src/HOL/Library/Multiset.thy
 author wenzelm Wed Mar 25 10:44:57 2015 +0100 (2015-03-25) changeset 59807 22bc39064290 parent 59625 aacdce52b2fc child 59813 6320064f22bb permissions -rw-r--r--
prefer local fixes;
1 (*  Title:      HOL/Library/Multiset.thy
2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
3     Author:     Andrei Popescu, TU Muenchen
4 *)
6 section {* (Finite) multisets *}
8 theory Multiset
9 imports Main
10 begin
12 subsection {* The type of multisets *}
14 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
16 typedef 'a multiset = "multiset :: ('a => nat) set"
17   morphisms count Abs_multiset
18   unfolding multiset_def
19 proof
20   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
21 qed
23 setup_lifting type_definition_multiset
25 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
26   "a :# M == 0 < count M a"
28 notation (xsymbols)
29   Melem (infix "\<in>#" 50)
31 lemma multiset_eq_iff:
32   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
33   by (simp only: count_inject [symmetric] fun_eq_iff)
35 lemma multiset_eqI:
36   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
37   using multiset_eq_iff by auto
39 text {*
40  \medskip Preservation of the representing set @{term multiset}.
41 *}
43 lemma const0_in_multiset:
44   "(\<lambda>a. 0) \<in> multiset"
45   by (simp add: multiset_def)
47 lemma only1_in_multiset:
48   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
49   by (simp add: multiset_def)
51 lemma union_preserves_multiset:
52   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
53   by (simp add: multiset_def)
55 lemma diff_preserves_multiset:
56   assumes "M \<in> multiset"
57   shows "(\<lambda>a. M a - N a) \<in> multiset"
58 proof -
59   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
60     by auto
61   with assms show ?thesis
62     by (auto simp add: multiset_def intro: finite_subset)
63 qed
65 lemma filter_preserves_multiset:
66   assumes "M \<in> multiset"
67   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
68 proof -
69   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
70     by auto
71   with assms show ?thesis
72     by (auto simp add: multiset_def intro: finite_subset)
73 qed
75 lemmas in_multiset = const0_in_multiset only1_in_multiset
76   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
79 subsection {* Representing multisets *}
81 text {* Multiset enumeration *}
83 instantiation multiset :: (type) cancel_comm_monoid_add
84 begin
86 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
87 by (rule const0_in_multiset)
89 abbreviation Mempty :: "'a multiset" ("{#}") where
90   "Mempty \<equiv> 0"
92 lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
93 by (rule union_preserves_multiset)
95 instance
96 by default (transfer, simp add: fun_eq_iff)+
98 end
100 lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
101 by (rule only1_in_multiset)
103 syntax
104   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
105 translations
106   "{#x, xs#}" == "{#x#} + {#xs#}"
107   "{#x#}" == "CONST single x"
109 lemma count_empty [simp]: "count {#} a = 0"
110   by (simp add: zero_multiset.rep_eq)
112 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
113   by (simp add: single.rep_eq)
116 subsection {* Basic operations *}
118 subsubsection {* Union *}
120 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
121   by (simp add: plus_multiset.rep_eq)
124 subsubsection {* Difference *}
126 instantiation multiset :: (type) comm_monoid_diff
127 begin
129 lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
130 by (rule diff_preserves_multiset)
132 instance
133 by default (transfer, simp add: fun_eq_iff)+
135 end
137 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
138   by (simp add: minus_multiset.rep_eq)
140 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
141   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
143 lemma diff_cancel[simp]: "A - A = {#}"
144   by (fact Groups.diff_cancel)
146 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
147   by (fact add_diff_cancel_right')
149 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
150   by (fact add_diff_cancel_left')
152 lemma diff_right_commute:
153   "(M::'a multiset) - N - Q = M - Q - N"
154   by (fact diff_right_commute)
157   "(M::'a multiset) - (N + Q) = M - N - Q"
158   by (rule sym) (fact diff_diff_add)
160 lemma insert_DiffM:
161   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
162   by (clarsimp simp: multiset_eq_iff)
164 lemma insert_DiffM2 [simp]:
165   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
166   by (clarsimp simp: multiset_eq_iff)
168 lemma diff_union_swap:
169   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
170   by (auto simp add: multiset_eq_iff)
172 lemma diff_union_single_conv:
173   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
174   by (simp add: multiset_eq_iff)
177 subsubsection {* Equality of multisets *}
