src/HOL/Finite_Set.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 49806 acb6fa98e310 child 51290 c48477e76de5 permissions -rw-r--r--
introduce order topology
1 (*  Title:      HOL/Finite_Set.thy
2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
3                 with contributions by Jeremy Avigad
4 *)
6 header {* Finite sets *}
8 theory Finite_Set
9 imports Option Power
10 begin
12 subsection {* Predicate for finite sets *}
14 inductive finite :: "'a set \<Rightarrow> bool"
15   where
16     emptyI [simp, intro!]: "finite {}"
17   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
19 simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}
21 lemma finite_induct [case_names empty insert, induct set: finite]:
22   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
23   assumes "finite F"
24   assumes "P {}"
25     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
26   shows "P F"
27 using `finite F`
28 proof induct
29   show "P {}" by fact
30   fix x F assume F: "finite F" and P: "P F"
31   show "P (insert x F)"
32   proof cases
33     assume "x \<in> F"
34     hence "insert x F = F" by (rule insert_absorb)
35     with P show ?thesis by (simp only:)
36   next
37     assume "x \<notin> F"
38     from F this P show ?thesis by (rule insert)
39   qed
40 qed
43 subsubsection {* Choice principles *}
45 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
46   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
47   shows "\<exists>a::'a. a \<notin> A"
48 proof -
49   from assms have "A \<noteq> UNIV" by blast
50   then show ?thesis by blast
51 qed
53 text {* A finite choice principle. Does not need the SOME choice operator. *}
55 lemma finite_set_choice:
56   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
57 proof (induct rule: finite_induct)
58   case empty then show ?case by simp
59 next
60   case (insert a A)
61   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
62   show ?case (is "EX f. ?P f")
63   proof
64     show "?P(%x. if x = a then b else f x)" using f ab by auto
65   qed
66 qed
69 subsubsection {* Finite sets are the images of initial segments of natural numbers *}
71 lemma finite_imp_nat_seg_image_inj_on:
72   assumes "finite A"
73   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
74 using assms
75 proof induct
76   case empty
77   show ?case
78   proof
79     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
80   qed
81 next
82   case (insert a A)
83   have notinA: "a \<notin> A" by fact
84   from insert.hyps obtain n f
85     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
86   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
87         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
88     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
89   thus ?case by blast
90 qed
92 lemma nat_seg_image_imp_finite:
93   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
94 proof (induct n arbitrary: A)
95   case 0 thus ?case by simp
96 next
97   case (Suc n)
98   let ?B = "f ` {i. i < n}"
99   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
100   show ?case
101   proof cases
102     assume "\<exists>k<n. f n = f k"
103     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
104     thus ?thesis using finB by simp
105   next
106     assume "\<not>(\<exists> k<n. f n = f k)"
107     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
108     thus ?thesis using finB by simp
109   qed
110 qed
112 lemma finite_conv_nat_seg_image:
113   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
114   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
116 lemma finite_imp_inj_to_nat_seg:
117   assumes "finite A"
118   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
119 proof -
120   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
121   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
122     by (auto simp:bij_betw_def)
123   let ?f = "the_inv_into {i. i<n} f"
124   have "inj_on ?f A & ?f ` A = {i. i<n}"
125     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
126   thus ?thesis by blast
127 qed
129 lemma finite_Collect_less_nat [iff]:
130   "finite {n::nat. n < k}"
131   by (fastforce simp: finite_conv_nat_seg_image)
133 lemma finite_Collect_le_nat [iff]:
134   "finite {n::nat. n \<le> k}"
135   by (simp add: le_eq_less_or_eq Collect_disj_eq)
138 subsubsection {* Finiteness and common set operations *}
140 lemma rev_finite_subset:
141   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
142 proof (induct arbitrary: A rule: finite_induct)
143   case empty
144   then show ?case by simp
145 next
146   case (insert x F A)
147   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
148   show "finite A"
149   proof cases
150     assume x: "x \<in> A"
151     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
152     with r have "finite (A - {x})" .
153     hence "finite (insert x (A - {x}))" ..
154     also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
155     finally show ?thesis .
156   next
157     show "A \<subseteq> F ==> ?thesis" by fact
158     assume "x \<notin> A"
159     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
160   qed
161 qed
163 lemma finite_subset:
164   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
165   by (rule rev_finite_subset)
167 lemma finite_UnI:
168   assumes "finite F" and "finite G"
169   shows "finite (F \<union> G)"
170   using assms by induct simp_all
172 lemma finite_Un [iff]:
173   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
174   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
176 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
177 proof -
178   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
179   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
180   then show ?thesis by simp
181 qed
183 lemma finite_Int [simp, intro]:
184   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
185   by (blast intro: finite_subset)
187 lemma finite_Collect_conjI [simp, intro]:
188   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
191 lemma finite_Collect_disjI [simp]:
192   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
195 lemma finite_Diff [simp, intro]:
196   "finite A \<Longrightarrow> finite (A - B)"
197   by (rule finite_subset, rule Diff_subset)
199 lemma finite_Diff2 [simp]:
200   assumes "finite B"
201   shows "finite (A - B) \<longleftrightarrow> finite A"
202 proof -
203   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
204   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
205   finally show ?thesis ..
206 qed
208 lemma finite_Diff_insert [iff]:
209   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
210 proof -
211   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
212   moreover have "A - insert a B = A - B - {a}" by auto
213   ultimately show ?thesis by simp
214 qed
216 lemma finite_compl[simp]:
217   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
220 lemma finite_Collect_not[simp]:
221   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
224 lemma finite_Union [simp, intro]:
225   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
226   by (induct rule: finite_induct) simp_all
228 lemma finite_UN_I [intro]:
229   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
230   by (induct rule: finite_induct) simp_all
232 lemma finite_UN [simp]:
233   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
234   by (blast intro: finite_subset)
236 lemma finite_Inter [intro]:
237   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
238   by (blast intro: Inter_lower finite_subset)
240 lemma finite_INT [intro]:
241   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
242   by (blast intro: INT_lower finite_subset)
244 lemma finite_imageI [simp, intro]:
245   "finite F \<Longrightarrow> finite (h ` F)"
246   by (induct rule: finite_induct) simp_all
248 lemma finite_image_set [simp]:
249   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
250   by (simp add: image_Collect [symmetric])
252 lemma finite_imageD:
253   assumes "finite (f ` A)" and "inj_on f A"
254   shows "finite A"
255 using assms
256 proof (induct "f ` A" arbitrary: A)
257   case empty then show ?case by simp
258 next
259   case (insert x B)
260   then have B_A: "insert x B = f ` A" by simp
261   then obtain y where "x = f y" and "y \<in> A" by blast
262   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
263   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)
264   moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
265   ultimately have "finite (A - {y})" by (rule insert.hyps)
266   then show "finite A" by simp
267 qed
269 lemma finite_surj:
270   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
271   by (erule finite_subset) (rule finite_imageI)
273 lemma finite_range_imageI:
274   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
275   by (drule finite_imageI) (simp add: range_composition)
277 lemma finite_subset_image:
278   assumes "finite B"
279   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
280 using assms
281 proof induct
282   case empty then show ?case by simp
283 next
284   case insert then show ?case
285     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
286        blast
287 qed
289 lemma finite_vimage_IntI:
290   "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
291   apply (induct rule: finite_induct)
292    apply simp_all
293   apply (subst vimage_insert)
294   apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
295   done
297 lemma finite_vimageI:
298   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
299   using finite_vimage_IntI[of F h UNIV] by auto
301 lemma finite_vimageD:
302   assumes fin: "finite (h -` F)" and surj: "surj h"
303   shows "finite F"
304 proof -
305   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
306   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
307   finally show "finite F" .
