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 => bool"
16 emptyI [simp, intro!]: "finite {}"
17 | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
19 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
20 assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
21 shows "\<exists>a::'a. a \<notin> A"
23 from assms have "A \<noteq> UNIV" by blast
27 lemma finite_induct [case_names empty insert, induct set: finite]:
29 P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
30 -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
33 insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
38 fix x F assume F: "finite F" and P: "P F"
42 hence "insert x F = F" by (rule insert_absorb)
43 with P show ?thesis by (simp only:)
46 from F this P show ?thesis by (rule insert)
51 lemma finite_ne_induct[case_names singleton insert, consumes 2]:
52 assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
53 \<lbrakk> \<And>x. P{x};
54 \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
55 \<Longrightarrow> P F"
58 case empty thus ?case by simp
64 thus ?thesis using `P {x}` by simp
66 assume "F \<noteq> {}"
67 thus ?thesis using insert by blast
71 lemma finite_subset_induct [consumes 2, case_names empty insert]:
72 assumes "finite F" and "F \<subseteq> A"
74 and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
77 from `finite F` and `F \<subseteq> A`
83 assume "finite F" and "x \<notin> F" and
84 P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
87 from i show "x \<in> A" by blast
88 from i have "F \<subseteq> A" by blast
90 show "finite F" by fact
91 show "x \<notin> F" by fact
97 text{* A finite choice principle. Does not need the SOME choice operator. *}
98 lemma finite_set_choice:
99 "finite A \<Longrightarrow> ALL x:A. (EX y. P x y) \<Longrightarrow> EX f. ALL x:A. P x (f x)"
100 proof (induct set: finite)
101 case empty thus ?case by simp
104 then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
105 show ?case (is "EX f. ?P f")
107 show "?P(%x. if x = a then b else f x)" using f ab by auto
112 text{* Finite sets are the images of initial segments of natural numbers: *}
114 lemma finite_imp_nat_seg_image_inj_on:
115 assumes fin: "finite A"
116 shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
121 proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp
125 have notinA: "a \<notin> A" by fact
126 from insert.hyps obtain n f
127 where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
128 hence "insert a A = f(n:=a) ` {i. i < Suc n}"
129 "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
130 by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
134 lemma nat_seg_image_imp_finite:
135 "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
137 case 0 thus ?case by simp
140 let ?B = "f ` {i. i < n}"
141 have finB: "finite ?B" by(rule Suc.hyps[OF refl])
144 assume "\<exists>k<n. f n = f k"
145 hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
146 thus ?thesis using finB by simp
148 assume "\<not>(\<exists> k<n. f n = f k)"
149 hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
150 thus ?thesis using finB by simp
154 lemma finite_conv_nat_seg_image:
155 "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
156 by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
158 lemma finite_imp_inj_to_nat_seg:
160 shows "EX f n::nat. f`A = {i. i<n} & inj_on f A"
162 from finite_imp_nat_seg_image_inj_on[OF `finite A`]
163 obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
164 by (auto simp:bij_betw_def)
165 let ?f = "the_inv_into {i. i<n} f"
166 have "inj_on ?f A & ?f ` A = {i. i<n}"
167 by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
168 thus ?thesis by blast
171 lemma finite_Collect_less_nat[iff]: "finite{n::nat. n<k}"
172 by(fastsimp simp: finite_conv_nat_seg_image)
174 text {* Finiteness and set theoretic constructions *}
176 lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
177 by (induct set: finite) simp_all
179 lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
180 -- {* Every subset of a finite set is finite. *}
183 thus "!!A. A \<subseteq> B ==> finite A"
189 have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+
192 assume x: "x \<in> A"
193 with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
194 with r have "finite (A - {x})" .
195 hence "finite (insert x (A - {x}))" ..
196 also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
197 finally show ?thesis .
199 show "A \<subseteq> F ==> ?thesis" by fact
200 assume "x \<notin> A"
201 with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
206 lemma rev_finite_subset: "finite B ==> A \<subseteq> B ==> finite A"
207 by (rule finite_subset)
209 lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
210 by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
212 lemma finite_Collect_disjI[simp]:
213 "finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})"
214 by(simp add:Collect_disj_eq)
216 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
217 -- {* The converse obviously fails. *}
218 by (blast intro: finite_subset)
220 lemma finite_Collect_conjI [simp, intro]:
221 "finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}"
222 -- {* The converse obviously fails. *}
223 by(simp add:Collect_conj_eq)
225 lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}"
226 by(simp add: le_eq_less_or_eq)
228 lemma finite_insert [simp]: "finite (insert a A) = finite A"
229 apply (subst insert_is_Un)
230 apply (simp only: finite_Un, blast)
233 lemma finite_Union[simp, intro]:
234 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
235 by (induct rule:finite_induct) simp_all
237 lemma finite_Inter[intro]: "EX A:M. finite(A) \<Longrightarrow> finite(Inter M)"
238 by (blast intro: Inter_lower finite_subset)
240 lemma finite_INT[intro]: "EX x:I. finite(A x) \<Longrightarrow> finite(INT x:I. A x)"
241 by (blast intro: INT_lower finite_subset)
243 lemma finite_empty_induct:
246 and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
253 assume c: "finite c" and b: "finite b"
254 and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
255 have "c \<subseteq> b ==> P (b - c)"
259 from P1 show ?case by simp
262 have "P (b - F - {x})"
264 from _ b show "finite (b - F)" by (rule finite_subset) blast
265 from insert show "x \<in> b - F" by simp
266 from insert show "P (b - F)" by simp
268 also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
272 then show ?thesis by this (simp_all add: assms)
274 then show ?thesis by simp
277 lemma finite_Diff [simp]: "finite A ==> finite (A - B)"
278 by (rule Diff_subset [THEN finite_subset])
280 lemma finite_Diff2 [simp]:
281 assumes "finite B" shows "finite (A - B) = finite A"
283 have "finite A \<longleftrightarrow> finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int)
284 also have "\<dots> \<longleftrightarrow> finite(A-B)" using `finite B` by(simp)
285 finally show ?thesis ..
288 lemma finite_compl[simp]:
289 "finite(A::'a set) \<Longrightarrow> finite(-A) = finite(UNIV::'a set)"
290 by(simp add:Compl_eq_Diff_UNIV)
292 lemma finite_Collect_not[simp]:
293 "finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)"
294 by(simp add:Collect_neg_eq)
296 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
297 apply (subst Diff_insert)
298 apply (case_tac "a : A - B")
299 apply (rule finite_insert [symmetric, THEN trans])
300 apply (subst insert_Diff, simp_all)
304 text {* Image and Inverse Image over Finite Sets *}
306 lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
307 -- {* The image of a finite set is finite. *}
308 by (induct set: finite) simp_all
310 lemma finite_image_set [simp]:
311 "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
312 by (simp add: image_Collect [symmetric])
314 lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
315 apply (frule finite_imageI)
316 apply (erule finite_subset, assumption)
319 lemma finite_range_imageI:
320 "finite (range g) ==> finite (range (%x. f (g x)))"
321 apply (drule finite_imageI, simp add: range_composition)
324 lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
326 have aux: "!!A. finite (A - {}) = finite A" by simp
329 thus "!!A. f`A = B ==> inj_on f A ==> finite A"
332 apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
334 apply (simp (no_asm_use) add: inj_on_def)
335 apply (blast dest!: aux [THEN iffD1], atomize)
336 apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
337 apply (frule subsetD [OF equalityD2 insertI1], clarify)
338 apply (rule_tac x = xa in bexI)
339 apply (simp_all add: inj_on_image_set_diff)
344 lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
345 -- {* The inverse image of a singleton under an injective function
346 is included in a singleton. *}
347 apply (auto simp add: inj_on_def)
348 apply (blast intro: the_equality [symmetric])
351 lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
352 -- {* The inverse image of a finite set under an injective function
354 apply (induct set: finite)
356 apply (subst vimage_insert)
357 apply (simp add: finite_subset [OF inj_vimage_singleton])
360 lemma finite_vimageD:
361 assumes fin: "finite (h -` F)" and surj: "surj h"
364 have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
365 also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
366 finally show "finite F" .