179 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
180   by (simp add: multiset_eq_iff)
182 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
183   by (auto simp add: multiset_eq_iff)
185 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
186   by (auto simp add: multiset_eq_iff)
188 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
189   by (auto simp add: multiset_eq_iff)
191 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
192   by (auto simp add: multiset_eq_iff)
194 lemma diff_single_trivial:
195   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
196   by (auto simp add: multiset_eq_iff)
198 lemma diff_single_eq_union:
199   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
200   by auto
202 lemma union_single_eq_diff:
203   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
204   by (auto dest: sym)
206 lemma union_single_eq_member:
207   "M + {#x#} = N \<Longrightarrow> x \<in># N"
208   by auto
210 lemma union_is_single:
211   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
212 proof
213   assume ?rhs then show ?lhs by auto
214 next
215   assume ?lhs then show ?rhs
216     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
217 qed
219 lemma single_is_union:
220   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
221   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
224   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
225 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
226 proof
227   assume ?rhs then show ?lhs
229     (drule sym, simp add: add.assoc [symmetric])
230 next
231   assume ?lhs
232   show ?rhs
233   proof (cases "a = b")
234     case True with `?lhs` show ?thesis by simp
235   next
236     case False
237     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
238     with False have "a \<in># N" by auto
239     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
240     moreover note False
241     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
242   qed
243 qed
245 lemma insert_noteq_member:
246   assumes BC: "B + {#b#} = C + {#c#}"
247    and bnotc: "b \<noteq> c"
248   shows "c \<in># B"
249 proof -
250   have "c \<in># C + {#c#}" by simp
251   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
252   then have "c \<in># B + {#b#}" using BC by simp
253   then show "c \<in># B" using nc by simp
254 qed
257   "(M + {#a#} = N + {#b#}) =
258     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
261 lemma multi_member_split:
262   "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
263   by (rule_tac x = "M - {#x#}" in exI, simp)
266   assumes "c \<in># B" and "b \<noteq> c"
267   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
268 proof -
269   from `c \<in># B` obtain A where B: "B = A + {#c#}"
270     by (blast dest: multi_member_split)
271   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
272   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
273     by (simp add: ac_simps)
274   then show ?thesis using B by simp
275 qed
278 subsubsection {* Pointwise ordering induced by count *}
280 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
281 begin
283 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)" .
285 lemmas mset_le_def = less_eq_multiset_def
287 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
288   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
290 instance
291   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
293 end
295 lemma mset_less_eqI:
296   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
297   by (simp add: mset_le_def)
299 lemma mset_le_exists_conv:
300   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
301 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
302 apply (auto intro: multiset_eq_iff [THEN iffD2])
303 done
305 instance multiset :: (type) ordered_cancel_comm_monoid_diff
306   by default (simp, fact mset_le_exists_conv)
308 lemma mset_le_mono_add_right_cancel [simp]:
309   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
310   by (fact add_le_cancel_right)
312 lemma mset_le_mono_add_left_cancel [simp]:
313   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
314   by (fact add_le_cancel_left)
317   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
318   by (fact add_mono)
320 lemma mset_le_add_left [simp]:
321   "(A::'a multiset) \<le> A + B"
322   unfolding mset_le_def by auto
324 lemma mset_le_add_right [simp]:
325   "B \<le> (A::'a multiset) + B"
326   unfolding mset_le_def by auto
328 lemma mset_le_single:
329   "a :# B \<Longrightarrow> {#a#} \<le> B"
330   by (simp add: mset_le_def)
332 lemma multiset_diff_union_assoc:
333   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
334   by (simp add: multiset_eq_iff mset_le_def)
336 lemma mset_le_multiset_union_diff_commute:
337   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
338 by (simp add: multiset_eq_iff mset_le_def)
340 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