308 qed
310 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
311   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
313 lemma finite_Collect_bex [simp]:
314   assumes "finite A"
315   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
316 proof -
317   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
318   with assms show ?thesis by simp
319 qed
321 lemma finite_Collect_bounded_ex [simp]:
322   assumes "finite {y. P y}"
323   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
324 proof -
325   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
326   with assms show ?thesis by simp
327 qed
329 lemma finite_Plus:
330   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
333 lemma finite_PlusD:
334   fixes A :: "'a set" and B :: "'b set"
335   assumes fin: "finite (A <+> B)"
336   shows "finite A" "finite B"
337 proof -
338   have "Inl ` A \<subseteq> A <+> B" by auto
339   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
340   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
341 next
342   have "Inr ` B \<subseteq> A <+> B" by auto
343   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
344   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
345 qed
347 lemma finite_Plus_iff [simp]:
348   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
349   by (auto intro: finite_PlusD finite_Plus)
351 lemma finite_Plus_UNIV_iff [simp]:
352   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
353   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
355 lemma finite_SigmaI [simp, intro]:
356   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
357   by (unfold Sigma_def) blast
359 lemma finite_cartesian_product:
360   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
361   by (rule finite_SigmaI)
363 lemma finite_Prod_UNIV:
364   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
365   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
367 lemma finite_cartesian_productD1:
368   assumes "finite (A \<times> B)" and "B \<noteq> {}"
369   shows "finite A"
370 proof -
371   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
372     by (auto simp add: finite_conv_nat_seg_image)
373   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
374   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
376   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
377   then show ?thesis
378     by (auto simp add: finite_conv_nat_seg_image)
379 qed
381 lemma finite_cartesian_productD2:
382   assumes "finite (A \<times> B)" and "A \<noteq> {}"
383   shows "finite B"
384 proof -
385   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
386     by (auto simp add: finite_conv_nat_seg_image)
387   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
388   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
390   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
391   then show ?thesis
392     by (auto simp add: finite_conv_nat_seg_image)
393 qed
395 lemma finite_prod:
396   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
397 by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV
398    dest: finite_cartesian_productD1 finite_cartesian_productD2)
400 lemma finite_Pow_iff [iff]:
401   "finite (Pow A) \<longleftrightarrow> finite A"
402 proof
403   assume "finite (Pow A)"
404   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
405   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
406 next
407   assume "finite A"
408   then show "finite (Pow A)"
409     by induct (simp_all add: Pow_insert)
410 qed
412 corollary finite_Collect_subsets [simp, intro]:
413   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
414   by (simp add: Pow_def [symmetric])
416 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
417 by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
419 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
420   by (blast intro: finite_subset [OF subset_Pow_Union])
423 subsubsection {* Further induction rules on finite sets *}
425 lemma finite_ne_induct [case_names singleton insert, consumes 2]:
426   assumes "finite F" and "F \<noteq> {}"
427   assumes "\<And>x. P {x}"
428     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
429   shows "P F"
430 using assms
431 proof induct
432   case empty then show ?case by simp
433 next
434   case (insert x F) then show ?case by cases auto
435 qed
437 lemma finite_subset_induct [consumes 2, case_names empty insert]:
438   assumes "finite F" and "F \<subseteq> A"
439   assumes empty: "P {}"
440     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
441   shows "P F"
442 using `finite F` `F \<subseteq> A`
443 proof induct
444   show "P {}" by fact
445 next
446   fix x F
447   assume "finite F" and "x \<notin> F" and
448     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
449   show "P (insert x F)"
450   proof (rule insert)
451     from i show "x \<in> A" by blast
452     from i have "F \<subseteq> A" by blast
453     with P show "P F" .
454     show "finite F" by fact
455     show "x \<notin> F" by fact
456   qed
457 qed
459 lemma finite_empty_induct:
460   assumes "finite A"
461   assumes "P A"
462     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
463   shows "P {}"
464 proof -
465   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
466   proof -
467     fix B :: "'a set"
468     assume "B \<subseteq> A"
469     with `finite A` have "finite B" by (rule rev_finite_subset)
470     from this `B \<subseteq> A` show "P (A - B)"
471     proof induct
472       case empty
473       from `P A` show ?case by simp
474     next
475       case (insert b B)
476       have "P (A - B - {b})"
477       proof (rule remove)
478         from `finite A` show "finite (A - B)" by induct auto
479         from insert show "b \<in> A - B" by simp
480         from insert show "P (A - B)" by simp
481       qed
482       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
483       finally show ?case .
484     qed
485   qed
486   then have "P (A - A)" by blast
487   then show ?thesis by simp
488 qed
491 subsection {* Class @{text finite}  *}
493 class finite =
494   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
495 begin
497 lemma finite [simp]: "finite (A \<Colon> 'a set)"
498   by (rule subset_UNIV finite_UNIV finite_subset)+
500 lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
501   by simp
503 end
505 instance prod :: (finite, finite) finite
506   by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
508 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
509   by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
511 instance "fun" :: (finite, finite) finite
512 proof
513   show "finite (UNIV :: ('a => 'b) set)"
514   proof (rule finite_imageD)
515     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
516     have "range ?graph \<subseteq> Pow UNIV" by simp
517     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
518       by (simp only: finite_Pow_iff finite)
519     ultimately show "finite (range ?graph)"
520       by (rule finite_subset)
521     show "inj ?graph" by (rule inj_graph)
522   qed
523 qed
525 instance bool :: finite
526   by default (simp add: UNIV_bool)
528 instance set :: (finite) finite
529   by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
531 instance unit :: finite
532   by default (simp add: UNIV_unit)
534 instance sum :: (finite, finite) finite
535   by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
537 lemma finite_option_UNIV [simp]:
538   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
539   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
541 instance option :: (finite) finite
542   by default (simp add: UNIV_option_conv)
545 subsection {* A basic fold functional for finite sets *}
547 text {* The intended behaviour is
548 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
549 if @{text f} is ``left-commutative'':
550 *}
552 locale comp_fun_commute =
553   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
554   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
555 begin
557 lemma fun_left_comm: "f x (f y z) = f y (f x z)"
558   using comp_fun_commute by (simp add: fun_eq_iff)
560 end
562 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
563 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
564   emptyI [intro]: "fold_graph f z {} z" |
565   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
566       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
568 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
570 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
571   "fold f z A = (THE y. fold_graph f z A y)"
573 text{*A tempting alternative for the definiens is
574 @{term "if finite A then THE y. fold_graph f z A y else e"}.
575 It allows the removal of finiteness assumptions from the theorems
576 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
577 The proofs become ugly. It is not worth the effort. (???) *}
579 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
580 by (induct rule: finite_induct) auto
583 subsubsection{*From @{const fold_graph} to @{term fold}*}
585 context comp_fun_commute
586 begin
588 lemma fold_graph_insertE_aux:
589   "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'"
590 proof (induct set: fold_graph)
591   case (insertI x A y) show ?case
592   proof (cases "x = a")
593     assume "x = a" with insertI show ?case by auto
594   next
595     assume "x \<noteq> a"
596     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
597       using insertI by auto
598     have "f x y = f a (f x y')"
599       unfolding y by (rule fun_left_comm)
600     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
601       using y' and `x \<noteq> a` and `x \<notin> A`
602       by (simp add: insert_Diff_if fold_graph.insertI)
603     ultimately show ?case by fast
604   qed
605 qed simp
607 lemma fold_graph_insertE:
608   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
609   obtains y where "v = f x y" and "fold_graph f z A y"
610 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
612 lemma fold_graph_determ:
613   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
614 proof (induct arbitrary: y set: fold_graph)
615   case (insertI x A y v)
616   from `fold_graph f z (insert x A) v` and `x \<notin> A`
617   obtain y' where "v = f x y'" and "fold_graph f z A y'"
618     by (rule fold_graph_insertE)
619   from `fold_graph f z A y'` have "y' = y" by (rule insertI)
620   with `v = f x y'` show "v = f x y" by simp
621 qed fast
623 lemma fold_equality:
624   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
625 by (unfold fold_def) (blast intro: fold_graph_determ)
627 lemma fold_graph_fold:
628   assumes "finite A"
629   shows "fold_graph f z A (fold f z A)"
630 proof -
631   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
632   moreover note fold_graph_determ
633   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
634   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
635   then show ?thesis by (unfold fold_def)
636 qed
638 text{* The base case for @{text fold}: *}
640 lemma (in -) fold_empty [simp]: "fold f z {} = z"
641 by (unfold fold_def) blast
643 text{* The various recursion equations for @{const fold}: *}
645 lemma fold_insert [simp]:
646   assumes "finite A" and "x \<notin> A"
647   shows "fold f z (insert x A) = f x (fold f z A)"
648 proof (rule fold_equality)
649   from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
650   with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
651 qed
653 lemma fold_fun_comm:
654   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
655 proof (induct rule: finite_induct)
656   case empty then show ?case by simp
657 next
658   case (insert y A) then show ?case
659     by (simp add: fun_left_comm[of x])
660 qed
662 lemma fold_insert2:
663   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
666 lemma fold_rec:
667   assumes "finite A" and "x \<in> A"
668   shows "fold f z A = f x (fold f z (A - {x}))"
669 proof -
670   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
671   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
672   also have "\<dots> = f x (fold f z (A - {x}))"
673     by (rule fold_insert) (simp add: `finite A`)+
674   finally show ?thesis .