369 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
370 unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
373 text {* The finite UNION of finite sets *}
375 lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
376 by (induct set: finite) simp_all
380 @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
383 @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
386 lemma finite_UN [simp]:
387 "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
388 by (blast intro: finite_UN_I finite_subset)
390 lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow>
391 finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})"
392 apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})")
396 lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \<Longrightarrow>
397 finite{x. EX y. P y & Q x y} = (ALL y. P y \<longrightarrow> finite{x. Q x y})"
398 apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})")
403 lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
404 by (simp add: Plus_def)
407 fixes A :: "'a set" and B :: "'b set"
408 assumes fin: "finite (A <+> B)"
409 shows "finite A" "finite B"
411 have "Inl ` A \<subseteq> A <+> B" by auto
412 hence "finite (Inl ` A :: ('a + 'b) set)" using fin by(rule finite_subset)
413 thus "finite A" by(rule finite_imageD)(auto intro: inj_onI)
415 have "Inr ` B \<subseteq> A <+> B" by auto
416 hence "finite (Inr ` B :: ('a + 'b) set)" using fin by(rule finite_subset)
417 thus "finite B" by(rule finite_imageD)(auto intro: inj_onI)
420 lemma finite_Plus_iff[simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
421 by(auto intro: finite_PlusD finite_Plus)
423 lemma finite_Plus_UNIV_iff[simp]:
424 "finite (UNIV :: ('a + 'b) set) =
425 (finite (UNIV :: 'a set) & finite (UNIV :: 'b set))"
426 by(subst UNIV_Plus_UNIV[symmetric])(rule finite_Plus_iff)
429 text {* Sigma of finite sets *}
431 lemma finite_SigmaI [simp]:
432 "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
433 by (unfold Sigma_def) (blast intro!: finite_UN_I)
435 lemma finite_cartesian_product: "[| finite A; finite B |] ==>
437 by (rule finite_SigmaI)
439 lemma finite_Prod_UNIV:
440 "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
441 apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
443 apply (erule finite_SigmaI, auto)
446 lemma finite_cartesian_productD1:
447 "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
448 apply (auto simp add: finite_conv_nat_seg_image)
449 apply (drule_tac x=n in spec)
450 apply (drule_tac x="fst o f" in spec)
451 apply (auto simp add: o_def)
452 prefer 2 apply (force dest!: equalityD2)
453 apply (drule equalityD1)
454 apply (rename_tac y x)
455 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
458 apply (rule_tac x=k in image_eqI, auto)
461 lemma finite_cartesian_productD2:
462 "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
463 apply (auto simp add: finite_conv_nat_seg_image)
464 apply (drule_tac x=n in spec)
465 apply (drule_tac x="snd o f" in spec)
466 apply (auto simp add: o_def)
467 prefer 2 apply (force dest!: equalityD2)
468 apply (drule equalityD1)
469 apply (rename_tac x y)
470 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
473 apply (rule_tac x=k in image_eqI, auto)
477 text {* The powerset of a finite set *}
479 lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
481 assume "finite (Pow A)"
482 with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
483 thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
486 thus "finite (Pow A)"
487 by induct (simp_all add: Pow_insert)
490 lemma finite_Collect_subsets[simp,intro]: "finite A \<Longrightarrow> finite{B. B \<subseteq> A}"
491 by(simp add: Pow_def[symmetric])
494 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
495 by(blast intro: finite_subset[OF subset_Pow_Union])
498 lemma finite_subset_image:
500 shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
501 using assms proof(induct)
502 case empty thus ?case by simp
504 case insert thus ?case
505 by (clarsimp simp del: image_insert simp add: image_insert[symmetric])
510 subsection {* Class @{text finite} *}
513 assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
516 lemma finite [simp]: "finite (A \<Colon> 'a set)"
517 by (rule subset_UNIV finite_UNIV finite_subset)+
521 lemma UNIV_unit [no_atp]:
522 "UNIV = {()}" by auto
524 instance unit :: finite proof
525 qed (simp add: UNIV_unit)
527 lemma UNIV_bool [no_atp]:
528 "UNIV = {False, True}" by auto
530 instance bool :: finite proof
531 qed (simp add: UNIV_bool)
533 instance prod :: (finite, finite) finite proof
534 qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
536 lemma finite_option_UNIV [simp]:
537 "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
538 by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
540 instance option :: (finite) finite proof
541 qed (simp add: UNIV_option_conv)
543 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
544 by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
546 instance "fun" :: (finite, finite) finite
548 show "finite (UNIV :: ('a => 'b) set)"
549 proof (rule finite_imageD)
550 let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
551 have "range ?graph \<subseteq> Pow UNIV" by simp
552 moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
553 by (simp only: finite_Pow_iff finite)
554 ultimately show "finite (range ?graph)"
555 by (rule finite_subset)
556 show "inj ?graph" by (rule inj_graph)
560 instance sum :: (finite, finite) finite proof
561 qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
564 subsection {* A basic fold functional for finite sets *}
566 text {* The intended behaviour is
567 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
568 if @{text f} is ``left-commutative'':
571 locale fun_left_comm =
572 fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
573 assumes fun_left_comm: "f x (f y z) = f y (f x z)"
576 text{* On a functional level it looks much nicer: *}
578 lemma fun_comp_comm: "f x \<circ> f y = f y \<circ> f x"
579 by (simp add: fun_left_comm expand_fun_eq)
583 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
584 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
585 emptyI [intro]: "fold_graph f z {} z" |
586 insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
587 \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
589 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
591 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
592 "fold f z A = (THE y. fold_graph f z A y)"
594 text{*A tempting alternative for the definiens is
595 @{term "if finite A then THE y. fold_graph f z A y else e"}.
