by simp
808 shows "SUPR A f = fold (sup \<circ> f) bot A"
809 using assms sup_SUPR_fold_sup [of A bot] by simp
814 subsection {* The derived combinator @{text fold_image} *}
816 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
817 where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
819 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
820 by (simp add:fold_image_def)
822 context ab_semigroup_mult
825 lemma fold_image_insert[simp]:
826 assumes "finite A" and "a \<notin> A"
827 shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
829 interpret comp_fun_commute "%x y. (g x) * y" proof
830 qed (simp add: fun_eq_iff mult_ac)
831 show ?thesis using assms by (simp add: fold_image_def)
834 lemma fold_image_reindex:
836 shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
837 using assms by induct auto
839 lemma fold_image_cong:
840 assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
841 shows "fold_image times g z A = fold_image times h z A"
844 have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
845 proof (induct arbitrary: C)
846 case empty then show ?case by simp
848 case (insert x F) then show ?case apply -
849 apply (simp add: subset_insert_iff, clarify)
850 apply (subgoal_tac "finite C")
851 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
852 apply (subgoal_tac "C = insert x (C - {x})")
855 apply (simp add: Ball_def del: insert_Diff_single)
858 with g_h show ?thesis by simp
863 context comm_monoid_mult
867 "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
868 apply (induct rule: finite_induct)
871 lemma fold_image_Un_Int:
872 "finite A ==> finite B ==>
873 fold_image times g 1 A * fold_image times g 1 B =
874 fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
875 apply (induct rule: finite_induct)
876 by (induct set: finite)
877 (auto simp add: mult_ac insert_absorb Int_insert_left)
879 lemma fold_image_Un_one:
880 assumes fS: "finite S" and fT: "finite T"
881 and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
882 shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
884 have "fold_image op * f 1 (S \<inter> T) = 1"
885 apply (rule fold_image_1)
886 using fS fT I0 by auto
887 with fold_image_Un_Int[OF fS fT] show ?thesis by simp
890 corollary fold_Un_disjoint:
891 "finite A ==> finite B ==> A Int B = {} ==>
892 fold_image times g 1 (A Un B) =
893 fold_image times g 1 A * fold_image times g 1 B"
894 by (simp add: fold_image_Un_Int)
896 lemma fold_image_UN_disjoint:
897 "\<lbrakk> finite I; ALL i:I. finite (A i);
898 ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
899 \<Longrightarrow> fold_image times g 1 (UNION I A) =
900 fold_image times (%i. fold_image times g 1 (A i)) 1 I"
901 apply (induct rule: finite_induct)
904 apply (subgoal_tac "ALL i:F. x \<noteq> i")
906 apply (subgoal_tac "A x Int UNION F A = {}")
908 apply (simp add: fold_Un_disjoint)
911 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
912 fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
913 fold_image times (split g) 1 (SIGMA x:A. B x)"
914 apply (subst Sigma_def)
915 apply (subst fold_image_UN_disjoint, assumption, simp)
917 apply (erule fold_image_cong)
918 apply (subst fold_image_UN_disjoint, simp, simp)
923 lemma fold_image_distrib: "finite A \<Longrightarrow>
924 fold_image times (%x. g x * h x) 1 A =
925 fold_image times g 1 A * fold_image times h 1 A"
926 by (erule finite_induct) (simp_all add: mult_ac)
928 lemma fold_image_related:
930 and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
931 and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
932 shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
933 using fS by (rule finite_subset_induct) (insert assms, auto)
935 lemma fold_image_eq_general:
936 assumes fS: "finite S"
937 and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
938 and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
939 shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
941 from h f12 have hS: "h ` S = S'" by auto
942 {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
943 from f12 h H have "x = y" by auto }
944 hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
945 from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
946 from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
947 also have "\<dots> = fold_image (op *) (f2 o h) e S"
948 using fold_image_reindex[OF fS hinj, of f2 e] .
949 also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
951 finally show ?thesis ..