675 qed
677 lemma fold_insert_remove:
678   assumes "finite A"
679   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
680 proof -
681   from `finite A` have "finite (insert x A)" by auto
682   moreover have "x \<in> insert x A" by auto
683   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
684     by (rule fold_rec)
685   then show ?thesis by simp
686 qed
688 text{* Other properties of @{const fold}: *}
690 lemma fold_image:
691   assumes "finite A" and "inj_on g A"
692   shows "fold f x (g ` A) = fold (f \<circ> g) x A"
693 using assms
694 proof induction
695   case (insert a F)
696     interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
697     from insert show ?case by auto
698 qed (simp)
700 end
702 lemma fold_cong:
703   assumes "comp_fun_commute f" "comp_fun_commute g"
704   assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
705     and "A = B" and "s = t"
706   shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
707 proof -
708   have "Finite_Set.fold f s A = Finite_Set.fold g s A"
709   using `finite A` cong proof (induct A)
710     case empty then show ?case by simp
711   next
712     case (insert x A)
713     interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
714     interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
715     from insert show ?case by simp
716   qed
717   with assms show ?thesis by simp
718 qed
721 text{* A simplified version for idempotent functions: *}
723 locale comp_fun_idem = comp_fun_commute +
724   assumes comp_fun_idem: "f x o f x = f x"
725 begin
727 lemma fun_left_idem: "f x (f x z) = f x z"
728   using comp_fun_idem by (simp add: fun_eq_iff)
730 lemma fold_insert_idem:
731   assumes fin: "finite A"
732   shows "fold f z (insert x A) = f x (fold f z A)"
733 proof cases
734   assume "x \<in> A"
735   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
736   then show ?thesis using assms by (simp add:fun_left_idem)
737 next
738   assume "x \<notin> A" then show ?thesis using assms by simp
739 qed
741 declare fold_insert[simp del] fold_insert_idem[simp]
743 lemma fold_insert_idem2:
744   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
747 end
750 subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
752 lemma (in comp_fun_commute) comp_comp_fun_commute:
753   "comp_fun_commute (f \<circ> g)"
754 proof
757 lemma (in comp_fun_idem) comp_comp_fun_idem:
758   "comp_fun_idem (f \<circ> g)"
759   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
762 lemma (in comp_fun_commute) comp_fun_commute_funpow:
763   "comp_fun_commute (\<lambda>x. f x ^^ g x)"
764 proof
765   fix y x
766   show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
767   proof (cases "x = y")
768     case True then show ?thesis by simp
769   next
770     case False show ?thesis
771     proof (induct "g x" arbitrary: g)
772       case 0 then show ?case by simp
773     next
774       case (Suc n g)
775       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
776       proof (induct "g y" arbitrary: g)
777         case 0 then show ?case by simp
778       next
779         case (Suc n g)
780         def h \<equiv> "\<lambda>z. g z - 1"
781         with Suc have "n = h y" by simp
782         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
783           by auto
784         from Suc h_def have "g y = Suc (h y)" by simp
785         then show ?case by (simp add: comp_assoc hyp)
787       qed
788       def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
789       with Suc have "n = h x" by simp
790       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
791         by auto
792       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
793       from Suc h_def have "g x = Suc (h x)" by simp
794       then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
796     qed
797   qed
798 qed
801 subsubsection {* Expressing set operations via @{const fold} *}
803 lemma comp_fun_idem_insert:
804   "comp_fun_idem insert"
805 proof
806 qed auto
808 lemma comp_fun_idem_remove:
809   "comp_fun_idem Set.remove"
810 proof
811 qed auto
813 lemma (in semilattice_inf) comp_fun_idem_inf:
814   "comp_fun_idem inf"
815 proof
816 qed (auto simp add: inf_left_commute)
818 lemma (in semilattice_sup) comp_fun_idem_sup:
819   "comp_fun_idem sup"
820 proof
821 qed (auto simp add: sup_left_commute)
823 lemma union_fold_insert:
824   assumes "finite A"
825   shows "A \<union> B = fold insert B A"
826 proof -
827   interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
828   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
829 qed
831 lemma minus_fold_remove:
832   assumes "finite A"
833   shows "B - A = fold Set.remove B A"
834 proof -
835   interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
836   from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
837   then show ?thesis ..
838 qed
840 lemma comp_fun_commute_filter_fold: "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
841 proof -
842   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
843   show ?thesis by default (auto simp: fun_eq_iff)
844 qed
846 lemma Set_filter_fold:
847   assumes "finite A"
848   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
849 using assms
850 by (induct A)
851   (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
853 lemma inter_Set_filter:
854   assumes "finite B"
855   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
856 using assms
857 by (induct B) (auto simp: Set.filter_def)
859 lemma image_fold_insert:
860   assumes "finite A"
861   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
862 using assms
863 proof -
864   interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
865   show ?thesis using assms by (induct A) auto
866 qed
868 lemma Ball_fold:
869   assumes "finite A"
870   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
871 using assms
872 proof -
873   interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
874   show ?thesis using assms by (induct A) auto
875 qed
877 lemma Bex_fold:
878   assumes "finite A"
879   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
880 using assms
881 proof -
882   interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
883   show ?thesis using assms by (induct A) auto
884 qed
886 lemma comp_fun_commute_Pow_fold:
887   "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
888   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
890 lemma Pow_fold:
891   assumes "finite A"
892   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
893 using assms
894 proof -
895   interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
896   show ?thesis using assms by (induct A) (auto simp: Pow_insert)
897 qed
899 lemma fold_union_pair:
900   assumes "finite B"
901   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
902 proof -
903   interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
904   show ?thesis using assms  by (induct B arbitrary: A) simp_all
905 qed
907 lemma comp_fun_commute_product_fold:
908   assumes "finite B"
909   shows "comp_fun_commute (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B)"
910 by default (auto simp: fold_union_pair[symmetric] assms)
912 lemma product_fold:
913   assumes "finite A"
914   assumes "finite B"
915   shows "A \<times> B = fold (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B) {} A"
916 using assms unfolding Sigma_def
917 by (induct A)
918   (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
921 context complete_lattice
922 begin
924 lemma inf_Inf_fold_inf:
925   assumes "finite A"
926   shows "inf B (Inf A) = fold inf B A"
927 proof -
928   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
929   from `finite A` show ?thesis by (induct A arbitrary: B)
931 qed
933 lemma sup_Sup_fold_sup:
934   assumes "finite A"
935   shows "sup B (Sup A) = fold sup B A"
936 proof -
937   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
938   from `finite A` show ?thesis by (induct A arbitrary: B)
940 qed
942 lemma Inf_fold_inf:
943   assumes "finite A"
944   shows "Inf A = fold inf top A"
945   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
947 lemma Sup_fold_sup:
948   assumes "finite A"
949   shows "Sup A = fold sup bot A"
950   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
952 lemma inf_INF_fold_inf:
953   assumes "finite A"
954   shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
955 proof (rule sym)
956   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
957   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
958   from `finite A` show "?fold = ?inf"
959     by (induct A arbitrary: B)
961 qed
963 lemma sup_SUP_fold_sup:
964   assumes "finite A"
965   shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
966 proof (rule sym)
967   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
968   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
969   from `finite A` show "?fold = ?sup"
970     by (induct A arbitrary: B)
972 qed
974 lemma INF_fold_inf:
975   assumes "finite A"
976   shows "INFI A f = fold (inf \<circ> f) top A"
977   using assms inf_INF_fold_inf [of A top] by simp
979 lemma SUP_fold_sup:
980   assumes "finite A"
981   shows "SUPR A f = fold (sup \<circ> f) bot A"
982   using assms sup_SUP_fold_sup [of A bot] by simp
984 end
987 subsection {* The derived combinator @{text fold_image} *}
989 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
990   where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
992 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
995 context ab_semigroup_mult
996 begin
998 lemma fold_image_insert[simp]:
999   assumes "finite A" and "a \<notin> A"
1000   shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
1001 proof -
1002   interpret comp_fun_commute "%x y. (g x) * y"
1003     by default (simp add: fun_eq_iff mult_ac)
1004   from assms show ?thesis by (simp add: fold_image_def)
1005 qed
1007 lemma fold_image_reindex:
1008   assumes "finite A"
1009   shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
1010   using assms by induct auto
1012 lemma fold_image_cong:
1013   assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
1014   shows "fold_image times g z A = fold_image times h z A"
1015 proof -
1016   from `finite A`
1017   have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
1018   proof (induct arbitrary: C)