596 It allows the removal of finiteness assumptions from the theorems
597 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
598 The proofs become ugly. It is not worth the effort. (???) *}
601 lemma Diff1_fold_graph:
602 "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
603 by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
605 lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
606 by (induct set: fold_graph) auto
608 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
609 by (induct set: finite) auto
612 subsubsection{*From @{const fold_graph} to @{term fold}*}
614 context fun_left_comm
617 lemma fold_graph_insertE_aux:
618 "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'"
619 proof (induct set: fold_graph)
620 case (insertI x A y) show ?case
621 proof (cases "x = a")
622 assume "x = a" with insertI show ?case by auto
624 assume "x \<noteq> a"
625 then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
626 using insertI by auto
627 have 1: "f x y = f a (f x y')"
628 unfolding y by (rule fun_left_comm)
629 have 2: "fold_graph f z (insert x A - {a}) (f x y')"
630 using y' and `x \<noteq> a` and `x \<notin> A`
631 by (simp add: insert_Diff_if fold_graph.insertI)
632 from 1 2 show ?case by fast
636 lemma fold_graph_insertE:
637 assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
638 obtains y where "v = f x y" and "fold_graph f z A y"
639 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
641 lemma fold_graph_determ:
642 "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
643 proof (induct arbitrary: y set: fold_graph)
644 case (insertI x A y v)
645 from `fold_graph f z (insert x A) v` and `x \<notin> A`
646 obtain y' where "v = f x y'" and "fold_graph f z A y'"
647 by (rule fold_graph_insertE)
648 from `fold_graph f z A y'` have "y' = y" by (rule insertI)
649 with `v = f x y'` show "v = f x y" by simp
653 "fold_graph f z A y \<Longrightarrow> fold f z A = y"
654 by (unfold fold_def) (blast intro: fold_graph_determ)
656 lemma fold_graph_fold: "finite A \<Longrightarrow> fold_graph f z A (fold f z A)"
660 apply (erule finite_imp_fold_graph)
661 apply (erule (1) fold_graph_determ)
664 text{* The base case for @{text fold}: *}
666 lemma (in -) fold_empty [simp]: "fold f z {} = z"
667 by (unfold fold_def) blast
669 text{* The various recursion equations for @{const fold}: *}
671 lemma fold_insert [simp]:
672 "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
673 apply (rule fold_equality)
674 apply (erule fold_graph.insertI)
675 apply (erule fold_graph_fold)
679 "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
680 proof (induct rule: finite_induct)
681 case empty then show ?case by simp
683 case (insert y A) then show ?case
684 by (simp add: fun_left_comm[of x])
688 "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
689 by (simp add: fold_fun_comm)
692 assumes "finite A" and "x \<in> A"
693 shows "fold f z A = f x (fold f z (A - {x}))"
695 have A: "A = insert x (A - {x})" using `x \<in> A` by blast
696 then have "fold f z A = fold f z (insert x (A - {x}))" by simp
697 also have "\<dots> = f x (fold f z (A - {x}))"
698 by (rule fold_insert) (simp add: `finite A`)+
699 finally show ?thesis .
702 lemma fold_insert_remove:
704 shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
706 from `finite A` have "finite (insert x A)" by auto
707 moreover have "x \<in> insert x A" by auto
708 ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
710 then show ?thesis by simp
715 text{* A simplified version for idempotent functions: *}
717 locale fun_left_comm_idem = fun_left_comm +
718 assumes fun_left_idem: "f x (f x z) = f x z"
721 text{* The nice version: *}
722 lemma fun_comp_idem : "f x o f x = f x"
723 by (simp add: fun_left_idem expand_fun_eq)
725 lemma fold_insert_idem:
726 assumes fin: "finite A"
727 shows "fold f z (insert x A) = f x (fold f z A)"
730 then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
731 then show ?thesis using assms by (simp add:fun_left_idem)
733 assume "x \<notin> A" then show ?thesis using assms by simp
736 declare fold_insert[simp del] fold_insert_idem[simp]
738 lemma fold_insert_idem2:
739 "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
740 by(simp add:fold_fun_comm)
745 subsubsection {* Expressing set operations via @{const fold} *}
747 lemma (in fun_left_comm) fun_left_comm_apply:
748 "fun_left_comm (\<lambda>x. f (g x))"
750 qed (simp_all add: fun_left_comm)
752 lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:
753 "fun_left_comm_idem (\<lambda>x. f (g x))"
754 by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales)
755 (simp_all add: fun_left_idem)
757 lemma fun_left_comm_idem_insert:
758 "fun_left_comm_idem insert"
762 lemma fun_left_comm_idem_remove:
763 "fun_left_comm_idem (\<lambda>x A. A - {x})"
767 lemma (in semilattice_inf) fun_left_comm_idem_inf:
768 "fun_left_comm_idem inf"
770 qed (auto simp add: inf_left_commute)
772 lemma (in semilattice_sup) fun_left_comm_idem_sup:
773 "fun_left_comm_idem sup"
775 qed (auto simp add: sup_left_commute)
777 lemma union_fold_insert:
779 shows "A \<union> B = fold insert B A"
781 interpret fun_left_comm_idem insert by (fact fun_left_comm_idem_insert)
782 from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
785 lemma minus_fold_remove:
787 shows "B - A = fold (\<lambda>x A. A - {x}) B A"
789 interpret fun_left_comm_idem "\<lambda>x A. A - {x}" by (fact fun_left_comm_idem_remove)
790 from `finite A` show ?thesis by (induct A arbitrary: B) auto
793 context complete_lattice
796 lemma inf_Inf_fold_inf:
798 shows "inf B (Inf A) = fold inf B A"
800 interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
801 from `finite A` show ?thesis by (induct A arbitrary: B)
802 (simp_all add: Inf_empty Inf_insert inf_commute fold_fun_comm)
805 lemma sup_Sup_fold_sup:
807 shows "sup B (Sup A) = fold sup B A"
809 interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
810 from `finite A` show ?thesis by (induct A arbitrary: B)
811 (simp_all add: Sup_empty Sup_insert sup_commute fold_fun_comm)
816 shows "Inf A = fold inf top A"
817 using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
821 shows "Sup A = fold sup bot A"
822 using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
824 lemma inf_INFI_fold_inf:
826 shows "inf B (INFI A f) = fold (\<lambda>A. inf (f A)) B A" (is "?inf = ?fold")
828 interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
829 interpret fun_left_comm_idem "\<lambda>A. inf (f A)" by (fact fun_left_comm_idem_apply)
830 from `finite A` show "?fold = ?inf"
831 by (induct A arbitrary: B)
832 (simp_all add: INFI_def Inf_empty Inf_insert inf_left_commute)
835 lemma sup_SUPR_fold_sup:
837 shows "sup B (SUPR A f) = fold (\<lambda>A. sup (f A)) B A" (is "?sup = ?fold")
839 interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
840 interpret fun_left_comm_idem "\<lambda>A. sup (f A)" by (fact fun_left_comm_idem_apply)
841 from `finite A` show "?fold = ?sup"
842 by (induct A arbitrary: B)
843 (simp_all add: SUPR_def Sup_empty Sup_insert sup_left_commute)
848 shows "INFI A f = fold (\<lambda>A. inf (f A)) top A"
849 using assms inf_INFI_fold_inf [of A top] by simp
853 shows "SUPR A f = fold (\<lambda>A. sup (f A)) bot A"
854 using assms sup_SUPR_fold_sup [of A bot] by simp
859 subsection {* The derived combinator @{text fold_image} *}
861 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
862 where "fold_image f g = fold (%x y. f (g x) y)"
864 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
865 by(simp add:fold_image_def)
867 context ab_semigroup_mult
870 lemma fold_image_insert[simp]:
871 assumes "finite A" and "a \<notin> A"
872 shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
874 interpret I: fun_left_comm "%x y. (g x) * y"
875 by unfold_locales (simp add: mult_ac)
876 show ?thesis using assms by(simp add:fold_image_def)
881 "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
882 apply (induct set: finite)
884 apply (simp add: mult_left_commute [of x])
887 lemma fold_nest_Un_Int:
888 "finite A ==> finite B
889 ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
890 apply (induct set: finite)
892 apply (simp add: fold_commute Int_insert_left insert_absorb)
895 lemma fold_nest_Un_disjoint:
896 "finite A ==> finite B ==> A Int B = {}
897 ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
898 by (simp add: fold_nest_Un_Int)
901 lemma fold_image_reindex:
902 assumes fin: "finite A"