954 lemma fold_image_eq_general_inverses:
955 assumes fS: "finite S"
956 and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
957 and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
958 shows "fold_image (op *) f e S = fold_image (op *) g e T"
959 (* metis solves it, but not yet available here *)
960 apply (rule fold_image_eq_general[OF fS, of T h g f e])
966 apply (drule hk) apply simp
968 apply (erule conjunct1[OF conjunct2[OF hk]])
977 subsection {* A fold functional for non-empty sets *}
979 text{* Does not require start value. *}
982 fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
983 for f :: "'a => 'a => 'a"
985 fold1Set_insertI [intro]:
986 "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
988 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
989 "fold1 f A == THE x. fold1Set f A x"
991 lemma fold1Set_nonempty:
992 "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
993 by(erule fold1Set.cases, simp_all)
995 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
997 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
1000 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
1001 by (blast elim: fold_graph.cases)
1003 lemma fold1_singleton [simp]: "fold1 f {a} = a"
1004 by (unfold fold1_def) blast
1006 lemma finite_nonempty_imp_fold1Set:
1007 "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
1008 apply (induct A rule: finite_induct)
1009 apply (auto dest: finite_imp_fold_graph [of _ f])
1012 text{*First, some lemmas about @{const fold_graph}.*}
1014 context ab_semigroup_mult
1017 lemma comp_fun_commute: "comp_fun_commute (op *)" proof
1018 qed (simp add: fun_eq_iff mult_ac)
1020 lemma fold_graph_insert_swap:
1021 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
1022 shows "fold_graph times z (insert b A) (z * y)"
1024 interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1025 from assms show ?thesis
1026 proof (induct rule: fold_graph.induct)
1027 case emptyI show ?case by (subst mult_commute [of z b], fast)
1029 case (insertI x A y)
1030 have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
1031 using insertI by force --{*how does @{term id} get unfolded?*}
1032 thus ?case by (simp add: insert_commute mult_ac)
1036 lemma fold_graph_permute_diff:
1037 assumes fold: "fold_graph times b A x"
1038 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
1040 proof (induct rule: fold_graph.induct)
1041 case emptyI thus ?case by simp
1043 case (insertI x A y)
1044 have "a = x \<or> a \<in> A" using insertI by simp
1048 with insertI show ?thesis
1049 by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
1051 assume ainA: "a \<in> A"
1052 hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
1053 using insertI by force
1055 have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
1056 using ainA insertI by blast
1057 ultimately show ?thesis by simp
1061 lemma fold1_eq_fold:
1062 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
1064 interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1065 from assms show ?thesis
1066 apply (simp add: fold1_def fold_def)
1067 apply (rule the_equality)
1068 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
1069 apply (rule sym, clarify)
1070 apply (case_tac "Aa=A")
1071 apply (best intro: fold_graph_determ)
1072 apply (subgoal_tac "fold_graph times a A x")
1073 apply (best intro: fold_graph_determ)
1074 apply (subgoal_tac "insert aa (Aa - {a}) = A")
1075 prefer 2 apply (blast elim: equalityE)
1076 apply (auto dest: fold_graph_permute_diff [where a=a])
1080 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
1083 apply (drule_tac x=x in spec)
1084 apply (drule_tac x="A-{x}" in spec, auto)
1088 assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
1089 shows "fold1 times (insert x A) = x * fold1 times A"
1091 interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1092 from nonempty obtain a A' where "A = insert a A' & a ~: A'"
1093 by (auto simp add: nonempty_iff)
1095 by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
1100 context ab_semigroup_idem_mult
1103 lemma comp_fun_idem: "comp_fun_idem (op *)" proof
1104 qed (simp_all add: fun_eq_iff mult_left_commute)
1106 lemma fold1_insert_idem [simp]:
1107 assumes nonempty: "A \<noteq> {}" and A: "finite A"
1108 shows "fold1 times (insert x A) = x * fold1 times A"
1110 interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
1111 by (rule comp_fun_idem)
1112 from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
1113 by (auto simp add: nonempty_iff)
1120 with A' a show ?thesis by simp
1122 assume "A' \<noteq> {}"
1123 with A A' a show ?thesis
1124 by (simp add: fold1_insert mult_assoc [symmetric])
1127 assume "a \<noteq> x"
1128 with A A' show ?thesis
1129 by (simp add: insert_commute fold1_eq_fold)
1133 lemma hom_fold1_commute:
1134 assumes hom: "!!x y. h (x * y) = h x * h y"
1135 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
1136 using N proof (induct rule: finite_ne_induct)
1137 case singleton thus ?case by simp
1140 then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
1141 also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
1142 also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
1143 also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
1144 using insert by(simp)
1145 also have "insert (h n) (h ` N) = h ` insert n N" by simp
1146 finally show ?case .