1019     case empty then show ?case by simp
1020   next
1021     case (insert x F) then show ?case apply -
1022     apply (simp add: subset_insert_iff, clarify)
1023     apply (subgoal_tac "finite C")
1024       prefer 2 apply (blast dest: finite_subset [rotated])
1025     apply (subgoal_tac "C = insert x (C - {x})")
1026       prefer 2 apply blast
1027     apply (erule ssubst)
1028     apply (simp add: Ball_def del: insert_Diff_single)
1029     done
1030   qed
1031   with g_h show ?thesis by simp
1032 qed
1034 end
1036 context comm_monoid_mult
1037 begin
1039 lemma fold_image_1:
1040   "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
1041   apply (induct rule: finite_induct)
1042   apply simp by auto
1044 lemma fold_image_Un_Int:
1045   "finite A ==> finite B ==>
1046     fold_image times g 1 A * fold_image times g 1 B =
1047     fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
1048   apply (induct rule: finite_induct)
1049 by (induct set: finite)
1050    (auto simp add: mult_ac insert_absorb Int_insert_left)
1052 lemma fold_image_Un_one:
1053   assumes fS: "finite S" and fT: "finite T"
1054   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
1055   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
1056 proof-
1057   have "fold_image op * f 1 (S \<inter> T) = 1"
1058     apply (rule fold_image_1)
1059     using fS fT I0 by auto
1060   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
1061 qed
1063 corollary fold_Un_disjoint:
1064   "finite A ==> finite B ==> A Int B = {} ==>
1065    fold_image times g 1 (A Un B) =
1066    fold_image times g 1 A * fold_image times g 1 B"
1069 lemma fold_image_UN_disjoint:
1070   "\<lbrakk> finite I; ALL i:I. finite (A i);
1071      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
1072    \<Longrightarrow> fold_image times g 1 (UNION I A) =
1073        fold_image times (%i. fold_image times g 1 (A i)) 1 I"
1074 apply (induct rule: finite_induct)
1075 apply simp
1076 apply atomize
1077 apply (subgoal_tac "ALL i:F. x \<noteq> i")
1078  prefer 2 apply blast
1079 apply (subgoal_tac "A x Int UNION F A = {}")
1080  prefer 2 apply blast
1082 done
1084 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
1085   fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
1086   fold_image times (split g) 1 (SIGMA x:A. B x)"
1087 apply (subst Sigma_def)
1088 apply (subst fold_image_UN_disjoint, assumption, simp)
1089  apply blast
1090 apply (erule fold_image_cong)
1091 apply (subst fold_image_UN_disjoint, simp, simp)
1092  apply blast
1093 apply simp
1094 done
1096 lemma fold_image_distrib: "finite A \<Longrightarrow>
1097    fold_image times (%x. g x * h x) 1 A =
1098    fold_image times g 1 A *  fold_image times h 1 A"
1099 by (erule finite_induct) (simp_all add: mult_ac)
1101 lemma fold_image_related:
1102   assumes Re: "R e e"
1103   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
1104   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
1105   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
1106   using fS by (rule finite_subset_induct) (insert assms, auto)
1108 lemma  fold_image_eq_general:
1109   assumes fS: "finite S"
1110   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
1111   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
1112   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
1113 proof-
1114   from h f12 have hS: "h ` S = S'" by auto
1115   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1116     from f12 h H  have "x = y" by auto }
1117   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1118   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1119   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
1120   also have "\<dots> = fold_image (op *) (f2 o h) e S"
1121     using fold_image_reindex[OF fS hinj, of f2 e] .
1122   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
1123     by blast
1124   finally show ?thesis ..
1125 qed
1127 lemma fold_image_eq_general_inverses:
1128   assumes fS: "finite S"
1129   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1130   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
1131   shows "fold_image (op *) f e S = fold_image (op *) g e T"
1132   (* metis solves it, but not yet available here *)
1133   apply (rule fold_image_eq_general[OF fS, of T h g f e])
1134   apply (rule ballI)
1135   apply (frule kh)
1136   apply (rule ex1I[])
1137   apply blast
1138   apply clarsimp
1139   apply (drule hk) apply simp
1140   apply (rule sym)
1141   apply (erule conjunct1[OF conjunct2[OF hk]])
1142   apply (rule ballI)
1143   apply (drule  hk)
1144   apply blast
1145   done
1147 end
1150 subsection {* A fold functional for non-empty sets *}
1152 text{* Does not require start value. *}
1154 inductive
1155   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
1156   for f :: "'a => 'a => 'a"
1157 where
1158   fold1Set_insertI [intro]:
1159    "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
1161 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
1162   "fold1 f A == THE x. fold1Set f A x"
1164 lemma fold1Set_nonempty:
1165   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
1166 by(erule fold1Set.cases, simp_all)
1168 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
1170 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
1173 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
1174 by (blast elim: fold_graph.cases)
1176 lemma fold1_singleton [simp]: "fold1 f {a} = a"
1177 by (unfold fold1_def) blast
1179 lemma finite_nonempty_imp_fold1Set:
1180   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
1181 apply (induct A rule: finite_induct)
1182 apply (auto dest: finite_imp_fold_graph [of _ f])
1183 done
1185 text{*First, some lemmas about @{const fold_graph}.*}
1187 context ab_semigroup_mult
1188 begin
1190 lemma comp_fun_commute: "comp_fun_commute (op *)"
1191   by default (simp add: fun_eq_iff mult_ac)
1193 lemma fold_graph_insert_swap:
1194 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
1195 shows "fold_graph times z (insert b A) (z * y)"
1196 proof -
1197   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1198 from assms show ?thesis
1199 proof (induct rule: fold_graph.induct)
1200   case emptyI show ?case by (subst mult_commute [of z b], fast)
1201 next
1202   case (insertI x A y)
1203     have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
1204       using insertI by force  --{*how does @{term id} get unfolded?*}
1205     thus ?case by (simp add: insert_commute mult_ac)
1206 qed
1207 qed
1209 lemma fold_graph_permute_diff:
1210 assumes fold: "fold_graph times b A x"
1211 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
1212 using fold
1213 proof (induct rule: fold_graph.induct)
1214   case emptyI thus ?case by simp
1215 next
1216   case (insertI x A y)
1217   have "a = x \<or> a \<in> A" using insertI by simp
1218   thus ?case
1219   proof
1220     assume "a = x"
1221     with insertI show ?thesis
1222       by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
1223   next
1224     assume ainA: "a \<in> A"
1225     hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
1226       using insertI by force
1227     moreover
1228     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
1229       using ainA insertI by blast
1230     ultimately show ?thesis by simp
1231   qed
1232 qed
1234 lemma fold1_eq_fold:
1235 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
1236 proof -
1237   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1238   from assms show ?thesis
1239 apply (simp add: fold1_def fold_def)
1240 apply (rule the_equality)
1241 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
1242 apply (rule sym, clarify)
1243 apply (case_tac "Aa=A")
1244  apply (best intro: fold_graph_determ)
1245 apply (subgoal_tac "fold_graph times a A x")
1246  apply (best intro: fold_graph_determ)
1247 apply (subgoal_tac "insert aa (Aa - {a}) = A")
1248  prefer 2 apply (blast elim: equalityE)
1249 apply (auto dest: fold_graph_permute_diff [where a=a])
1250 done
1251 qed
1253 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
1254 apply safe
1255  apply simp
1256  apply (drule_tac x=x in spec)
1257  apply (drule_tac x="A-{x}" in spec, auto)
1258 done
1260 lemma fold1_insert:
1261   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
1262   shows "fold1 times (insert x A) = x * fold1 times A"
1263 proof -
1264   interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
1265   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
1266     by (auto simp add: nonempty_iff)
1267   with A show ?thesis
1268     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
1269 qed
1271 end
1273 context ab_semigroup_idem_mult
1274 begin
1276 lemma comp_fun_idem: "comp_fun_idem (op *)"
1277   by default (simp_all add: fun_eq_iff mult_left_commute)
1279 lemma fold1_insert_idem [simp]:
1280   assumes nonempty: "A \<noteq> {}" and A: "finite A"
1281   shows "fold1 times (insert x A) = x * fold1 times A"
1282 proof -
1283   interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
1284     by (rule comp_fun_idem)
1285   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
1286     by (auto simp add: nonempty_iff)
1287   show ?thesis
1288   proof cases
1289     assume a: "a = x"
1290     show ?thesis
1291     proof cases
1292       assume "A' = {}"
1293       with A' a show ?thesis by simp
1294     next
1295       assume "A' \<noteq> {}"
1296       with A A' a show ?thesis
1297         by (simp add: fold1_insert mult_assoc [symmetric])
1298     qed
1299   next
1300     assume "a \<noteq> x"
1301     with A A' show ?thesis
1302       by (simp add: insert_commute fold1_eq_fold)
1303   qed
1304 qed
1306 lemma hom_fold1_commute:
1307 assumes hom: "!!x y. h (x * y) = h x * h y"
1308 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
1309 using N
1310 proof (induct rule: finite_ne_induct)
1311   case singleton thus ?case by simp
1312 next
1313   case (insert n N)
1314   then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
1315   also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
1316   also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
1317   also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
1318     using insert by(simp)
1319   also have "insert (h n) (h ` N) = h ` insert n N" by simp
1320   finally show ?case .