341 by(simp add: mset_le_def)
343 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
344 apply (clarsimp simp: mset_le_def mset_less_def)
345 apply (erule_tac x=x in allE)
346 apply auto
347 done
349 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
350 apply (clarsimp simp: mset_le_def mset_less_def)
351 apply (erule_tac x = x in allE)
352 apply auto
353 done
355 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
356 apply (rule conjI)
357  apply (simp add: mset_lessD)
358 apply (clarsimp simp: mset_le_def mset_less_def)
359 apply safe
360  apply (erule_tac x = a in allE)
361  apply (auto split: split_if_asm)
362 done
364 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
365 apply (rule conjI)
366  apply (simp add: mset_leD)
367 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
368 done
370 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
371   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
373 lemma empty_le[simp]: "{#} \<le> A"
374   unfolding mset_le_exists_conv by auto
376 lemma le_empty[simp]: "(M \<le> {#}) = (M = {#})"
377   unfolding mset_le_exists_conv by auto
379 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
380   by (auto simp: mset_le_def mset_less_def)
382 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
383   by simp
386   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
387   by (fact add_less_imp_less_right)
389 lemma mset_less_empty_nonempty:
390   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
391   by (auto simp: mset_le_def mset_less_def)
393 lemma mset_less_diff_self:
394   "c \<in># B \<Longrightarrow> B - {#c#} < B"
395   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
398 subsubsection {* Intersection *}
400 instantiation multiset :: (type) semilattice_inf
401 begin
403 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
404   multiset_inter_def: "inf_multiset A B = A - (A - B)"
406 instance
407 proof -
408   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
409   show "OFCLASS('a multiset, semilattice_inf_class)"
410     by default (auto simp add: multiset_inter_def mset_le_def aux)
411 qed
413 end
415 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
416   "multiset_inter \<equiv> inf"
418 lemma multiset_inter_count [simp]:
419   "count (A #\<inter> B) x = min (count A x) (count B x)"
420   by (simp add: multiset_inter_def)
422 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
423   by (rule multiset_eqI) auto
425 lemma multiset_union_diff_commute:
426   assumes "B #\<inter> C = {#}"
427   shows "A + B - C = A - C + B"
428 proof (rule multiset_eqI)
429   fix x
430   from assms have "min (count B x) (count C x) = 0"
431     by (auto simp add: multiset_eq_iff)
432   then have "count B x = 0 \<or> count C x = 0"
433     by auto
434   then show "count (A + B - C) x = count (A - C + B) x"
435     by auto
436 qed
438 lemma empty_inter [simp]:
439   "{#} #\<inter> M = {#}"
440   by (simp add: multiset_eq_iff)
442 lemma inter_empty [simp]:
443   "M #\<inter> {#} = {#}"
444   by (simp add: multiset_eq_iff)
447   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
448   by (simp add: multiset_eq_iff)
451   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
452   by (simp add: multiset_eq_iff)
455   "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
456   by (simp add: multiset_eq_iff)
459   "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
460   by (simp add: multiset_eq_iff)
463 subsubsection {* Bounded union *}
465 instantiation multiset :: (type) semilattice_sup
466 begin
468 definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
469   "sup_multiset A B = A + (B - A)"
471 instance
472 proof -
473   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" by arith
474   show "OFCLASS('a multiset, semilattice_sup_class)"
475     by default (auto simp add: sup_multiset_def mset_le_def aux)
476 qed
478 end
480 abbreviation sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<union>" 70) where
481   "sup_multiset \<equiv> sup"
483 lemma sup_multiset_count [simp]:
484   "count (A #\<union> B) x = max (count A x) (count B x)"
485   by (simp add: sup_multiset_def)
487 lemma empty_sup [simp]:
488   "{#} #\<union> M = M"
489   by (simp add: multiset_eq_iff)
491 lemma sup_empty [simp]:
492   "M #\<union> {#} = M"
493   by (simp add: multiset_eq_iff)
496   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
497   by (simp add: multiset_eq_iff)
500   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
501   by (simp add: multiset_eq_iff)
504   "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
505   by (simp add: multiset_eq_iff)
508   "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
509   by (simp add: multiset_eq_iff)
512 subsubsection {* Filter (with comprehension syntax) *}
514 text {* Multiset comprehension *}
516 lift_definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
517 by (rule filter_preserves_multiset)