903 shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
904 using fin by induct auto
908 Fusion theorem, as described in Graham Hutton's paper,
909 A Tutorial on the Universality and Expressiveness of Fold,
910 JFP 9:4 (355-372), 1999.
914 assumes "ab_semigroup_mult g"
915 assumes fin: "finite A"
916 and hyp: "\<And>x y. h (g x y) = times x (h y)"
917 shows "h (fold g j w A) = fold times j (h w) A"
919 class_interpret ab_semigroup_mult [g] by fact
920 show ?thesis using fin hyp by (induct set: finite) simp_all
924 lemma fold_image_cong:
925 "finite A \<Longrightarrow>
926 (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
927 apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C")
929 apply (erule finite_induct, simp)
930 apply (simp add: subset_insert_iff, clarify)
931 apply (subgoal_tac "finite C")
932 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
933 apply (subgoal_tac "C = insert x (C - {x})")
937 apply (erule (1) notE impE)
938 apply (simp add: Ball_def del: insert_Diff_single)
943 context comm_monoid_mult
947 "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
948 apply (induct set: finite)
951 lemma fold_image_Un_Int:
952 "finite A ==> finite B ==>
953 fold_image times g 1 A * fold_image times g 1 B =
954 fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
955 by (induct set: finite)
956 (auto simp add: mult_ac insert_absorb Int_insert_left)
958 lemma fold_image_Un_one:
959 assumes fS: "finite S" and fT: "finite T"
960 and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
961 shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
963 have "fold_image op * f 1 (S \<inter> T) = 1"
964 apply (rule fold_image_1)
965 using fS fT I0 by auto
966 with fold_image_Un_Int[OF fS fT] show ?thesis by simp
969 corollary fold_Un_disjoint:
970 "finite A ==> finite B ==> A Int B = {} ==>
971 fold_image times g 1 (A Un B) =
972 fold_image times g 1 A * fold_image times g 1 B"
973 by (simp add: fold_image_Un_Int)
975 lemma fold_image_UN_disjoint:
976 "\<lbrakk> finite I; ALL i:I. finite (A i);
977 ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
978 \<Longrightarrow> fold_image times g 1 (UNION I A) =
979 fold_image times (%i. fold_image times g 1 (A i)) 1 I"
980 apply (induct set: finite, simp, atomize)
981 apply (subgoal_tac "ALL i:F. x \<noteq> i")
983 apply (subgoal_tac "A x Int UNION F A = {}")
985 apply (simp add: fold_Un_disjoint)
988 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
989 fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
990 fold_image times (split g) 1 (SIGMA x:A. B x)"
991 apply (subst Sigma_def)
992 apply (subst fold_image_UN_disjoint, assumption, simp)
994 apply (erule fold_image_cong)
995 apply (subst fold_image_UN_disjoint, simp, simp)
1000 lemma fold_image_distrib: "finite A \<Longrightarrow>
1001 fold_image times (%x. g x * h x) 1 A =
1002 fold_image times g 1 A * fold_image times h 1 A"
1003 by (erule finite_induct) (simp_all add: mult_ac)
1005 lemma fold_image_related:
1007 and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
1008 and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
1009 shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
1010 using fS by (rule finite_subset_induct) (insert assms, auto)
1012 lemma fold_image_eq_general:
1013 assumes fS: "finite S"
1014 and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
1015 and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
1016 shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
1018 from h f12 have hS: "h ` S = S'" by auto
1019 {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1020 from f12 h H have "x = y" by auto }
1021 hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1022 from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1023 from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
1024 also have "\<dots> = fold_image (op *) (f2 o h) e S"
1025 using fold_image_reindex[OF fS hinj, of f2 e] .
1026 also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
1028 finally show ?thesis ..
1031 lemma fold_image_eq_general_inverses:
1032 assumes fS: "finite S"
1033 and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1034 and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
1035 shows "fold_image (op *) f e S = fold_image (op *) g e T"
1036 (* metis solves it, but not yet available here *)
1037 apply (rule fold_image_eq_general[OF fS, of T h g f e])
1043 apply (drule hk) apply simp
1045 apply (erule conjunct1[OF conjunct2[OF hk]])
1054 subsection {* A fold functional for non-empty sets *}
1056 text{* Does not require start value. *}
1059 fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
1060 for f :: "'a => 'a => 'a"
1062 fold1Set_insertI [intro]:
1063 "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
1065 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
1066 "fold1 f A == THE x. fold1Set f A x"
1068 lemma fold1Set_nonempty:
1069 "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
1070 by(erule fold1Set.cases, simp_all)
1072 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
1074 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
1077 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
1078 by (blast elim: fold_graph.cases)
1080 lemma fold1_singleton [simp]: "fold1 f {a} = a"
1081 by (unfold fold1_def) blast
1083 lemma finite_nonempty_imp_fold1Set:
1084 "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
1085 apply (induct A rule: finite_induct)
1086 apply (auto dest: finite_imp_fold_graph [of _ f])
1089 text{*First, some lemmas about @{const fold_graph}.*}
1091 context ab_semigroup_mult
1094 lemma fun_left_comm: "fun_left_comm(op *)"
1095 by unfold_locales (simp add: mult_ac)
1097 lemma fold_graph_insert_swap:
1098 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
1099 shows "fold_graph times z (insert b A) (z * y)"
1101 interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
1102 from assms show ?thesis
1103 proof (induct rule: fold_graph.induct)
1104 case emptyI show ?case by (subst mult_commute [of z b], fast)
1106 case (insertI x A y)
1107 have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
1108 using insertI by force --{*how does @{term id} get unfolded?*}
1109 thus ?case by (simp add: insert_commute mult_ac)
1113 lemma fold_graph_permute_diff:
1114 assumes fold: "fold_graph times b A x"
1115 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
1117 proof (induct rule: fold_graph.induct)
1118 case emptyI thus ?case by simp
1120 case (insertI x A y)
1121 have "a = x \<or> a \<in> A" using insertI by simp
1125 with insertI show ?thesis
1126 by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
1128 assume ainA: "a \<in> A"
1129 hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
1130 using insertI by force
1132 have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
1133 using ainA insertI by blast
1134 ultimately show ?thesis by simp
1138 lemma fold1_eq_fold:
1139 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
1141 interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
1142 from assms show ?thesis
1143 apply (simp add: fold1_def fold_def)
1144 apply (rule the_equality)
1145 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
1146 apply (rule sym, clarify)
1147 apply (case_tac "Aa=A")
1148 apply (best intro: fold_graph_determ)
1149 apply (subgoal_tac "fold_graph times a A x")
1150 apply (best intro: fold_graph_determ)
1151 apply (subgoal_tac "insert aa (Aa - {a}) = A")
1152 prefer 2 apply (blast elim: equalityE)
1153 apply (auto dest: fold_graph_permute_diff [where a=a])
1157 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
1160 apply (drule_tac x=x in spec)
1161 apply (drule_tac x="A-{x}" in spec, auto)
1165 assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
1166 shows "fold1 times (insert x A) = x * fold1 times A"
1168 interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
1169 from nonempty obtain a A' where "A = insert a A' & a ~: A'"
1170 by (auto simp add: nonempty_iff)
1172 by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
1177 context ab_semigroup_idem_mult
1180 lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"
1181 apply unfold_locales
1182 apply (rule mult_left_commute)
1183 apply (rule mult_left_idem)
1186 lemma fold1_insert_idem [simp]:
1187 assumes nonempty: "A \<noteq> {}" and A: "finite A"
1188 shows "fold1 times (insert x A) = x * fold1 times A"
1190 interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
1191 by (rule fun_left_comm_idem)
1192 from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
1193 by (auto simp add: nonempty_iff)
1200 with prems show ?thesis by simp
1202 assume "A' \<noteq> {}"
1203 with prems show ?thesis
1204 by (simp add: fold1_insert mult_assoc [symmetric])
1207 assume "a \<noteq> x"
1208 with prems show ?thesis
1209 by (simp add: insert_commute fold1_eq_fold)
1213 lemma hom_fold1_commute:
1214 assumes hom: "!!x y. h (x * y) = h x * h y"
1215 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
1216 using N proof (induct rule: finite_ne_induct)
1217 case singleton thus ?case by simp
1220 then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
1221 also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
1222 also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
1223 also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
1224 using insert by(simp)
1225 also have "insert (h n) (h ` N) = h ` insert n N" by simp
1226 finally show ?case .