1321 qed
1323 lemma fold1_eq_fold_idem:
1324   assumes "finite A"
1325   shows "fold1 times (insert a A) = fold times a A"
1326 proof (cases "a \<in> A")
1327   case False
1328   with assms show ?thesis by (simp add: fold1_eq_fold)
1329 next
1330   interpret comp_fun_idem times by (fact comp_fun_idem)
1331   case True then obtain b B
1332     where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
1333   with assms have "finite B" by auto
1334   then have "fold times a (insert a B) = fold times (a * a) B"
1335     using `a \<notin> B` by (rule fold_insert2)
1336   then show ?thesis
1337     using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
1338 qed
1340 end
1343 text{* Now the recursion rules for definitions: *}
1345 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
1346 by simp
1348 lemma (in ab_semigroup_mult) fold1_insert_def:
1349   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1352 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
1353   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1354 by simp
1356 subsubsection{* Determinacy for @{term fold1Set} *}
1358 (*Not actually used!!*)
1359 (*
1360 context ab_semigroup_mult
1361 begin
1363 lemma fold_graph_permute:
1364   "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
1365    ==> fold_graph times id a (insert b A) x"
1366 apply (cases "a=b")
1367 apply (auto dest: fold_graph_permute_diff)
1368 done
1370 lemma fold1Set_determ:
1371   "fold1Set times A x ==> fold1Set times A y ==> y = x"
1372 proof (clarify elim!: fold1Set.cases)
1373   fix A x B y a b
1374   assume Ax: "fold_graph times id a A x"
1375   assume By: "fold_graph times id b B y"
1376   assume anotA:  "a \<notin> A"
1377   assume bnotB:  "b \<notin> B"
1378   assume eq: "insert a A = insert b B"
1379   show "y=x"
1380   proof cases
1381     assume same: "a=b"
1382     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
1383     thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
1384   next
1385     assume diff: "a\<noteq>b"
1386     let ?D = "B - {a}"
1387     have B: "B = insert a ?D" and A: "A = insert b ?D"
1388      and aB: "a \<in> B" and bA: "b \<in> A"
1389       using eq anotA bnotB diff by (blast elim!:equalityE)+
1390     with aB bnotB By
1391     have "fold_graph times id a (insert b ?D) y"
1392       by (auto intro: fold_graph_permute simp add: insert_absorb)
1393     moreover
1394     have "fold_graph times id a (insert b ?D) x"
1395       by (simp add: A [symmetric] Ax)
1396     ultimately show ?thesis by (blast intro: fold_graph_determ)
1397   qed
1398 qed
1400 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
1401   by (unfold fold1_def) (blast intro: fold1Set_determ)
1403 end
1404 *)
1406 declare
1407   empty_fold_graphE [rule del]  fold_graph.intros [rule del]
1408   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
1409   -- {* No more proofs involve these relations. *}
1411 subsubsection {* Lemmas about @{text fold1} *}
1413 context ab_semigroup_mult
1414 begin
1416 lemma fold1_Un:
1417 assumes A: "finite A" "A \<noteq> {}"
1418 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
1419        fold1 times (A Un B) = fold1 times A * fold1 times B"
1420 using A by (induct rule: finite_ne_induct)
1423 lemma fold1_in:
1424   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
1425   shows "fold1 times A \<in> A"
1426 using A
1427 proof (induct rule:finite_ne_induct)
1428   case singleton thus ?case by simp
1429 next
1430   case insert thus ?case using elem by (force simp add:fold1_insert)
1431 qed
1433 end
1435 lemma (in ab_semigroup_idem_mult) fold1_Un2:
1436 assumes A: "finite A" "A \<noteq> {}"
1437 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
1438        fold1 times (A Un B) = fold1 times A * fold1 times B"
1439 using A
1440 proof(induct rule:finite_ne_induct)
1441   case singleton thus ?case by simp
1442 next
1443   case insert thus ?case by (simp add: mult_assoc)
1444 qed
1447 subsection {* Locales as mini-packages for fold operations *}
1449 subsubsection {* The natural case *}
1451 locale folding =
1452   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
1453   fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
1454   assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
1455   assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
1456 begin
1458 lemma empty [simp]:
1459   "F {} = id"
1460   by (simp add: eq_fold fun_eq_iff)
1462 lemma insert [simp]:
1463   assumes "finite A" and "x \<notin> A"
1464   shows "F (insert x A) = F A \<circ> f x"
1465 proof -
1466   interpret comp_fun_commute f
1467     by default (insert comp_fun_commute, simp add: fun_eq_iff)
1468   from fold_insert2 assms
1469   have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
1470   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
1471 qed
1473 lemma remove:
1474   assumes "finite A" and "x \<in> A"
1475   shows "F A = F (A - {x}) \<circ> f x"
1476 proof -
1477   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1478     by (auto dest: mk_disjoint_insert)
1479   moreover from `finite A` this have "finite B" by simp
1480   ultimately show ?thesis by simp
1481 qed
1483 lemma insert_remove:
1484   assumes "finite A"
1485   shows "F (insert x A) = F (A - {x}) \<circ> f x"
1486   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1488 lemma commute_left_comp:
1489   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
1490   by (simp add: o_assoc comp_fun_commute)
1492 lemma comp_fun_commute':
1493   assumes "finite A"
1494   shows "f x \<circ> F A = F A \<circ> f x"
1495   using assms by (induct A)
1496     (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: comp_assoc comp_fun_commute)
1498 lemma commute_left_comp':
1499   assumes "finite A"
1500   shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
1501   using assms by (simp add: o_assoc comp_fun_commute')
1503 lemma comp_fun_commute'':
1504   assumes "finite A" and "finite B"
1505   shows "F B \<circ> F A = F A \<circ> F B"
1506   using assms by (induct A)
1509 lemma commute_left_comp'':
1510   assumes "finite A" and "finite B"
1511   shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
1512   using assms by (simp add: o_assoc comp_fun_commute'')
1514 lemmas comp_fun_commutes = comp_assoc comp_fun_commute commute_left_comp
1515   comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
1517 lemma union_inter:
1518   assumes "finite A" and "finite B"
1519   shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
1520   using assms by (induct A)
1521     (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
1524 lemma union:
1525   assumes "finite A" and "finite B"
1526   and "A \<inter> B = {}"
1527   shows "F (A \<union> B) = F A \<circ> F B"
1528 proof -
1529   from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
1530   with `A \<inter> B = {}` show ?thesis by simp
1531 qed
1533 end
1536 subsubsection {* The natural case with idempotency *}
1538 locale folding_idem = folding +
1539   assumes idem_comp: "f x \<circ> f x = f x"
1540 begin
1542 lemma idem_left_comp:
1543   "f x \<circ> (f x \<circ> g) = f x \<circ> g"
1544   by (simp add: o_assoc idem_comp)
1546 lemma in_comp_idem:
1547   assumes "finite A" and "x \<in> A"
1548   shows "F A \<circ> f x = F A"
1549 using assms by (induct A)
1552 lemma subset_comp_idem:
1553   assumes "finite A" and "B \<subseteq> A"
1554   shows "F A \<circ> F B = F A"
1555 proof -
1556   from assms have "finite B" by (blast dest: finite_subset)
1557   then show ?thesis using `B \<subseteq> A` by (induct B)
1558     (simp_all add: o_assoc in_comp_idem `finite A`)
1559 qed
1561 declare insert [simp del]
1563 lemma insert_idem [simp]:
1564   assumes "finite A"
1565   shows "F (insert x A) = F A \<circ> f x"
1566   using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
1568 lemma union_idem:
1569   assumes "finite A" and "finite B"
1570   shows "F (A \<union> B) = F A \<circ> F B"
1571 proof -
1572   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1573   then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
1574   with assms show ?thesis by (simp add: union_inter)
1575 qed
1577 end
1580 subsubsection {* The image case with fixed function *}
1582 no_notation times (infixl "*" 70)
1583 no_notation Groups.one ("1")
1585 locale folding_image_simple = comm_monoid +
1586   fixes g :: "('b \<Rightarrow> 'a)"
1587   fixes F :: "'b set \<Rightarrow> 'a"
1588   assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
1589 begin
1591 lemma empty [simp]:
1592   "F {} = 1"
1595 lemma insert [simp]:
1596   assumes "finite A" and "x \<notin> A"
1597   shows "F (insert x A) = g x * F A"
1598 proof -