519 hide_const (open) filter
521 lemma count_filter [simp]:
522   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
523   by (simp add: filter.rep_eq)
525 lemma filter_empty [simp]:
526   "Multiset.filter P {#} = {#}"
527   by (rule multiset_eqI) simp
529 lemma filter_single [simp]:
530   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
531   by (rule multiset_eqI) simp
533 lemma filter_union [simp]:
534   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
535   by (rule multiset_eqI) simp
537 lemma filter_diff [simp]:
538   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
539   by (rule multiset_eqI) simp
541 lemma filter_inter [simp]:
542   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
543   by (rule multiset_eqI) simp
545 lemma multiset_filter_subset[simp]: "Multiset.filter f M \<le> M"
546   unfolding less_eq_multiset.rep_eq by auto
548 lemma multiset_filter_mono: assumes "A \<le> B"
549   shows "Multiset.filter f A \<le> Multiset.filter f B"
550 proof -
551   from assms[unfolded mset_le_exists_conv]
552   obtain C where B: "B = A + C" by auto
553   show ?thesis unfolding B by auto
554 qed
556 syntax
557   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
558 syntax (xsymbol)
559   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
560 translations
561   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
564 subsubsection {* Set of elements *}
566 definition set_of :: "'a multiset => 'a set" where
567   "set_of M = {x. x :# M}"
569 lemma set_of_empty [simp]: "set_of {#} = {}"
570 by (simp add: set_of_def)
572 lemma set_of_single [simp]: "set_of {#b#} = {b}"
573 by (simp add: set_of_def)
575 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
576 by (auto simp add: set_of_def)
578 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
579 by (auto simp add: set_of_def multiset_eq_iff)
581 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
582 by (auto simp add: set_of_def)
584 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
585 by (auto simp add: set_of_def)
587 lemma finite_set_of [iff]: "finite (set_of M)"
588   using count [of M] by (simp add: multiset_def set_of_def)
590 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
591   unfolding set_of_def[symmetric] by simp
593 lemma set_of_mono: "A \<le> B \<Longrightarrow> set_of A \<subseteq> set_of B"
594   by (metis mset_leD subsetI mem_set_of_iff)
596 subsubsection {* Size *}
598 definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
600 lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
601   by (auto simp: wcount_def add_mult_distrib)
603 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
604   "size_multiset f M = setsum (wcount f M) (set_of M)"
606 lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
608 instantiation multiset :: (type) size begin
609 definition size_multiset where
610   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
611 instance ..
612 end
614 lemmas size_multiset_overloaded_eq =
615   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
617 lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
618 by (simp add: size_multiset_def)
620 lemma size_empty [simp]: "size {#} = 0"
623 lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
624 by (simp add: size_multiset_eq)
626 lemma size_single [simp]: "size {#b#} = 1"
629 lemma setsum_wcount_Int:
630   "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_of N) = setsum (wcount f N) A"
631 apply (induct rule: finite_induct)
632  apply simp
633 apply (simp add: Int_insert_left set_of_def wcount_def)
634 done
636 lemma size_multiset_union [simp]:
637   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
638 apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
639 apply (subst Int_commute)
640 apply (simp add: setsum_wcount_Int)
641 done
643 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
646 lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
647 by (auto simp add: size_multiset_eq multiset_eq_iff)
649 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
652 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
653 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
655 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
656 apply (unfold size_multiset_overloaded_eq)
657 apply (drule setsum_SucD)
658 apply auto
659 done
661 lemma size_eq_Suc_imp_eq_union:
662   assumes "size M = Suc n"
663   shows "\<exists>a N. M = N + {#a#}"
664 proof -
665   from assms obtain a where "a \<in># M"
666     by (erule size_eq_Suc_imp_elem [THEN exE])
667   then have "M = M - {#a#} + {#a#}" by simp
668   then show ?thesis by blast
669 qed
672 subsection {* Induction and case splits *}
674 theorem multiset_induct [case_names empty add, induct type: multiset]:
675   assumes empty: "P {#}"
676   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
677   shows "P M"
678 proof (induct n \<equiv> "size M" arbitrary: M)