1229 lemma fold1_eq_fold_idem:
1231 shows "fold1 times (insert a A) = fold times a A"
1232 proof (cases "a \<in> A")
1234 with assms show ?thesis by (simp add: fold1_eq_fold)
1236 interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
1237 case True then obtain b B
1238 where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
1239 with assms have "finite B" by auto
1240 then have "fold times a (insert a B) = fold times (a * a) B"
1241 using `a \<notin> B` by (rule fold_insert2)
1243 using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
1249 text{* Now the recursion rules for definitions: *}
1251 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
1254 lemma (in ab_semigroup_mult) fold1_insert_def:
1255 "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1256 by (simp add:fold1_insert)
1258 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
1259 "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
1262 subsubsection{* Determinacy for @{term fold1Set} *}
1264 (*Not actually used!!*)
1266 context ab_semigroup_mult
1269 lemma fold_graph_permute:
1270 "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
1271 ==> fold_graph times id a (insert b A) x"
1273 apply (auto dest: fold_graph_permute_diff)
1276 lemma fold1Set_determ:
1277 "fold1Set times A x ==> fold1Set times A y ==> y = x"
1278 proof (clarify elim!: fold1Set.cases)
1280 assume Ax: "fold_graph times id a A x"
1281 assume By: "fold_graph times id b B y"
1282 assume anotA: "a \<notin> A"
1283 assume bnotB: "b \<notin> B"
1284 assume eq: "insert a A = insert b B"
1288 hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
1289 thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
1291 assume diff: "a\<noteq>b"
1293 have B: "B = insert a ?D" and A: "A = insert b ?D"
1294 and aB: "a \<in> B" and bA: "b \<in> A"
1295 using eq anotA bnotB diff by (blast elim!:equalityE)+
1297 have "fold_graph times id a (insert b ?D) y"
1298 by (auto intro: fold_graph_permute simp add: insert_absorb)
1300 have "fold_graph times id a (insert b ?D) x"
1301 by (simp add: A [symmetric] Ax)
1302 ultimately show ?thesis by (blast intro: fold_graph_determ)
1306 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
1307 by (unfold fold1_def) (blast intro: fold1Set_determ)
1313 empty_fold_graphE [rule del] fold_graph.intros [rule del]
1314 empty_fold1SetE [rule del] insert_fold1SetE [rule del]
1315 -- {* No more proofs involve these relations. *}
1317 subsubsection {* Lemmas about @{text fold1} *}
1319 context ab_semigroup_mult
1323 assumes A: "finite A" "A \<noteq> {}"
1324 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
1325 fold1 times (A Un B) = fold1 times A * fold1 times B"
1326 using A by (induct rule: finite_ne_induct)
1327 (simp_all add: fold1_insert mult_assoc)
1330 assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
1331 shows "fold1 times A \<in> A"
1333 proof (induct rule:finite_ne_induct)
1334 case singleton thus ?case by simp
1336 case insert thus ?case using elem by (force simp add:fold1_insert)
1341 lemma (in ab_semigroup_idem_mult) fold1_Un2:
1342 assumes A: "finite A" "A \<noteq> {}"
1343 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
1344 fold1 times (A Un B) = fold1 times A * fold1 times B"
1346 proof(induct rule:finite_ne_induct)
1347 case singleton thus ?case by simp
1349 case insert thus ?case by (simp add: mult_assoc)
1353 subsection {* Locales as mini-packages for fold operations *}
1355 subsubsection {* The natural case *}
1358 fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
1359 fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
1360 assumes commute_comp: "f y \<circ> f x = f x \<circ> f y"
1361 assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
1366 by (simp add: eq_fold expand_fun_eq)
1368 lemma insert [simp]:
1369 assumes "finite A" and "x \<notin> A"
1370 shows "F (insert x A) = F A \<circ> f x"
1372 interpret fun_left_comm f proof
1373 qed (insert commute_comp, simp add: expand_fun_eq)
1374 from fold_insert2 assms
1375 have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
1376 with `finite A` show ?thesis by (simp add: eq_fold expand_fun_eq)
1380 assumes "finite A" and "x \<in> A"
1381 shows "F A = F (A - {x}) \<circ> f x"
1383 from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1384 by (auto dest: mk_disjoint_insert)
1385 moreover from `finite A` this have "finite B" by simp
1386 ultimately show ?thesis by simp
1389 lemma insert_remove:
1391 shows "F (insert x A) = F (A - {x}) \<circ> f x"
1392 using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1394 lemma commute_left_comp:
1395 "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
1396 by (simp add: o_assoc commute_comp)
1398 lemma commute_comp':
1400 shows "f x \<circ> F A = F A \<circ> f x"
1401 using assms by (induct A)
1402 (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] commute_comp)
1404 lemma commute_left_comp':
1406 shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
1407 using assms by (simp add: o_assoc commute_comp')
1409 lemma commute_comp'':
1410 assumes "finite A" and "finite B"
1411 shows "F B \<circ> F A = F A \<circ> F B"
1412 using assms by (induct A)
1413 (simp_all add: o_assoc, simp add: o_assoc [symmetric] commute_comp')
1415 lemma commute_left_comp'':
1416 assumes "finite A" and "finite B"
1417 shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
1418 using assms by (simp add: o_assoc commute_comp'')
1420 lemmas commute_comps = o_assoc [symmetric] commute_comp commute_left_comp
1421 commute_comp' commute_left_comp' commute_comp'' commute_left_comp''
1424 assumes "finite A" and "finite B"
1425 shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
1426 using assms by (induct A)
1427 (simp_all del: o_apply add: insert_absorb Int_insert_left commute_comps,
1431 assumes "finite A" and "finite B"
1432 and "A \<inter> B = {}"
1433 shows "F (A \<union> B) = F A \<circ> F B"
1435 from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
1436 with `A \<inter> B = {}` show ?thesis by simp
1442 subsubsection {* The natural case with idempotency *}
1444 locale folding_idem = folding +
1445 assumes idem_comp: "f x \<circ> f x = f x"
1448 lemma idem_left_comp:
1449 "f x \<circ> (f x \<circ> g) = f x \<circ> g"
1450 by (simp add: o_assoc idem_comp)
1453 assumes "finite A" and "x \<in> A"
1454 shows "F A \<circ> f x = F A"
1455 using assms by (induct A)
1456 (auto simp add: commute_comps idem_comp, simp add: commute_left_comp' [symmetric] commute_comp')
1458 lemma subset_comp_idem:
1459 assumes "finite A" and "B \<subseteq> A"
1460 shows "F A \<circ> F B = F A"
1462 from assms have "finite B" by (blast dest: finite_subset)
1463 then show ?thesis using `B \<subseteq> A` by (induct B)
1464 (simp_all add: o_assoc in_comp_idem `finite A`)
1467 declare insert [simp del]
1469 lemma insert_idem [simp]:
1471 shows "F (insert x A) = F A \<circ> f x"
1472 using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
1475 assumes "finite A" and "finite B"
1476 shows "F (A \<union> B) = F A \<circ> F B"
1478 from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1479 then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
1480 with assms show ?thesis by (simp add: union_inter)
1486 subsubsection {* The image case with fixed function *}
1488 no_notation times (infixl "*" 70)
1489 no_notation Groups.one ("1")
1491 locale folding_image_simple = comm_monoid +
1492 fixes g :: "('b \<Rightarrow> 'a)"
1493 fixes F :: "'b set \<Rightarrow> 'a"
1494 assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
1499 by (simp add: eq_fold_g)
1501 lemma insert [simp]:
1502 assumes "finite A" and "x \<notin> A"
1503 shows "F (insert x A) = g x * F A"
1505 interpret fun_left_comm "%x y. (g x) * y" proof
1506 qed (simp add: ac_simps)
1507 with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
1508 by (simp add: fold_image_def)