1599   interpret comp_fun_commute "%x y. (g x) * y"
1600     by default (simp add: ac_simps fun_eq_iff)
1601   from assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
1603   with `finite A` show ?thesis by (simp add: eq_fold_g)
1604 qed
1606 lemma remove:
1607   assumes "finite A" and "x \<in> A"
1608   shows "F A = g x * F (A - {x})"
1609 proof -
1610   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1611     by (auto dest: mk_disjoint_insert)
1612   moreover from `finite A` this have "finite B" by simp
1613   ultimately show ?thesis by simp
1614 qed
1616 lemma insert_remove:
1617   assumes "finite A"
1618   shows "F (insert x A) = g x * F (A - {x})"
1619   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1621 lemma neutral:
1622   assumes "finite A" and "\<forall>x\<in>A. g x = 1"
1623   shows "F A = 1"
1624   using assms by (induct A) simp_all
1626 lemma union_inter:
1627   assumes "finite A" and "finite B"
1628   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1629 using assms proof (induct A)
1630   case empty then show ?case by simp
1631 next
1632   case (insert x A) then show ?case
1633     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
1634 qed
1636 corollary union_inter_neutral:
1637   assumes "finite A" and "finite B"
1638   and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
1639   shows "F (A \<union> B) = F A * F B"
1640   using assms by (simp add: union_inter [symmetric] neutral)
1642 corollary union_disjoint:
1643   assumes "finite A" and "finite B"
1644   assumes "A \<inter> B = {}"
1645   shows "F (A \<union> B) = F A * F B"
1646   using assms by (simp add: union_inter_neutral)
1648 end
1651 subsubsection {* The image case with flexible function *}
1653 locale folding_image = comm_monoid +
1654   fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
1655   assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
1657 sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
1658 qed (fact eq_fold)
1660 context folding_image
1661 begin
1663 lemma reindex: (* FIXME polymorhism *)
1664   assumes "finite A" and "inj_on h A"
1665   shows "F g (h ` A) = F (g \<circ> h) A"
1666   using assms by (induct A) auto
1668 lemma cong:
1669   assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
1670   shows "F g A = F h A"
1671 proof -
1672   from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
1673   apply - apply (erule finite_induct) apply simp
1674   apply (simp add: subset_insert_iff, clarify)
1675   apply (subgoal_tac "finite C")
1676   prefer 2 apply (blast dest: finite_subset [rotated])
1677   apply (subgoal_tac "C = insert x (C - {x})")
1678   prefer 2 apply blast
1679   apply (erule ssubst)
1680   apply (drule spec)
1681   apply (erule (1) notE impE)
1682   apply (simp add: Ball_def del: insert_Diff_single)
1683   done
1684   with assms show ?thesis by simp
1685 qed
1687 lemma UNION_disjoint:
1688   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
1689   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
1690   shows "F g (UNION I A) = F (F g \<circ> A) I"
1691 apply (insert assms)
1692 apply (induct rule: finite_induct)
1693 apply simp
1694 apply atomize
1695 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
1696  prefer 2 apply blast
1697 apply (subgoal_tac "A x Int UNION Fa A = {}")
1698  prefer 2 apply blast
1700 done
1702 lemma distrib:
1703   assumes "finite A"
1704   shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
1705   using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
1707 lemma related:
1708   assumes Re: "R 1 1"
1709   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
1710   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
1711   shows "R (F h S) (F g S)"
1712   using fS by (rule finite_subset_induct) (insert assms, auto)
1714 lemma eq_general:
1715   assumes fS: "finite S"
1716   and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
1717   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
1718   shows "F f1 S = F f2 S'"
1719 proof-
1720   from h f12 have hS: "h ` S = S'" by blast
1721   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1722     from f12 h H  have "x = y" by auto }
1723   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1724   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1725   from hS have "F f2 S' = F f2 (h ` S)" by simp
1726   also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
1727   also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
1728     by blast
1729   finally show ?thesis ..
1730 qed
1732 lemma eq_general_inverses:
1733   assumes fS: "finite S"
1734   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1735   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
1736   shows "F j S = F g T"
1737   (* metis solves it, but not yet available here *)
1738   apply (rule eq_general [OF fS, of T h g j])
1739   apply (rule ballI)
1740   apply (frule kh)
1741   apply (rule ex1I[])
1742   apply blast
1743   apply clarsimp
1744   apply (drule hk) apply simp
1745   apply (rule sym)
1746   apply (erule conjunct1[OF conjunct2[OF hk]])
1747   apply (rule ballI)
1748   apply (drule hk)
1749   apply blast
1750   done
1752 end
1755 subsubsection {* The image case with fixed function and idempotency *}
1757 locale folding_image_simple_idem = folding_image_simple +
1758   assumes idem: "x * x = x"
1760 sublocale folding_image_simple_idem < semilattice: semilattice proof
1761 qed (fact idem)
1763 context folding_image_simple_idem
1764 begin
1766 lemma in_idem:
1767   assumes "finite A" and "x \<in> A"
1768   shows "g x * F A = F A"
1769   using assms by (induct A) (auto simp add: left_commute)
1771 lemma subset_idem:
1772   assumes "finite A" and "B \<subseteq> A"
1773   shows "F B * F A = F A"
1774 proof -
1775   from assms have "finite B" by (blast dest: finite_subset)
1776   then show ?thesis using `B \<subseteq> A` by (induct B)
1777     (auto simp add: assoc in_idem `finite A`)
1778 qed
1780 declare insert [simp del]
1782 lemma insert_idem [simp]:
1783   assumes "finite A"
1784   shows "F (insert x A) = g x * F A"
1785   using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
1787 lemma union_idem:
1788   assumes "finite A" and "finite B"
1789   shows "F (A \<union> B) = F A * F B"
1790 proof -
1791   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1792   then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
1793   with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1794 qed
1796 end
1799 subsubsection {* The image case with flexible function and idempotency *}
1801 locale folding_image_idem = folding_image +
1802   assumes idem: "x * x = x"
1804 sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
1805 qed (fact idem)
1808 subsubsection {* The neutral-less case *}
1810 locale folding_one = abel_semigroup +
1811   fixes F :: "'a set \<Rightarrow> 'a"
1812   assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
1813 begin
1815 lemma singleton [simp]:
1816   "F {x} = x"
1819 lemma eq_fold':
1820   assumes "finite A" and "x \<notin> A"
1821   shows "F (insert x A) = fold (op *) x A"
1822 proof -
1823   interpret ab_semigroup_mult "op *" by default (simp_all add: ac_simps)
1824   from assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
1825 qed
1827 lemma insert [simp]:
1828   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
1829   shows "F (insert x A) = x * F A"
1830 proof -
1831   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
1832   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
1833   with `finite A` have "finite B" by simp
1834   interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
1835   qed (simp_all add: fun_eq_iff ac_simps)
1836   from `finite B` fold.comp_fun_commute' [of B x]
1837     have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
1838   then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
1839   from `finite B` * fold.insert [of B b]
1840     have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
1841   then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
1842   from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
1843 qed
1845 lemma remove:
1846   assumes "finite A" and "x \<in> A"
1847   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
1848 proof -
1849   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
1850   with assms show ?thesis by simp
1851 qed
1853 lemma insert_remove:
1854   assumes "finite A"
1855   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
1856   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
1858 lemma union_disjoint:
1859   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
1860   shows "F (A \<union> B) = F A * F B"