679   case 0 thus "P M" by (simp add: empty)
680 next
681   case (Suc k)
682   obtain N x where "M = N + {#x#}"
683     using `Suc k = size M` [symmetric]
684     using size_eq_Suc_imp_eq_union by fast
685   with Suc add show "P M" by simp
686 qed
688 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
689 by (induct M) auto
691 lemma multiset_cases [cases type]:
692   obtains (empty) "M = {#}"
693     | (add) N x where "M = N + {#x#}"
694   using assms by (induct M) simp_all
696 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
697 by (cases "B = {#}") (auto dest: multi_member_split)
699 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
700 apply (subst multiset_eq_iff)
701 apply auto
702 done
704 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
705 proof (induct A arbitrary: B)
706   case (empty M)
707   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
708   then obtain M' x where "M = M' + {#x#}"
709     by (blast dest: multi_nonempty_split)
710   then show ?case by simp
711 next
712   case (add S x T)
713   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
714   have SxsubT: "S + {#x#} < T" by fact
715   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
716   then obtain T' where T: "T = T' + {#x#}"
717     by (blast dest: multi_member_split)
718   then have "S < T'" using SxsubT
719     by (blast intro: mset_less_add_bothsides)
720   then have "size S < size T'" using IH by simp
721   then show ?case using T by simp
722 qed
725 subsubsection {* Strong induction and subset induction for multisets *}
727 text {* Well-foundedness of strict subset relation *}
729 lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M < N}"
730 apply (rule wf_measure [THEN wf_subset, where f1=size])
731 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
732 done
734 lemma full_multiset_induct [case_names less]:
735 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
736 shows "P B"
737 apply (rule wf_less_mset_rel [THEN wf_induct])
738 apply (rule ih, auto)
739 done
741 lemma multi_subset_induct [consumes 2, case_names empty add]:
742 assumes "F \<le> A"
743   and empty: "P {#}"
744   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
745 shows "P F"
746 proof -
747   from `F \<le> A`
748   show ?thesis
749   proof (induct F)
750     show "P {#}" by fact
751   next
752     fix x F
753     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
754     show "P (F + {#x#})"
755     proof (rule insert)
756       from i show "x \<in># A" by (auto dest: mset_le_insertD)
757       from i have "F \<le> A" by (auto dest: mset_le_insertD)
758       with P show "P F" .
759     qed
760   qed
761 qed
764 subsection {* The fold combinator *}
766 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
767 where
768   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
770 lemma fold_mset_empty [simp]:
771   "fold f s {#} = s"
772   by (simp add: fold_def)
774 context comp_fun_commute
775 begin
777 lemma fold_mset_insert:
778   "fold f s (M + {#x#}) = f x (fold f s M)"
779 proof -
780   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
781     by (fact comp_fun_commute_funpow)
782   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
783     by (fact comp_fun_commute_funpow)
784   show ?thesis
785   proof (cases "x \<in> set_of M")
786     case False
787     then have *: "count (M + {#x#}) x = 1" by simp
788     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
789       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
790       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
791     with False * show ?thesis
792       by (simp add: fold_def del: count_union)
793   next
794     case True
795     def N \<equiv> "set_of M - {x}"
796     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
797     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
798       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
799       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
800     with * show ?thesis by (simp add: fold_def del: count_union) simp
801   qed
802 qed
804 corollary fold_mset_single [simp]:
805   "fold f s {#x#} = f x s"
806 proof -
807   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
808   then show ?thesis by simp
809 qed
811 lemma fold_mset_fun_left_comm:
812   "f x (fold f s M) = fold f (f x s) M"
813   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
815 lemma fold_mset_union [simp]:
816   "fold f s (M + N) = fold f (fold f s M) N"
817 proof (induct M)
818   case empty then show ?case by simp
819 next
820   case (add M x)
821   have "M + {#x#} + N = (M + N) + {#x#}"
822     by (simp add: ac_simps)
823   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
824 qed
826 lemma fold_mset_fusion:
827   assumes "comp_fun_commute g"
828   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold g w A) = fold f (h w) A" (is "PROP ?P")
829 proof -
830   interpret comp_fun_commute g by (fact assms)