1149 lemma fold1_eq_fold_idem:
1151 shows "fold1 times (insert a A) = fold times a A"
1152 proof (cases "a \<in> A")
1154 with assms show ?thesis by (simp add: fold1_eq_fold)
1156 interpret comp_fun_idem times by (fact comp_fun_idem)
1157 case True then obtain b B
1158 where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
1159 with assms have "finite B" by auto
1160 then have "fold times a (insert a B) = fold times (a * a) B"
1161 using `a \<notin> B` by (rule fold_insert2)
1163 using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
1169 text{* Now the recursion rules for definitions: *}
1171 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
1174 lemma (in ab_semigroup_mult) fold1_insert_def:
1175 "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1176 by (simp add:fold1_insert)
1178 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
1179 "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1182 subsubsection{* Determinacy for @{term fold1Set} *}
1184 (*Not actually used!!*)
1186 context ab_semigroup_mult
1189 lemma fold_graph_permute:
1190 "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
1191 ==> fold_graph times id a (insert b A) x"
1193 apply (auto dest: fold_graph_permute_diff)
1196 lemma fold1Set_determ:
1197 "fold1Set times A x ==> fold1Set times A y ==> y = x"
1198 proof (clarify elim!: fold1Set.cases)
1200 assume Ax: "fold_graph times id a A x"
1201 assume By: "fold_graph times id b B y"
1202 assume anotA: "a \<notin> A"
1203 assume bnotB: "b \<notin> B"
1204 assume eq: "insert a A = insert b B"
1208 hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
1209 thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
1211 assume diff: "a\<noteq>b"
1213 have B: "B = insert a ?D" and A: "A = insert b ?D"
1214 and aB: "a \<in> B" and bA: "b \<in> A"
1215 using eq anotA bnotB diff by (blast elim!:equalityE)+
1217 have "fold_graph times id a (insert b ?D) y"
1218 by (auto intro: fold_graph_permute simp add: insert_absorb)
1220 have "fold_graph times id a (insert b ?D) x"
1221 by (simp add: A [symmetric] Ax)
1222 ultimately show ?thesis by (blast intro: fold_graph_determ)
1226 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
1227 by (unfold fold1_def) (blast intro: fold1Set_determ)
1233 empty_fold_graphE [rule del] fold_graph.intros [rule del]
1234 empty_fold1SetE [rule del] insert_fold1SetE [rule del]
1235 -- {* No more proofs involve these relations. *}
1237 subsubsection {* Lemmas about @{text fold1} *}
1239 context ab_semigroup_mult
1243 assumes A: "finite A" "A \<noteq> {}"
1244 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
1245 fold1 times (A Un B) = fold1 times A * fold1 times B"
1246 using A by (induct rule: finite_ne_induct)
1247 (simp_all add: fold1_insert mult_assoc)
1250 assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
1251 shows "fold1 times A \<in> A"
1253 proof (induct rule:finite_ne_induct)
1254 case singleton thus ?case by simp
1256 case insert thus ?case using elem by (force simp add:fold1_insert)
1261 lemma (in ab_semigroup_idem_mult) fold1_Un2:
1262 assumes A: "finite A" "A \<noteq> {}"
1263 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
1264 fold1 times (A Un B) = fold1 times A * fold1 times B"
1266 proof(induct rule:finite_ne_induct)
1267 case singleton thus ?case by simp
1269 case insert thus ?case by (simp add: mult_assoc)
1273 subsection {* Locales as mini-packages for fold operations *}
1275 subsubsection {* The natural case *}
1278 fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
1279 fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
1280 assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
1281 assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
1286 by (simp add: eq_fold fun_eq_iff)
1288 lemma insert [simp]:
1289 assumes "finite A" and "x \<notin> A"
1290 shows "F (insert x A) = F A \<circ> f x"
1292 interpret comp_fun_commute f proof
1293 qed (insert comp_fun_commute, simp add: fun_eq_iff)
1294 from fold_insert2 assms
1295 have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
1296 with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
1300 assumes "finite A" and "x \<in> A"
1301 shows "F A = F (A - {x}) \<circ> f x"
1303 from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1304 by (auto dest: mk_disjoint_insert)
1305 moreover from `finite A` this have "finite B" by simp
1306 ultimately show ?thesis by simp
1309 lemma insert_remove:
1311 shows "F (insert x A) = F (A - {x}) \<circ> f x"
1312 using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1314 lemma commute_left_comp:
1315 "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
1316 by (simp add: o_assoc comp_fun_commute)
1318 lemma comp_fun_commute':
1320 shows "f x \<circ> F A = F A \<circ> f x"
1321 using assms by (induct A)
1322 (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] comp_fun_commute)
1324 lemma commute_left_comp':
1326 shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
1327 using assms by (simp add: o_assoc comp_fun_commute')
1329 lemma comp_fun_commute'':
1330 assumes "finite A" and "finite B"
1331 shows "F B \<circ> F A = F A \<circ> F B"
1332 using assms by (induct A)
1333 (simp_all add: o_assoc, simp add: o_assoc [symmetric] comp_fun_commute')
1335 lemma commute_left_comp'':
1336 assumes "finite A" and "finite B"
1337 shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
1338 using assms by (simp add: o_assoc comp_fun_commute'')
1340 lemmas comp_fun_commutes = o_assoc [symmetric] comp_fun_commute commute_left_comp
1341 comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
1344 assumes "finite A" and "finite B"
1345 shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
1346 using assms by (induct A)
1347 (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
1351 assumes "finite A" and "finite B"