1861   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
1863 lemma union_inter:
1864   assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
1865   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1866 proof -
1867   from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
1868   from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
1869     case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
1870   next
1871     case (insert x A) show ?case proof (cases "x \<in> B")
1872       case True then have "B \<noteq> {}" by auto
1873       with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
1874         (simp_all add: insert_absorb ac_simps union_disjoint)
1875     next
1876       case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
1877       moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
1878         by auto
1879       ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
1880     qed
1881   qed
1882 qed
1884 lemma closed:
1885   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
1886   shows "F A \<in> A"
1887 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
1888   case singleton then show ?case by simp
1889 next
1890   case insert with elem show ?case by force
1891 qed
1893 end
1896 subsubsection {* The neutral-less case with idempotency *}
1898 locale folding_one_idem = folding_one +
1899   assumes idem: "x * x = x"
1901 sublocale folding_one_idem < semilattice: semilattice proof
1902 qed (fact idem)
1904 context folding_one_idem
1905 begin
1907 lemma in_idem:
1908   assumes "finite A" and "x \<in> A"
1909   shows "x * F A = F A"
1910 proof -
1911   from assms have "A \<noteq> {}" by auto
1912   with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
1913 qed
1915 lemma subset_idem:
1916   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
1917   shows "F B * F A = F A"
1918 proof -
1919   from assms have "finite B" by (blast dest: finite_subset)
1920   then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
1921     (simp_all add: assoc in_idem `finite A`)
1922 qed
1924 lemma eq_fold_idem':
1925   assumes "finite A"
1926   shows "F (insert a A) = fold (op *) a A"
1927 proof -
1928   interpret ab_semigroup_idem_mult "op *" by default (simp_all add: ac_simps)
1929   from assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
1930 qed
1932 lemma insert_idem [simp]:
1933   assumes "finite A" and "A \<noteq> {}"
1934   shows "F (insert x A) = x * F A"
1935 proof (cases "x \<in> A")
1936   case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
1937 next
1938   case True
1939   from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
1940 qed
1942 lemma union_idem:
1943   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
1944   shows "F (A \<union> B) = F A * F B"
1945 proof (cases "A \<inter> B = {}")
1946   case True with assms show ?thesis by (simp add: union_disjoint)
1947 next
1948   case False
1949   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1950   with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
1951   with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1952 qed
1954 lemma hom_commute:
1955   assumes hom: "\<And>x y. h (x * y) = h x * h y"
1956   and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
1957 using N proof (induct rule: finite_ne_induct)
1958   case singleton thus ?case by simp
1959 next
1960   case (insert n N)
1961   then have "h (F (insert n N)) = h (n * F N)" by simp
1962   also have "\<dots> = h n * h (F N)" by (rule hom)
1963   also have "h (F N) = F (h ` N)" by(rule insert)
1964   also have "h n * \<dots> = F (insert (h n) (h ` N))"
1965     using insert by(simp)
1966   also have "insert (h n) (h ` N) = h ` insert n N" by simp
1967   finally show ?case .
1968 qed
1970 end
1972 notation times (infixl "*" 70)
1973 notation Groups.one ("1")
1976 subsection {* Finite cardinality *}
1978 text {* This definition, although traditional, is ugly to work with:
1979 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
1980 But now that we have @{text fold_image} things are easy:
1981 *}
1983 definition card :: "'a set \<Rightarrow> nat" where
1984   "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
1986 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
1989 lemma card_infinite [simp]:
1990   "\<not> finite A \<Longrightarrow> card A = 0"
1993 lemma card_empty:
1994   "card {} = 0"
1995   by (fact card.empty)
1997 lemma card_insert_disjoint:
1998   "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
1999   by simp
2001 lemma card_insert_if:
2002   "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
2003   by auto (simp add: card.insert_remove card.remove)
2005 lemma card_ge_0_finite:
2006   "card A > 0 \<Longrightarrow> finite A"
2007   by (rule ccontr) simp
2009 lemma card_0_eq [simp, no_atp]:
2010   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
2011   by (auto dest: mk_disjoint_insert)
2013 lemma finite_UNIV_card_ge_0:
2014   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
2015   by (rule ccontr) simp
2017 lemma card_eq_0_iff:
2018   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
2019   by auto
2021 lemma card_gt_0_iff:
2022   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
2023   by (simp add: neq0_conv [symmetric] card_eq_0_iff)
2025 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
2026 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
2027 apply(simp del:insert_Diff_single)
2028 done
2030 lemma card_Diff_singleton:
2031   "finite A ==> x: A ==> card (A - {x}) = card A - 1"
2032 by (simp add: card_Suc_Diff1 [symmetric])
2034 lemma card_Diff_singleton_if:
2035   "finite A ==> card (A - {x}) = (if x : A then card A - 1 else card A)"
2038 lemma card_Diff_insert[simp]:
2039 assumes "finite A" and "a:A" and "a ~: B"
2040 shows "card(A - insert a B) = card(A - B) - 1"
2041 proof -
2042   have "A - insert a B = (A - B) - {a}" using assms by blast
2043   then show ?thesis using assms by(simp add:card_Diff_singleton)
2044 qed
2046 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
2047 by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
2049 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
2052 lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
2053 by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
2055 lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
2056 using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
2058 lemma card_mono:
2059   assumes "finite B" and "A \<subseteq> B"
2060   shows "card A \<le> card B"
2061 proof -
2062   from assms have "finite A" by (auto intro: finite_subset)
2063   then show ?thesis using assms proof (induct A arbitrary: B)
2064     case empty then show ?case by simp
2065   next
2066     case (insert x A)
2067     then have "x \<in> B" by simp
2068     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
2069     with insert.hyps have "card A \<le> card (B - {x})" by auto
2070     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
2071   qed
2072 qed
2074 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
2075 apply (induct rule: finite_induct)
2076 apply simp
2077 apply clarify
2078 apply (subgoal_tac "finite A & A - {x} <= F")
2079  prefer 2 apply (blast intro: finite_subset, atomize)
2080 apply (drule_tac x = "A - {x}" in spec)
2082 apply (case_tac "card A", auto)
2083 done
2085 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
2086 apply (simp add: psubset_eq linorder_not_le [symmetric])
2087 apply (blast dest: card_seteq)
2088 done
2090 lemma card_Un_Int: "finite A ==> finite B
2091     ==> card A + card B = card (A Un B) + card (A Int B)"
2092   by (fact card.union_inter [symmetric])
2094 lemma card_Un_disjoint: "finite A ==> finite B
2095     ==> A Int B = {} ==> card (A Un B) = card A + card B"
2096   by (fact card.union_disjoint)
2098 lemma card_Diff_subset:
2099   assumes "finite B" and "B \<subseteq> A"
2100   shows "card (A - B) = card A - card B"
2101 proof (cases "finite A")
2102   case False with assms show ?thesis by simp
2103 next
2104   case True with assms show ?thesis by (induct B arbitrary: A) simp_all
2105 qed
2107 lemma card_Diff_subset_Int:
2108   assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
2109 proof -
2110   have "A - B = A - A \<inter> B" by auto
2111   thus ?thesis
2112     by (simp add: card_Diff_subset AB)
2113 qed
2115 lemma diff_card_le_card_Diff:
2116 assumes "finite B" shows "card A - card B \<le> card(A - B)"
2117 proof-
2118   have "card A - card B \<le> card A - card (A \<inter> B)"
2119     using card_mono[OF assms Int_lower2, of A] by arith
2120   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
2121   finally show ?thesis .