831   show "PROP ?P" by (induct A) auto
832 qed
834 end
836 text {*
837   A note on code generation: When defining some function containing a
838   subterm @{term "fold F"}, code generation is not automatic. When
839   interpreting locale @{text left_commutative} with @{text F}, the
840   would be code thms for @{const fold} become thms like
841   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
842   contains defined symbols, i.e.\ is not a code thm. Hence a separate
843   constant with its own code thms needs to be introduced for @{text
844   F}. See the image operator below.
845 *}
848 subsection {* Image *}
850 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
851   "image_mset f = fold (plus o single o f) {#}"
853 lemma comp_fun_commute_mset_image:
854   "comp_fun_commute (plus o single o f)"
855 proof
856 qed (simp add: ac_simps fun_eq_iff)
858 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
859   by (simp add: image_mset_def)
861 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
862 proof -
863   interpret comp_fun_commute "plus o single o f"
864     by (fact comp_fun_commute_mset_image)
865   show ?thesis by (simp add: image_mset_def)
866 qed
868 lemma image_mset_union [simp]:
869   "image_mset f (M + N) = image_mset f M + image_mset f N"
870 proof -
871   interpret comp_fun_commute "plus o single o f"
872     by (fact comp_fun_commute_mset_image)
873   show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
874 qed
876 corollary image_mset_insert:
877   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
878   by simp
880 lemma set_of_image_mset [simp]:
881   "set_of (image_mset f M) = image f (set_of M)"
882   by (induct M) simp_all
884 lemma size_image_mset [simp]:
885   "size (image_mset f M) = size M"
886   by (induct M) simp_all
888 lemma image_mset_is_empty_iff [simp]:
889   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
890   by (cases M) auto
892 syntax
893   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
894       ("({#_/. _ :# _#})")
895 translations
896   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
898 syntax
899   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
900       ("({#_/ | _ :# _./ _#})")
901 translations
902   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
904 text {*
905   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
906   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
907   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
908   @{term "{#x+x|x:#M. x<c#}"}.
909 *}
911 functor image_mset: image_mset
912 proof -
913   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
914   proof
915     fix A
916     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
917       by (induct A) simp_all
918   qed
919   show "image_mset id = id"
920   proof
921     fix A
922     show "image_mset id A = id A"
923       by (induct A) simp_all
924   qed
925 qed
927 declare image_mset.identity [simp]
930 subsection {* Further conversions *}
932 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
933   "multiset_of [] = {#}" |
934   "multiset_of (a # x) = multiset_of x + {# a #}"
936 lemma in_multiset_in_set:
937   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
938   by (induct xs) simp_all
940 lemma count_multiset_of:
941   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
942   by (induct xs) simp_all
944 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
945 by (induct x) auto
947 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
948 by (induct x) auto
950 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
951 by (induct x) auto
953 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
954 by (induct xs) auto
956 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
957   by (induct xs) simp_all
959 lemma multiset_of_append [simp]:
960   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
961   by (induct xs arbitrary: ys) (auto simp: ac_simps)
963 lemma multiset_of_filter:
964   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
965   by (induct xs) simp_all
967 lemma multiset_of_rev [simp]:
968   "multiset_of (rev xs) = multiset_of xs"
969   by (induct xs) simp_all
971 lemma surj_multiset_of: "surj multiset_of"
972 apply (unfold surj_def)
973 apply (rule allI)
974 apply (rule_tac M = y in multiset_induct)
975  apply auto
976 apply (rule_tac x = "x # xa" in exI)
977 apply auto
978 done
980 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
981 by (induct x) auto
983 lemma distinct_count_atmost_1:
984   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
985 apply (induct x, simp, rule iffI, simp_all)
986 apply (rename_tac a b)
987 apply (rule conjI)
988 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
989 apply (erule_tac x = a in allE, simp, clarify)
990 apply (erule_tac x = aa in allE, simp)
991 done
993 lemma multiset_of_eq_setD:
994   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
995 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
997 lemma set_eq_iff_multiset_of_eq_distinct:
998   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
999     (set x = set y) = (multiset_of x = multiset_of y)"
1000 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
1002 lemma set_eq_iff_multiset_of_remdups_eq:
1003    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
1004 apply (rule iffI)
1005 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
1006 apply (drule distinct_remdups [THEN distinct_remdups
1007       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
1008 apply simp
1009 done
1011 lemma multiset_of_compl_union [simp]:
1012   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
1013   by (induct xs) (auto simp: ac_simps)
1015 lemma count_multiset_of_length_filter:
1016   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
1017   by (induct xs) auto