1352 and "A \<inter> B = {}"
1353 shows "F (A \<union> B) = F A \<circ> F B"
1355 from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
1356 with `A \<inter> B = {}` show ?thesis by simp
1362 subsubsection {* The natural case with idempotency *}
1364 locale folding_idem = folding +
1365 assumes idem_comp: "f x \<circ> f x = f x"
1368 lemma idem_left_comp:
1369 "f x \<circ> (f x \<circ> g) = f x \<circ> g"
1370 by (simp add: o_assoc idem_comp)
1373 assumes "finite A" and "x \<in> A"
1374 shows "F A \<circ> f x = F A"
1375 using assms by (induct A)
1376 (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
1378 lemma subset_comp_idem:
1379 assumes "finite A" and "B \<subseteq> A"
1380 shows "F A \<circ> F B = F A"
1382 from assms have "finite B" by (blast dest: finite_subset)
1383 then show ?thesis using `B \<subseteq> A` by (induct B)
1384 (simp_all add: o_assoc in_comp_idem `finite A`)
1387 declare insert [simp del]
1389 lemma insert_idem [simp]:
1391 shows "F (insert x A) = F A \<circ> f x"
1392 using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
1395 assumes "finite A" and "finite B"
1396 shows "F (A \<union> B) = F A \<circ> F B"
1398 from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1399 then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
1400 with assms show ?thesis by (simp add: union_inter)
1406 subsubsection {* The image case with fixed function *}
1408 no_notation times (infixl "*" 70)
1409 no_notation Groups.one ("1")
1411 locale folding_image_simple = comm_monoid +
1412 fixes g :: "('b \<Rightarrow> 'a)"
1413 fixes F :: "'b set \<Rightarrow> 'a"
1414 assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
1419 by (simp add: eq_fold_g)
1421 lemma insert [simp]:
1422 assumes "finite A" and "x \<notin> A"
1423 shows "F (insert x A) = g x * F A"
1425 interpret comp_fun_commute "%x y. (g x) * y" proof
1426 qed (simp add: ac_simps fun_eq_iff)
1427 with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
1428 by (simp add: fold_image_def)
1429 with `finite A` show ?thesis by (simp add: eq_fold_g)
1433 assumes "finite A" and "x \<in> A"
1434 shows "F A = g x * F (A - {x})"
1436 from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1437 by (auto dest: mk_disjoint_insert)
1438 moreover from `finite A` this have "finite B" by simp
1439 ultimately show ?thesis by simp
1442 lemma insert_remove:
1444 shows "F (insert x A) = g x * F (A - {x})"
1445 using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1448 assumes "finite A" and "\<forall>x\<in>A. g x = 1"
1450 using assms by (induct A) simp_all
1453 assumes "finite A" and "finite B"
1454 shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1455 using assms proof (induct A)
1456 case empty then show ?case by simp
1458 case (insert x A) then show ?case
1459 by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
1462 corollary union_inter_neutral:
1463 assumes "finite A" and "finite B"
1464 and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
1465 shows "F (A \<union> B) = F A * F B"
1466 using assms by (simp add: union_inter [symmetric] neutral)
1468 corollary union_disjoint:
1469 assumes "finite A" and "finite B"
1470 assumes "A \<inter> B = {}"
1471 shows "F (A \<union> B) = F A * F B"
1472 using assms by (simp add: union_inter_neutral)
1477 subsubsection {* The image case with flexible function *}
1479 locale folding_image = comm_monoid +
1480 fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
1481 assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
1483 sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
1486 context folding_image
1489 lemma reindex: (* FIXME polymorhism *)
1490 assumes "finite A" and "inj_on h A"
1491 shows "F g (h ` A) = F (g \<circ> h) A"
1492 using assms by (induct A) auto
1495 assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
1496 shows "F g A = F h A"
1498 from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
1499 apply - apply (erule finite_induct) apply simp
1500 apply (simp add: subset_insert_iff, clarify)
1501 apply (subgoal_tac "finite C")
1502 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
1503 apply (subgoal_tac "C = insert x (C - {x})")
1504 prefer 2 apply blast
1505 apply (erule ssubst)
1507 apply (erule (1) notE impE)
1508 apply (simp add: Ball_def del: insert_Diff_single)
1510 with assms show ?thesis by simp
1513 lemma UNION_disjoint:
1514 assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
1515 and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
1516 shows "F g (UNION I A) = F (F g \<circ> A) I"
1517 apply (insert assms)
1518 apply (induct rule: finite_induct)
1521 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
1522 prefer 2 apply blast
1523 apply (subgoal_tac "A x Int UNION Fa A = {}")
1524 prefer 2 apply blast
1525 apply (simp add: union_disjoint)
1530 shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
1531 using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
1535 and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
1536 and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
1537 shows "R (F h S) (F g S)"
1538 using fS by (rule finite_subset_induct) (insert assms, auto)
1541 assumes fS: "finite S"
1542 and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
1543 and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
1544 shows "F f1 S = F f2 S'"
1546 from h f12 have hS: "h ` S = S'" by blast
1547 {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1548 from f12 h H have "x = y" by auto }
1549 hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1550 from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1551 from hS have "F f2 S' = F f2 (h ` S)" by simp
1552 also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
1553 also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
1555 finally show ?thesis ..