1509 with `finite A` show ?thesis by (simp add: eq_fold_g)
1513 assumes "finite A" and "x \<in> A"
1514 shows "F A = g x * F (A - {x})"
1516 from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
1517 by (auto dest: mk_disjoint_insert)
1518 moreover from `finite A` this have "finite B" by simp
1519 ultimately show ?thesis by simp
1522 lemma insert_remove:
1524 shows "F (insert x A) = g x * F (A - {x})"
1525 using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
1528 assumes "finite A" and "\<forall>x\<in>A. g x = 1"
1530 using assms by (induct A) simp_all
1533 assumes "finite A" and "finite B"
1534 shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1535 using assms proof (induct A)
1536 case empty then show ?case by simp
1538 case (insert x A) then show ?case
1539 by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
1542 corollary union_inter_neutral:
1543 assumes "finite A" and "finite B"
1544 and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
1545 shows "F (A \<union> B) = F A * F B"
1546 using assms by (simp add: union_inter [symmetric] neutral)
1548 corollary union_disjoint:
1549 assumes "finite A" and "finite B"
1550 assumes "A \<inter> B = {}"
1551 shows "F (A \<union> B) = F A * F B"
1552 using assms by (simp add: union_inter_neutral)
1557 subsubsection {* The image case with flexible function *}
1559 locale folding_image = comm_monoid +
1560 fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
1561 assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
1563 sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
1566 context folding_image
1569 lemma reindex: (* FIXME polymorhism *)
1570 assumes "finite A" and "inj_on h A"
1571 shows "F g (h ` A) = F (g \<circ> h) A"
1572 using assms by (induct A) auto
1575 assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
1576 shows "F g A = F h A"
1578 from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
1579 apply - apply (erule finite_induct) apply simp
1580 apply (simp add: subset_insert_iff, clarify)
1581 apply (subgoal_tac "finite C")
1582 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
1583 apply (subgoal_tac "C = insert x (C - {x})")
1584 prefer 2 apply blast
1585 apply (erule ssubst)
1587 apply (erule (1) notE impE)
1588 apply (simp add: Ball_def del: insert_Diff_single)
1590 with assms show ?thesis by simp
1593 lemma UNION_disjoint:
1594 assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
1595 and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
1596 shows "F g (UNION I A) = F (F g \<circ> A) I"
1597 apply (insert assms)
1598 apply (induct set: finite, simp, atomize)
1599 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
1600 prefer 2 apply blast
1601 apply (subgoal_tac "A x Int UNION Fa A = {}")
1602 prefer 2 apply blast
1603 apply (simp add: union_disjoint)
1608 shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
1609 using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
1613 and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
1614 and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
1615 shows "R (F h S) (F g S)"
1616 using fS by (rule finite_subset_induct) (insert assms, auto)
1619 assumes fS: "finite S"
1620 and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
1621 and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
1622 shows "F f1 S = F f2 S'"
1624 from h f12 have hS: "h ` S = S'" by blast
1625 {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1626 from f12 h H have "x = y" by auto }
1627 hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1628 from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1629 from hS have "F f2 S' = F f2 (h ` S)" by simp
1630 also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
1631 also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
1633 finally show ?thesis ..
1636 lemma eq_general_inverses:
1637 assumes fS: "finite S"
1638 and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1639 and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
1640 shows "F j S = F g T"
1641 (* metis solves it, but not yet available here *)
1642 apply (rule eq_general [OF fS, of T h g j])
1648 apply (drule hk) apply simp
1650 apply (erule conjunct1[OF conjunct2[OF hk]])
1659 subsubsection {* The image case with fixed function and idempotency *}
1661 locale folding_image_simple_idem = folding_image_simple +
1662 assumes idem: "x * x = x"
1664 sublocale folding_image_simple_idem < semilattice proof
1667 context folding_image_simple_idem
1671 assumes "finite A" and "x \<in> A"
1672 shows "g x * F A = F A"
1673 using assms by (induct A) (auto simp add: left_commute)
1676 assumes "finite A" and "B \<subseteq> A"
1677 shows "F B * F A = F A"
1679 from assms have "finite B" by (blast dest: finite_subset)
1680 then show ?thesis using `B \<subseteq> A` by (induct B)
1681 (auto simp add: assoc in_idem `finite A`)
1684 declare insert [simp del]
1686 lemma insert_idem [simp]:
1688 shows "F (insert x A) = g x * F A"
1689 using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
1692 assumes "finite A" and "finite B"
1693 shows "F (A \<union> B) = F A * F B"
1695 from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1696 then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
1697 with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1703 subsubsection {* The image case with flexible function and idempotency *}
1705 locale folding_image_idem = folding_image +
1706 assumes idem: "x * x = x"
1708 sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
1712 subsubsection {* The neutral-less case *}
1714 locale folding_one = abel_semigroup +
1715 fixes F :: "'a set \<Rightarrow> 'a"
1716 assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
1719 lemma singleton [simp]:
1721 by (simp add: eq_fold)
1724 assumes "finite A" and "x \<notin> A"
1725 shows "F (insert x A) = fold (op *) x A"
1727 interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
1728 with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
1731 lemma insert [simp]:
1732 assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
1733 shows "F (insert x A) = x * F A"
1735 from `A \<noteq> {}` obtain b where "b \<in> A" by blast
1736 then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
1737 with `finite A` have "finite B" by simp
1738 interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
1739 qed (simp_all add: expand_fun_eq ac_simps)
1740 thm fold.commute_comp' [of B b, simplified expand_fun_eq, simplified]
1741 from `finite B` fold.commute_comp' [of B x]
1742 have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
1743 then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: expand_fun_eq commute)
1744 from `finite B` * fold.insert [of B b]
1745 have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
1746 then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: expand_fun_eq)
1747 from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
1751 assumes "finite A" and "x \<in> A"
1752 shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
1754 from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
1755 with assms show ?thesis by simp
1758 lemma insert_remove:
1760 shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
1761 using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
1763 lemma union_disjoint:
1764 assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
1765 shows "F (A \<union> B) = F A * F B"
1766 using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
1769 assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
1770 shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
1772 from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
1773 from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
1774 case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
1776 case (insert x A) show ?case proof (cases "x \<in> B")
1777 case True then have "B \<noteq> {}" by auto