1019 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
1020 apply (induct ls arbitrary: i)
1021  apply simp
1022 apply (case_tac i)
1023  apply auto
1024 done
1026 lemma multiset_of_remove1[simp]:
1027   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
1028 by (induct xs) (auto simp add: multiset_eq_iff)
1030 lemma multiset_of_eq_length:
1031   assumes "multiset_of xs = multiset_of ys"
1032   shows "length xs = length ys"
1033   using assms by (metis size_multiset_of)
1035 lemma multiset_of_eq_length_filter:
1036   assumes "multiset_of xs = multiset_of ys"
1037   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
1038   using assms by (metis count_multiset_of)
1040 lemma fold_multiset_equiv:
1041   assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
1042     and equiv: "multiset_of xs = multiset_of ys"
1043   shows "List.fold f xs = List.fold f ys"
1044 using f equiv [symmetric]
1045 proof (induct xs arbitrary: ys)
1046   case Nil then show ?case by simp
1047 next
1048   case (Cons x xs)
1049   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
1050   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
1051     by (rule Cons.prems(1)) (simp_all add: *)
1052   moreover from * have "x \<in> set ys" by simp
1053   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
1054   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
1055   ultimately show ?case by simp
1056 qed
1058 lemma multiset_of_insort [simp]:
1059   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
1060   by (induct xs) (simp_all add: ac_simps)
1062 lemma in_multiset_of:
1063   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
1064   by (induct xs) simp_all
1066 lemma multiset_of_map:
1067   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
1068   by (induct xs) simp_all
1070 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
1071 where
1072   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
1074 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
1075 where
1076   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
1077 proof -
1078   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
1079   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
1080   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
1081 qed
1083 lemma count_multiset_of_set [simp]:
1084   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
1085   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
1086   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
1087 proof -
1088   { fix A
1089     assume "x \<notin> A"
1090     have "count (multiset_of_set A) x = 0"
1091     proof (cases "finite A")
1092       case False then show ?thesis by simp
1093     next
1094       case True from True `x \<notin> A` show ?thesis by (induct A) auto
1095     qed
1096   } note * = this
1097   then show "PROP ?P" "PROP ?Q" "PROP ?R"
1098   by (auto elim!: Set.set_insert)
1099 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
1101 context linorder
1102 begin
1104 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
1105 where
1106   "sorted_list_of_multiset M = fold insort [] M"
1108 lemma sorted_list_of_multiset_empty [simp]:
1109   "sorted_list_of_multiset {#} = []"
1110   by (simp add: sorted_list_of_multiset_def)
1112 lemma sorted_list_of_multiset_singleton [simp]:
1113   "sorted_list_of_multiset {#x#} = [x]"
1114 proof -
1115   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1116   show ?thesis by (simp add: sorted_list_of_multiset_def)
1117 qed
1119 lemma sorted_list_of_multiset_insert [simp]:
1120   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
1121 proof -
1122   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1123   show ?thesis by (simp add: sorted_list_of_multiset_def)
1124 qed
1126 end
1128 lemma multiset_of_sorted_list_of_multiset [simp]:
1129   "multiset_of (sorted_list_of_multiset M) = M"
1130   by (induct M) simp_all
1132 lemma sorted_list_of_multiset_multiset_of [simp]:
1133   "sorted_list_of_multiset (multiset_of xs) = sort xs"
1134   by (induct xs) simp_all
1136 lemma finite_set_of_multiset_of_set:
1137   assumes "finite A"
1138   shows "set_of (multiset_of_set A) = A"
1139   using assms by (induct A) simp_all
1141 lemma infinite_set_of_multiset_of_set:
1142   assumes "\<not> finite A"
1143   shows "set_of (multiset_of_set A) = {}"
1144   using assms by simp
1146 lemma set_sorted_list_of_multiset [simp]:
1147   "set (sorted_list_of_multiset M) = set_of M"
1148   by (induct M) (simp_all add: set_insort)
1150 lemma sorted_list_of_multiset_of_set [simp]:
1151   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
1152   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
1155 subsection {* Big operators *}
1157 no_notation times (infixl "*" 70)
1158 no_notation Groups.one ("1")
1160 locale comm_monoid_mset = comm_monoid
1161 begin
1163 definition F :: "'a multiset \<Rightarrow> 'a"
1164 where
1165   eq_fold: "F M = Multiset.fold f 1 M"
1167 lemma empty [simp]:
1168   "F {#} = 1"
1169   by (simp add: eq_fold)
1171 lemma singleton [simp]:
1172   "F {#x#} = x"
1173 proof -
1174   interpret comp_fun_commute
1175     by default (simp add: fun_eq_iff left_commute)
1176   show ?thesis by (simp add: eq_fold)
1177 qed
1179 lemma union [simp]:
1180   "F (M + N) = F M * F N"
1181 proof -
1182   interpret comp_fun_commute f
1183     by default (simp add: fun_eq_iff left_commute)
1184   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
1185 qed
1187 end
1189 notation times (infixl "*" 70)
1190 notation Groups.one ("1")
1193 begin
1195 definition msetsum :: "'a multiset \<Rightarrow> 'a"
1196 where
1197   "msetsum = comm_monoid_mset.F plus 0"
1199 sublocale msetsum!: comm_monoid_mset plus 0
1200 where
1201   "comm_monoid_mset.F plus 0 = msetsum"
1202 proof -
1203   show "comm_monoid_mset plus 0" ..