1558 lemma eq_general_inverses:
1559 assumes fS: "finite S"
1560 and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1561 and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
1562 shows "F j S = F g T"
1563 (* metis solves it, but not yet available here *)
1564 apply (rule eq_general [OF fS, of T h g j])
1570 apply (drule hk) apply simp
1572 apply (erule conjunct1[OF conjunct2[OF hk]])
1581 subsubsection {* The image case with fixed function and idempotency *}
1583 locale folding_image_simple_idem = folding_image_simple +
1584 assumes idem: "x * x = x"
1586 sublocale folding_image_simple_idem < semilattice proof
1589 context folding_image_simple_idem
1593 assumes "finite A" and "x \<in> A"
1594 shows "g x * F A = F A"
1595 using assms by (induct A) (auto simp add: left_commute)
1598 assumes "finite A" and "B \<subseteq> A"
1599 shows "F B * F A = F A"
1601 from assms have "finite B" by (blast dest: finite_subset)
1602 then show ?thesis using `B \<subseteq> A` by (induct B)
1603 (auto simp add: assoc in_idem `finite A`)
1606 declare insert [simp del]
1608 lemma insert_idem [simp]:
1610 shows "F (insert x A) = g x * F A"
1611 using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
1614 assumes "finite A" and "finite B"
1615 shows "F (A \<union> B) = F A * F B"
1617 from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1618 then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
1619 with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1625 subsubsection {* The image case with flexible function and idempotency *}
1627 locale folding_image_idem = folding_image +
1628 assumes idem: "x * x = x"
1630 sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
1634 subsubsection {* The neutral-less case *}
1636 locale folding_one = abel_semigroup +
1637 fixes F :: "'a set \<Rightarrow> 'a"
1638 assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
1641 lemma singleton [simp]:
1643 by (simp add: eq_fold)
1646 assumes "finite A" and "x \<notin> A"
1647 shows "F (insert x A) = fold (op *) x A"
1649 interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
1650 with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
1653 lemma insert [simp]:
1654 assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
1655 shows "F (insert x A) = x * F A"
1657 from `A \<noteq> {}` obtain b where "b \<in> A" by blast
1658 then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
1659 with `finite A` have "finite B" by simp
1660 interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
1661 qed (simp_all add: fun_eq_iff ac_simps)
1662 thm fold.comp_fun_commute' [of B b, simplified fun_eq_iff, simplified]
1663 from `finite B` fold.comp_fun_commute' [of B x]
1664 have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
1665 then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
1666 from `finite B` * fold.insert [of B b]
1667 have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
1668 then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
1669 from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
1673 assumes "finite A" and "x \<in> A"
1674 shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
1676 from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
1677 with assms show ?thesis by simp
1680 lemma insert_remove:
1682 shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
1683 using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
1685 lemma union_disjoint:
1686 assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
1687 shows "F (A \<union> B) = F A * F B"
1688 using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
1691 assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
1692 shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1694 from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
1695 from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
1696 case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
1698 case (insert x A) show ?case proof (cases "x \<in> B")
1699 case True then have "B \<noteq> {}" by auto
1700 with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
1701 (simp_all add: insert_absorb ac_simps union_disjoint)
1703 case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
1704 moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
1706 ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
1712 assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
1714 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
1715 case singleton then show ?case by simp
1717 case insert with elem show ?case by force
1723 subsubsection {* The neutral-less case with idempotency *}
1725 locale folding_one_idem = folding_one +
1726 assumes idem: "x * x = x"
1728 sublocale folding_one_idem < semilattice proof
1731 context folding_one_idem
1735 assumes "finite A" and "x \<in> A"
1736 shows "x * F A = F A"
1738 from assms have "A \<noteq> {}" by auto
1739 with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
1743 assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
1744 shows "F B * F A = F A"
1746 from assms have "finite B" by (blast dest: finite_subset)
1747 then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
1748 (simp_all add: assoc in_idem `finite A`)
1751 lemma eq_fold_idem':
1753 shows "F (insert a A) = fold (op *) a A"
1755 interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
1756 with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
1759 lemma insert_idem [simp]:
1760 assumes "finite A" and "A \<noteq> {}"
1761 shows "F (insert x A) = x * F A"
1762 proof (cases "x \<in> A")
1763 case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
1766 from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
1770 assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
1771 shows "F (A \<union> B) = F A * F B"
1772 proof (cases "A \<inter> B = {}")
1773 case True with assms show ?thesis by (simp add: union_disjoint)
1776 from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1777 with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
1778 with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1782 assumes hom: "\<And>x y. h (x * y) = h x * h y"
1783 and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
1784 using N proof (induct rule: finite_ne_induct)
1785 case singleton thus ?case by simp
1788 then have "h (F (insert n N)) = h (n * F N)" by simp
1789 also have "\<dots> = h n * h (F N)" by (rule hom)
1790 also have "h (F N) = F (h ` N)" by(rule insert)
1791 also have "h n * \<dots> = F (insert (h n) (h ` N))"
1792 using insert by(simp)
1793 also have "insert (h n) (h ` N) = h ` insert n N" by simp
1794 finally show ?case .