1778 with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
1779 (simp_all add: insert_absorb ac_simps union_disjoint)
1781 case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
1782 moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
1784 ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
1790 assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
1792 using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
1793 case singleton then show ?case by simp
1795 case insert with elem show ?case by force
1801 subsubsection {* The neutral-less case with idempotency *}
1803 locale folding_one_idem = folding_one +
1804 assumes idem: "x * x = x"
1806 sublocale folding_one_idem < semilattice proof
1809 context folding_one_idem
1813 assumes "finite A" and "x \<in> A"
1814 shows "x * F A = F A"
1816 from assms have "A \<noteq> {}" by auto
1817 with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
1821 assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
1822 shows "F B * F A = F A"
1824 from assms have "finite B" by (blast dest: finite_subset)
1825 then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
1826 (simp_all add: assoc in_idem `finite A`)
1829 lemma eq_fold_idem':
1831 shows "F (insert a A) = fold (op *) a A"
1833 interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
1834 with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
1837 lemma insert_idem [simp]:
1838 assumes "finite A" and "A \<noteq> {}"
1839 shows "F (insert x A) = x * F A"
1840 proof (cases "x \<in> A")
1841 case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
1844 from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
1848 assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
1849 shows "F (A \<union> B) = F A * F B"
1850 proof (cases "A \<inter> B = {}")
1851 case True with assms show ?thesis by (simp add: union_disjoint)
1854 from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
1855 with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
1856 with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
1860 assumes hom: "\<And>x y. h (x * y) = h x * h y"
1861 and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
1862 using N proof (induct rule: finite_ne_induct)
1863 case singleton thus ?case by simp
1866 then have "h (F (insert n N)) = h (n * F N)" by simp
1867 also have "\<dots> = h n * h (F N)" by (rule hom)
1868 also have "h (F N) = F (h ` N)" by(rule insert)
1869 also have "h n * \<dots> = F (insert (h n) (h ` N))"
1870 using insert by(simp)
1871 also have "insert (h n) (h ` N) = h ` insert n N" by simp
1872 finally show ?case .
1877 notation times (infixl "*" 70)
1878 notation Groups.one ("1")
1881 subsection {* Finite cardinality *}
1883 text {* This definition, although traditional, is ugly to work with:
1884 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
1885 But now that we have @{text fold_image} things are easy:
1888 definition card :: "'a set \<Rightarrow> nat" where
1889 "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
1891 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
1892 qed (simp add: card_def)
1894 lemma card_infinite [simp]:
1895 "\<not> finite A \<Longrightarrow> card A = 0"
1896 by (simp add: card_def)
1900 by (fact card.empty)
1902 lemma card_insert_disjoint:
1903 "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
1906 lemma card_insert_if:
1907 "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
1908 by auto (simp add: card.insert_remove card.remove)
1910 lemma card_ge_0_finite:
1911 "card A > 0 \<Longrightarrow> finite A"
1912 by (rule ccontr) simp
1914 lemma card_0_eq [simp, no_atp]:
1915 "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
1916 by (auto dest: mk_disjoint_insert)
1918 lemma finite_UNIV_card_ge_0:
1919 "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
1920 by (rule ccontr) simp
1922 lemma card_eq_0_iff:
1923 "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
1926 lemma card_gt_0_iff:
1927 "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
1928 by (simp add: neq0_conv [symmetric] card_eq_0_iff)
1930 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
1931 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
1932 apply(simp del:insert_Diff_single)
1935 lemma card_Diff_singleton:
1936 "finite A ==> x: A ==> card (A - {x}) = card A - 1"
1937 by (simp add: card_Suc_Diff1 [symmetric])
1939 lemma card_Diff_singleton_if:
1940 "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
1941 by (simp add: card_Diff_singleton)
1943 lemma card_Diff_insert[simp]:
1944 assumes "finite A" and "a:A" and "a ~: B"
1945 shows "card(A - insert a B) = card(A - B) - 1"
1947 have "A - insert a B = (A - B) - {a}" using assms by blast
1948 then show ?thesis using assms by(simp add:card_Diff_singleton)
1951 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
1952 by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
1954 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
1955 by (simp add: card_insert_if)
1958 assumes "finite B" and "A \<subseteq> B"
1959 shows "card A \<le> card B"
1961 from assms have "finite A" by (auto intro: finite_subset)
1962 then show ?thesis using assms proof (induct A arbitrary: B)
1963 case empty then show ?case by simp
1966 then have "x \<in> B" by simp
1967 from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
1968 with insert.hyps have "card A \<le> card (B - {x})" by auto
1969 with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
1973 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
1974 apply (induct set: finite, simp, clarify)
1975 apply (subgoal_tac "finite A & A - {x} <= F")
1976 prefer 2 apply (blast intro: finite_subset, atomize)
1977 apply (drule_tac x = "A - {x}" in spec)
1978 apply (simp add: card_Diff_singleton_if split add: split_if_asm)
1979 apply (case_tac "card A", auto)
1982 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
1983 apply (simp add: psubset_eq linorder_not_le [symmetric])
1984 apply (blast dest: card_seteq)
1987 lemma card_Un_Int: "finite A ==> finite B
1988 ==> card A + card B = card (A Un B) + card (A Int B)"
1989 by (fact card.union_inter [symmetric])
1991 lemma card_Un_disjoint: "finite A ==> finite B
1992 ==> A Int B = {} ==> card (A Un B) = card A + card B"
1993 by (fact card.union_disjoint)
1995 lemma card_Diff_subset:
1996 assumes "finite B" and "B \<subseteq> A"
1997 shows "card (A - B) = card A - card B"
1998 proof (cases "finite A")
1999 case False with assms show ?thesis by simp
2001 case True with assms show ?thesis by (induct B arbitrary: A) simp_all
2004 lemma card_Diff_subset_Int:
2005 assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
2007 have "A - B = A - A \<inter> B" by auto
2009 by (simp add: card_Diff_subset AB)
2012 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
2013 apply (rule Suc_less_SucD)
2014 apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
2017 lemma card_Diff2_less:
2018 "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
2019 apply (case_tac "x = y")
2020 apply (simp add: card_Diff1_less del:card_Diff_insert)
2021 apply (rule less_trans)
2022 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
2025 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
2026 apply (case_tac "x : A")
2027 apply (simp_all add: card_Diff1_less less_imp_le)
2030 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
2031 by (erule psubsetI, blast)
2033 lemma insert_partition:
2034 "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
2035 \<Longrightarrow> x \<inter> \<Union> F = {}"
2038 lemma finite_psubset_induct[consumes 1, case_names psubset]:
2039 assumes fin: "finite A"
2040 and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
2043 proof (induct A taking: card rule: measure_induct_rule)
2045 have fin: "finite A" by fact
2046 have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