2122 qed
2124 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
2125 apply (rule Suc_less_SucD)
2126 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
2127 done
2129 lemma card_Diff2_less:
2130   "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
2131 apply (case_tac "x = y")
2132  apply (simp add: card_Diff1_less del:card_Diff_insert)
2133 apply (rule less_trans)
2134  prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
2135 done
2137 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
2138 apply (case_tac "x : A")
2139  apply (simp_all add: card_Diff1_less less_imp_le)
2140 done
2142 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
2143 by (erule psubsetI, blast)
2145 lemma insert_partition:
2146   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
2147   \<Longrightarrow> x \<inter> \<Union> F = {}"
2148 by auto
2150 lemma finite_psubset_induct[consumes 1, case_names psubset]:
2151   assumes fin: "finite A"
2152   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
2153   shows "P A"
2154 using fin
2155 proof (induct A taking: card rule: measure_induct_rule)
2156   case (less A)
2157   have fin: "finite A" by fact
2158   have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
2159   { fix B
2160     assume asm: "B \<subset> A"
2161     from asm have "card B < card A" using psubset_card_mono fin by blast
2162     moreover
2163     from asm have "B \<subseteq> A" by auto
2164     then have "finite B" using fin finite_subset by blast
2165     ultimately
2166     have "P B" using ih by simp
2167   }
2168   with fin show "P A" using major by blast
2169 qed
2171 text{* main cardinality theorem *}
2172 lemma card_partition [rule_format]:
2173   "finite C ==>
2174      finite (\<Union> C) -->
2175      (\<forall>c\<in>C. card c = k) -->
2176      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
2177      k * card(C) = card (\<Union> C)"
2178 apply (erule finite_induct, simp)
2179 apply (simp add: card_Un_disjoint insert_partition
2180        finite_subset [of _ "\<Union> (insert x F)"])
2181 done
2183 lemma card_eq_UNIV_imp_eq_UNIV:
2184   assumes fin: "finite (UNIV :: 'a set)"
2185   and card: "card A = card (UNIV :: 'a set)"
2186   shows "A = (UNIV :: 'a set)"
2187 proof
2188   show "A \<subseteq> UNIV" by simp
2189   show "UNIV \<subseteq> A"
2190   proof
2191     fix x
2192     show "x \<in> A"
2193     proof (rule ccontr)
2194       assume "x \<notin> A"
2195       then have "A \<subset> UNIV" by auto
2196       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
2197       with card show False by simp
2198     qed
2199   qed
2200 qed
2202 text{*The form of a finite set of given cardinality*}
2204 lemma card_eq_SucD:
2205 assumes "card A = Suc k"
2206 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
2207 proof -
2208   have fin: "finite A" using assms by (auto intro: ccontr)
2209   moreover have "card A \<noteq> 0" using assms by auto
2210   ultimately obtain b where b: "b \<in> A" by auto
2211   show ?thesis
2212   proof (intro exI conjI)
2213     show "A = insert b (A-{b})" using b by blast
2214     show "b \<notin> A - {b}" by blast
2215     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
2216       using assms b fin by(fastforce dest:mk_disjoint_insert)+
2217   qed
2218 qed
2220 lemma card_Suc_eq:
2221   "(card A = Suc k) =
2222    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
2223 apply(rule iffI)
2224  apply(erule card_eq_SucD)
2225 apply(auto)
2226 apply(subst card_insert)
2227  apply(auto intro:ccontr)
2228 done
2230 lemma card_le_Suc_iff: "finite A \<Longrightarrow>
2231   Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
2232 by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
2233   dest: subset_singletonD split: nat.splits if_splits)
2235 lemma finite_fun_UNIVD2:
2236   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
2237   shows "finite (UNIV :: 'b set)"
2238 proof -
2239   from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
2240     by (rule finite_imageI)
2241   moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
2242     by (rule UNIV_eq_I) auto
2243   ultimately show "finite (UNIV :: 'b set)" by simp
2244 qed
2246 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
2247   unfolding UNIV_unit by simp
2249 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
2250   unfolding UNIV_bool by simp
2253 subsubsection {* Cardinality of image *}
2255 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
2256 apply (induct rule: finite_induct)
2257  apply simp
2258 apply (simp add: le_SucI card_insert_if)
2259 done
2261 lemma card_image:
2262   assumes "inj_on f A"
2263   shows "card (f ` A) = card A"
2264 proof (cases "finite A")
2265   case True then show ?thesis using assms by (induct A) simp_all
2266 next
2267   case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
2268   with False show ?thesis by simp
2269 qed
2271 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
2272 by(auto simp: card_image bij_betw_def)
2274 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
2275 by (simp add: card_seteq card_image)
2277 lemma eq_card_imp_inj_on:
2278   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
2279 apply (induct rule:finite_induct)
2280 apply simp
2281 apply(frule card_image_le[where f = f])
2283 done
2285 lemma inj_on_iff_eq_card:
2286   "finite A ==> inj_on f A = (card(f ` A) = card A)"
2287 by(blast intro: card_image eq_card_imp_inj_on)
2290 lemma card_inj_on_le:
2291   "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
2292 apply (subgoal_tac "finite A")
2293  apply (force intro: card_mono simp add: card_image [symmetric])
2294 apply (blast intro: finite_imageD dest: finite_subset)
2295 done
2297 lemma card_bij_eq:
2298   "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
2299      finite A; finite B |] ==> card A = card B"
2300 by (auto intro: le_antisym card_inj_on_le)
2302 lemma bij_betw_finite:
2303   assumes "bij_betw f A B"
2304   shows "finite A \<longleftrightarrow> finite B"
2305 using assms unfolding bij_betw_def
2306 using finite_imageD[of f A] by auto
2309 subsubsection {* Pigeonhole Principles *}
2311 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
2312 by (auto dest: card_image less_irrefl_nat)
2314 lemma pigeonhole_infinite:
2315 assumes  "~ finite A" and "finite(f`A)"
2316 shows "EX a0:A. ~finite{a:A. f a = f a0}"
2317 proof -
2318   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
2319   proof(induct "f`A" arbitrary: A rule: finite_induct)
2320     case empty thus ?case by simp
2321   next
2322     case (insert b F)
2323     show ?case
2324     proof cases
2325       assume "finite{a:A. f a = b}"
2326       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
2327       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
2328       finally have "~ finite({a:A. f a \<noteq> b})" .
2329       from insert(3)[OF _ this]
2330       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
2331     next
2332       assume 1: "~finite{a:A. f a = b}"
2333       hence "{a \<in> A. f a = b} \<noteq> {}" by force
2334       thus ?thesis using 1 by blast
2335     qed
2336   qed
2337   from this[OF assms(2,1)] show ?thesis .
2338 qed
2340 lemma pigeonhole_infinite_rel:
2341 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
2342 shows "EX b:B. ~finite{a:A. R a b}"
2343 proof -
2344    let ?F = "%a. {b:B. R a b}"
2345    from finite_Pow_iff[THEN iffD2, OF `finite B`]
2346    have "finite(?F ` A)" by(blast intro: rev_finite_subset)
2347    from pigeonhole_infinite[where f = ?F, OF assms(1) this]
2348    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
2349    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
2350    { assume "finite{a:A. R a b0}"
2351      then have "finite {a\<in>A. ?F a = ?F a0}"
2352        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
2353    }
2354    with 1 `b0 : B` show ?thesis by blast
2355 qed
2358 subsubsection {* Cardinality of sums *}
2360 lemma card_Plus:
2361   assumes "finite A" and "finite B"
2362   shows "card (A <+> B) = card A + card B"
2363 proof -
2364   have "Inl`A \<inter> Inr`B = {}" by fast
2365   with assms show ?thesis
2366     unfolding Plus_def
2367     by (simp add: card_Un_disjoint card_image)
2368 qed
2370 lemma card_Plus_conv_if:
2371   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
2372   by (auto simp add: card_Plus)
2375 subsubsection {* Cardinality of the Powerset *}
2377 lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
2378 apply (induct rule: finite_induct)
2380 apply (subst card_Un_disjoint, blast)
2381   apply (blast, blast)
2382 apply (subgoal_tac "inj_on (insert x) (Pow F)")
2383  apply (subst mult_2)
2384  apply (simp add: card_image Pow_insert)
2385 apply (unfold inj_on_def)
2386 apply (blast elim!: equalityE)
2387 done
2389 text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
2391 lemma dvd_partition:
2392   "finite (Union C) ==>
2393     ALL c : C. k dvd card c ==>
2394     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
2395   k dvd card (Union C)"
2396 apply (frule finite_UnionD)
2397 apply (rotate_tac -1)
2398 apply (induct rule: finite_induct)
2399 apply simp_all
2400 apply clarify
2401 apply (subst card_Un_disjoint)
2402    apply (auto simp add: disjoint_eq_subset_Compl)
2403 done
2406 subsubsection {* Relating injectivity and surjectivity *}
2408 lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
2409 apply(rule eq_card_imp_inj_on, assumption)
2410 apply(frule finite_imageI)
2411 apply(drule (1) card_seteq)
2412  apply(erule card_image_le)
2413 apply simp
2414 done
2416 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
2417 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
2418 by (blast intro: finite_surj_inj subset_UNIV)
2420 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
2421 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
2422 by(fastforce simp:surj_def dest!: endo_inj_surj)
2424 corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
2425 proof
2426   assume "finite(UNIV::nat set)"
2427   with finite_UNIV_inj_surj[of Suc]
2428   show False by simp (blast dest: Suc_neq_Zero surjD)
2429 qed
2431 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
2432 lemma infinite_UNIV_char_0[no_atp]:
2433   "\<not> finite (UNIV::'a::semiring_char_0 set)"
2434 proof
2435   assume "finite (UNIV::'a set)"
2436   with subset_UNIV have "finite (range of_nat::'a set)"
2437     by (rule finite_subset)
2438   moreover have "inj (of_nat::nat \<Rightarrow> 'a)"