1799 notation times (infixl "*" 70)
1800 notation Groups.one ("1")
1803 subsection {* Finite cardinality *}
1805 text {* This definition, although traditional, is ugly to work with:
1806 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
1807 But now that we have @{text fold_image} things are easy:
1810 definition card :: "'a set \<Rightarrow> nat" where
1811 "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
1813 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
1814 qed (simp add: card_def)
1816 lemma card_infinite [simp]:
1817 "\<not> finite A \<Longrightarrow> card A = 0"
1818 by (simp add: card_def)
1822 by (fact card.empty)
1824 lemma card_insert_disjoint:
1825 "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
1828 lemma card_insert_if:
1829 "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
1830 by auto (simp add: card.insert_remove card.remove)
1832 lemma card_ge_0_finite:
1833 "card A > 0 \<Longrightarrow> finite A"
1834 by (rule ccontr) simp
1836 lemma card_0_eq [simp, no_atp]:
1837 "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
1838 by (auto dest: mk_disjoint_insert)
1840 lemma finite_UNIV_card_ge_0:
1841 "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
1842 by (rule ccontr) simp
1844 lemma card_eq_0_iff:
1845 "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
1848 lemma card_gt_0_iff:
1849 "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
1850 by (simp add: neq0_conv [symmetric] card_eq_0_iff)
1852 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
1853 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
1854 apply(simp del:insert_Diff_single)
1857 lemma card_Diff_singleton:
1858 "finite A ==> x: A ==> card (A - {x}) = card A - 1"
1859 by (simp add: card_Suc_Diff1 [symmetric])
1861 lemma card_Diff_singleton_if:
1862 "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
1863 by (simp add: card_Diff_singleton)
1865 lemma card_Diff_insert[simp]:
1866 assumes "finite A" and "a:A" and "a ~: B"
1867 shows "card(A - insert a B) = card(A - B) - 1"
1869 have "A - insert a B = (A - B) - {a}" using assms by blast
1870 then show ?thesis using assms by(simp add:card_Diff_singleton)
1873 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
1874 by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
1876 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
1877 by (simp add: card_insert_if)
1879 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
1880 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
1882 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
1883 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
1886 assumes "finite B" and "A \<subseteq> B"
1887 shows "card A \<le> card B"
1889 from assms have "finite A" by (auto intro: finite_subset)
1890 then show ?thesis using assms proof (induct A arbitrary: B)
1891 case empty then show ?case by simp
1894 then have "x \<in> B" by simp
1895 from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
1896 with insert.hyps have "card A \<le> card (B - {x})" by auto
1897 with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
1901 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
1902 apply (induct rule: finite_induct)
1905 apply (subgoal_tac "finite A & A - {x} <= F")
1906 prefer 2 apply (blast intro: finite_subset, atomize)
1907 apply (drule_tac x = "A - {x}" in spec)
1908 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
1909 apply (case_tac "card A", auto)
1912 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
1913 apply (simp add: psubset_eq linorder_not_le [symmetric])
1914 apply (blast dest: card_seteq)
1917 lemma card_Un_Int: "finite A ==> finite B
1918 ==> card A + card B = card (A Un B) + card (A Int B)"
1919 by (fact card.union_inter [symmetric])
1921 lemma card_Un_disjoint: "finite A ==> finite B
1922 ==> A Int B = {} ==> card (A Un B) = card A + card B"
1923 by (fact card.union_disjoint)
1925 lemma card_Diff_subset:
1926 assumes "finite B" and "B \<subseteq> A"
1927 shows "card (A - B) = card A - card B"
1928 proof (cases "finite A")
1929 case False with assms show ?thesis by simp
1931 case True with assms show ?thesis by (induct B arbitrary: A) simp_all
1934 lemma card_Diff_subset_Int:
1935 assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
1937 have "A - B = A - A \<inter> B" by auto
1939 by (simp add: card_Diff_subset AB)
1942 lemma diff_card_le_card_Diff:
1943 assumes "finite B" shows "card A - card B \<le> card(A - B)"
1945 have "card A - card B \<le> card A - card (A \<inter> B)"
1946 using card_mono[OF assms Int_lower2, of A] by arith
1947 also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
1948 finally show ?thesis .