2048 assume asm: "B \<subset> A"
2049 from asm have "card B < card A" using psubset_card_mono fin by blast
2051 from asm have "B \<subseteq> A" by auto
2052 then have "finite B" using fin finite_subset by blast
2054 have "P B" using ih by simp
2056 with fin show "P A" using major by blast
2059 text{* main cardinality theorem *}
2060 lemma card_partition [rule_format]:
2062 finite (\<Union> C) -->
2063 (\<forall>c\<in>C. card c = k) -->
2064 (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
2065 k * card(C) = card (\<Union> C)"
2066 apply (erule finite_induct, simp)
2067 apply (simp add: card_Un_disjoint insert_partition
2068 finite_subset [of _ "\<Union> (insert x F)"])
2071 lemma card_eq_UNIV_imp_eq_UNIV:
2072 assumes fin: "finite (UNIV :: 'a set)"
2073 and card: "card A = card (UNIV :: 'a set)"
2074 shows "A = (UNIV :: 'a set)"
2076 show "A \<subseteq> UNIV" by simp
2077 show "UNIV \<subseteq> A"
2082 assume "x \<notin> A"
2083 then have "A \<subset> UNIV" by auto
2084 with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
2085 with card show False by simp
2090 text{*The form of a finite set of given cardinality*}
2093 assumes "card A = Suc k"
2094 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
2096 have fin: "finite A" using assms by (auto intro: ccontr)
2097 moreover have "card A \<noteq> 0" using assms by auto
2098 ultimately obtain b where b: "b \<in> A" by auto
2100 proof (intro exI conjI)
2101 show "A = insert b (A-{b})" using b by blast
2102 show "b \<notin> A - {b}" by blast
2103 show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
2104 using assms b fin by(fastsimp dest:mk_disjoint_insert)+
2110 (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
2112 apply(erule card_eq_SucD)
2114 apply(subst card_insert)
2115 apply(auto intro:ccontr)
2118 lemma finite_fun_UNIVD2:
2119 assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
2120 shows "finite (UNIV :: 'b set)"
2122 from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
2123 by(rule finite_imageI)
2124 moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
2125 by(rule UNIV_eq_I) auto
2126 ultimately show "finite (UNIV :: 'b set)" by simp
2129 lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
2130 unfolding UNIV_unit by simp
2133 subsubsection {* Cardinality of image *}
2135 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
2136 apply (induct set: finite)
2138 apply (simp add: le_SucI card_insert_if)
2142 assumes "inj_on f A"
2143 shows "card (f ` A) = card A"
2144 proof (cases "finite A")
2145 case True then show ?thesis using assms by (induct A) simp_all
2147 case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
2148 with False show ?thesis by simp
2151 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
2152 by(auto simp: card_image bij_betw_def)
2154 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
2155 by (simp add: card_seteq card_image)
2157 lemma eq_card_imp_inj_on:
2158 "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
2159 apply (induct rule:finite_induct)
2161 apply(frule card_image_le[where f = f])
2162 apply(simp add:card_insert_if split:if_splits)
2165 lemma inj_on_iff_eq_card:
2166 "finite A ==> inj_on f A = (card(f ` A) = card A)"
2167 by(blast intro: card_image eq_card_imp_inj_on)
2170 lemma card_inj_on_le:
2171 "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
2172 apply (subgoal_tac "finite A")
2173 apply (force intro: card_mono simp add: card_image [symmetric])
2174 apply (blast intro: finite_imageD dest: finite_subset)
2178 "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
2179 finite A; finite B |] ==> card A = card B"
2180 by (auto intro: le_antisym card_inj_on_le)
2183 subsubsection {* Pigeonhole Principles *}
2185 lemma pigeonhole: "finite(A) \<Longrightarrow> card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
2186 by (auto dest: card_image less_irrefl_nat)
2188 lemma pigeonhole_infinite:
2189 assumes "~ finite A" and "finite(f`A)"
2190 shows "EX a0:A. ~finite{a:A. f a = f a0}"
2192 have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
2193 proof(induct "f`A" arbitrary: A rule: finite_induct)
2194 case empty thus ?case by simp
2199 assume "finite{a:A. f a = b}"
2200 hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
2201 also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
2202 finally have "~ finite({a:A. f a \<noteq> b})" .
2203 from insert(3)[OF _ this]
2204 show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
2206 assume 1: "~finite{a:A. f a = b}"
2207 hence "{a \<in> A. f a = b} \<noteq> {}" by force
2208 thus ?thesis using 1 by blast
2211 from this[OF assms(2,1)] show ?thesis .
2214 lemma pigeonhole_infinite_rel:
2215 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
2216 shows "EX b:B. ~finite{a:A. R a b}"
2218 let ?F = "%a. {b:B. R a b}"
2219 from finite_Pow_iff[THEN iffD2, OF `finite B`]
2220 have "finite(?F ` A)" by(blast intro: rev_finite_subset)
2221 from pigeonhole_infinite[where f = ?F, OF assms(1) this]
2222 obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
2223 obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
2224 { assume "finite{a:A. R a b0}"
2225 then have "finite {a\<in>A. ?F a = ?F a0}"
2226 using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
2228 with 1 `b0 : B` show ?thesis by blast
2232 subsubsection {* Cardinality of sums *}
2235 assumes "finite A" and "finite B"
2236 shows "card (A <+> B) = card A + card B"
2238 have "Inl`A \<inter> Inr`B = {}" by fast
2239 with assms show ?thesis
2241 by (simp add: card_Un_disjoint card_image)
2244 lemma card_Plus_conv_if:
2245 "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
2246 by (auto simp add: card_Plus)
2249 subsubsection {* Cardinality of the Powerset *}
2251 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *)
2252 apply (induct set: finite)
2253 apply (simp_all add: Pow_insert)
2254 apply (subst card_Un_disjoint, blast)
2255 apply (blast intro: finite_imageI, blast)
2256 apply (subgoal_tac "inj_on (insert x) (Pow F)")
2257 apply (simp add: card_image Pow_insert)
2258 apply (unfold inj_on_def)
2259 apply (blast elim!: equalityE)
2262 text {* Relates to equivalence classes. Based on a theorem of F. Kammüller. *}
2264 lemma dvd_partition:
2265 "finite (Union C) ==>
2266 ALL c : C. k dvd card c ==>
2267 (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
2268 k dvd card (Union C)"
2269 apply(frule finite_UnionD)
2270 apply(rotate_tac -1)
2271 apply (induct set: finite, simp_all, clarify)
2272 apply (subst card_Un_disjoint)
2273 apply (auto simp add: disjoint_eq_subset_Compl)
2277 subsubsection {* Relating injectivity and surjectivity *}
2279 lemma finite_surj_inj: "finite(A) \<Longrightarrow> A <= f`A \<Longrightarrow> inj_on f A"
2280 apply(rule eq_card_imp_inj_on, assumption)
2281 apply(frule finite_imageI)
2282 apply(drule (1) card_seteq)
2283 apply(erule card_image_le)
2287 lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
2288 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
2289 by (blast intro: finite_surj_inj subset_UNIV dest:surj_range)
2291 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
2292 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
2293 by(fastsimp simp:surj_def dest!: endo_inj_surj)
2295 corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
2297 assume "finite(UNIV::nat set)"
2298 with finite_UNIV_inj_surj[of Suc]
2299 show False by simp (blast dest: Suc_neq_Zero surjD)
2302 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
2303 lemma infinite_UNIV_char_0[no_atp]:
2304 "\<not> finite (UNIV::'a::semiring_char_0 set)"
2306 assume "finite (UNIV::'a set)"
2307 with subset_UNIV have "finite (range of_nat::'a set)"
2308 by (rule finite_subset)
2309 moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
2310 by (simp add: inj_on_def)
2311 ultimately have "finite (UNIV::nat set)"
2312 by (rule finite_imageD)