1951 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
1952 apply (rule Suc_less_SucD)
1953 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
1956 lemma card_Diff2_less:
1957 "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
1958 apply (case_tac "x = y")
1959 apply (simp add: card_Diff1_less del:card_Diff_insert)
1960 apply (rule less_trans)
1961 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
1964 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
1965 apply (case_tac "x : A")
1966 apply (simp_all add: card_Diff1_less less_imp_le)
1969 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
1970 by (erule psubsetI, blast)
1972 lemma insert_partition:
1973 "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
1974 \<Longrightarrow> x \<inter> \<Union> F = {}"
1977 lemma finite_psubset_induct[consumes 1, case_names psubset]:
1978 assumes fin: "finite A"
1979 and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
1982 proof (induct A taking: card rule: measure_induct_rule)
1984 have fin: "finite A" by fact
1985 have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
1987 assume asm: "B \<subset> A"
1988 from asm have "card B < card A" using psubset_card_mono fin by blast
1990 from asm have "B \<subseteq> A" by auto
1991 then have "finite B" using fin finite_subset by blast
1993 have "P B" using ih by simp
1995 with fin show "P A" using major by blast
1998 text{* main cardinality theorem *}
1999 lemma card_partition [rule_format]:
2001 finite (\<Union> C) -->
2002 (\<forall>c\<in>C. card c = k) -->
2003 (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
2004 k * card(C) = card (\<Union> C)"
2005 apply (erule finite_induct, simp)
2006 apply (simp add: card_Un_disjoint insert_partition
2007 finite_subset [of _ "\<Union> (insert x F)"])
2010 lemma card_eq_UNIV_imp_eq_UNIV:
2011 assumes fin: "finite (UNIV :: 'a set)"
2012 and card: "card A = card (UNIV :: 'a set)"
2013 shows "A = (UNIV :: 'a set)"
2015 show "A \<subseteq> UNIV" by simp
2016 show "UNIV \<subseteq> A"
2021 assume "x \<notin> A"
2022 then have "A \<subset> UNIV" by auto
2023 with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
2024 with card show False by simp
2029 text{*The form of a finite set of given cardinality*}
2032 assumes "card A = Suc k"
2033 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
2035 have fin: "finite A" using assms by (auto intro: ccontr)
2036 moreover have "card A \<noteq> 0" using assms by auto
2037 ultimately obtain b where b: "b \<in> A" by auto
2039 proof (intro exI conjI)
2040 show "A = insert b (A-{b})" using b by blast
2041 show "b \<notin> A - {b}" by blast
2042 show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
2043 using assms b fin by(fastsimp dest:mk_disjoint_insert)+
2049 (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
2051 apply(erule card_eq_SucD)
2053 apply(subst card_insert)
2054 apply(auto intro:ccontr)
2057 lemma finite_fun_UNIVD2:
2058 assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
2059 shows "finite (UNIV :: 'b set)"
2061 from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
2062 by(rule finite_imageI)
2063 moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
2064 by(rule UNIV_eq_I) auto
2065 ultimately show "finite (UNIV :: 'b set)" by simp
2068 lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
2069 unfolding UNIV_unit by simp
2072 subsubsection {* Cardinality of image *}
2074 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
2075 apply (induct rule: finite_induct)
2077 apply (simp add: le_SucI card_insert_if)
2081 assumes "inj_on f A"
2082 shows "card (f ` A) = card A"
2083 proof (cases "finite A")
2084 case True then show ?thesis using assms by (induct A) simp_all
2086 case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
2087 with False show ?thesis by simp
2090 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
2091 by(auto simp: card_image bij_betw_def)
2093 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
2094 by (simp add: card_seteq card_image)
2096 lemma eq_card_imp_inj_on:
2097 "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
2098 apply (induct rule:finite_induct)
2100 apply(frule card_image_le[where f = f])
2101 apply(simp add:card_insert_if split:if_splits)
2104 lemma inj_on_iff_eq_card:
2105 "finite A ==> inj_on f A = (card(f ` A) = card A)"
2106 by(blast intro: card_image eq_card_imp_inj_on)
2109 lemma card_inj_on_le:
2110 "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
2111 apply (subgoal_tac "finite A")
2112 apply (force intro: card_mono simp add: card_image [symmetric])
2113 apply (blast intro: finite_imageD dest: finite_subset)
2117 "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
2118 finite A; finite B |] ==> card A = card B"
2119 by (auto intro: le_antisym card_inj_on_le)
2121 lemma bij_betw_finite:
2122 assumes "bij_betw f A B"
2123 shows "finite A \<longleftrightarrow> finite B"
2124 using assms unfolding bij_betw_def
2125 using finite_imageD[of f A] by auto
2128 subsubsection {* Pigeonhole Principles *}
2130 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
2131 by (auto dest: card_image less_irrefl_nat)
2133 lemma pigeonhole_infinite:
2134 assumes "~ finite A" and "finite(f`A)"
2135 shows "EX a0:A. ~finite{a:A. f a = f a0}"
2137 have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
2138 proof(induct "f`A" arbitrary: A rule: finite_induct)
2139 case empty thus ?case by simp
2144 assume "finite{a:A. f a = b}"
2145 hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
2146 also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
2147 finally have "~ finite({a:A. f a \<noteq> b})" .
2148 from insert(3)[OF _ this]
2149 show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
2151 assume 1: "~finite{a:A. f a = b}"
2152 hence "{a \<in> A. f a = b} \<noteq> {}" by force
2153 thus ?thesis using 1 by blast
2156 from this[OF assms(2,1)] show ?thesis .
2159 lemma pigeonhole_infinite_rel:
2160 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
2161 shows "EX b:B. ~finite{a:A. R a b}"
2163 let ?F = "%a. {b:B. R a b}"
2164 from finite_Pow_iff[THEN iffD2, OF `finite B`]
2165 have "finite(?F ` A)" by(blast intro: rev_finite_subset)
2166 from pigeonhole_infinite[where f = ?F, OF assms(1) this]
2167 obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
2168 obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
2169 { assume "finite{a:A. R a b0}"
2170 then have "finite {a\<in>A. ?F a = ?F a0}"
2171 using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
2173 with 1 `b0 : B` show ?thesis by blast
2177 subsubsection {* Cardinality of sums *}
2180 assumes "finite A" and "finite B"
2181 shows "card (A <+> B) = card A + card B"
2183 have "Inl`A \<inter> Inr`B = {}" by fast
2184 with assms show ?thesis
2186 by (simp add: card_Un_disjoint card_image)
2189 lemma card_Plus_conv_if:
2190 "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
2191 by (auto simp add: card_Plus)
2194 subsubsection {* Cardinality of the Powerset *}
2196 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *)
2197 apply (induct rule: finite_induct)
2198 apply (simp_all add: Pow_insert)
2199 apply (subst card_Un_disjoint, blast)
2200 apply (blast, blast)
2201 apply (subgoal_tac "inj_on (insert x) (Pow F)")
2202 apply (simp add: card_image Pow_insert)
2203 apply (unfold inj_on_def)
2204 apply (blast elim!: equalityE)
2207 text {* Relates to equivalence classes. Based on a theorem of F. Kamm\"uller. *}
2209 lemma dvd_partition:
2210 "finite (Union C) ==>
2211 ALL c : C. k dvd card c ==>
2212 (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
2213 k dvd card (Union C)"
2214 apply (frule finite_UnionD)
2215 apply (rotate_tac -1)
2216 apply (induct rule: finite_induct)
2219 apply (subst card_Un_disjoint)
2220 apply (auto simp add: disjoint_eq_subset_Compl)
2224 subsubsection {* Relating injectivity and surjectivity *}
2226 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
2227 apply(rule eq_card_imp_inj_on, assumption)
2228 apply(frule finite_imageI)
2229 apply(drule (1) card_seteq)
2230 apply(erule card_image_le)
2234 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
2235 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
2236 by (blast intro: finite_surj_inj subset_UNIV)
2238 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
2239 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
2240 by(fastsimp simp:surj_def dest!: endo_inj_surj)
2242 corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
2244 assume "finite(UNIV::nat set)"
2245 with finite_UNIV_inj_surj[of Suc]
2246 show False by simp (blast dest: Suc_neq_Zero surjD)
2249 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
2250 lemma infinite_UNIV_char_0[no_atp]:
2251 "\<not> finite (UNIV::'a::semiring_char_0 set)"
2253 assume "finite (UNIV::'a set)"
2254 with subset_UNIV have "finite (range of_nat::'a set)"
2255 by (rule finite_subset)
2256 moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
2257 by (simp add: inj_on_def)
2258 ultimately have "finite (UNIV::nat set)"
2259 by (rule finite_imageD)