1204   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
1205 qed
1207 lemma setsum_unfold_msetsum:
1208   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
1209   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1211 end
1213 syntax
1214   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1215       ("(3SUM _:#_. _)" [0, 51, 10] 10)
1217 syntax (xsymbols)
1218   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1219       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1221 syntax (HTML output)
1222   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1223       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1225 translations
1226   "SUM i :# A. b" == "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"
1228 context comm_monoid_mult
1229 begin
1231 definition msetprod :: "'a multiset \<Rightarrow> 'a"
1232 where
1233   "msetprod = comm_monoid_mset.F times 1"
1235 sublocale msetprod!: comm_monoid_mset times 1
1236 where
1237   "comm_monoid_mset.F times 1 = msetprod"
1238 proof -
1239   show "comm_monoid_mset times 1" ..
1240   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
1241 qed
1243 lemma msetprod_empty:
1244   "msetprod {#} = 1"
1245   by (fact msetprod.empty)
1247 lemma msetprod_singleton:
1248   "msetprod {#x#} = x"
1249   by (fact msetprod.singleton)
1251 lemma msetprod_Un:
1252   "msetprod (A + B) = msetprod A * msetprod B"
1253   by (fact msetprod.union)
1255 lemma setprod_unfold_msetprod:
1256   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
1257   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1259 lemma msetprod_multiplicity:
1260   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
1261   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
1263 end
1265 syntax
1266   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1267       ("(3PROD _:#_. _)" [0, 51, 10] 10)
1269 syntax (xsymbols)
1270   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1271       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1273 syntax (HTML output)
1274   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1275       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1277 translations
1278   "PROD i :# A. b" == "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"
1280 lemma (in comm_semiring_1) dvd_msetprod:
1281   assumes "x \<in># A"
1282   shows "x dvd msetprod A"
1283 proof -
1284   from assms have "A = (A - {#x#}) + {#x#}" by simp
1285   then obtain B where "A = B + {#x#}" ..
1286   then show ?thesis by simp
1287 qed
1290 subsection {* Cardinality *}
1292 definition mcard :: "'a multiset \<Rightarrow> nat"
1293 where
1294   "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
1296 lemma mcard_empty [simp]:
1297   "mcard {#} = 0"
1298   by (simp add: mcard_def)
1300 lemma mcard_singleton [simp]:
1301   "mcard {#a#} = Suc 0"
1302   by (simp add: mcard_def)
1304 lemma mcard_plus [simp]:
1305   "mcard (M + N) = mcard M + mcard N"
1306   by (simp add: mcard_def)
1308 lemma mcard_empty_iff [simp]:
1309   "mcard M = 0 \<longleftrightarrow> M = {#}"
1310   by (induct M) simp_all
1312 lemma mcard_unfold_setsum:
1313   "mcard M = setsum (count M) (set_of M)"
1314 proof (induct M)
1315   case empty then show ?case by simp
1316 next
1317   case (add M x) then show ?case
1318     by (cases "x \<in> set_of M")
1319       (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
1320 qed
1322 lemma size_eq_mcard:
1323   "size = mcard"
1324   by (simp add: fun_eq_iff size_multiset_overloaded_eq mcard_unfold_setsum)
1326 lemma mcard_multiset_of:
1327   "mcard (multiset_of xs) = length xs"
1328   by (induct xs) simp_all
1330 lemma mcard_mono: assumes "A \<le> B"
1331   shows "mcard A \<le> mcard B"
1332 proof -
1333   from assms[unfolded mset_le_exists_conv]
1334   obtain C where B: "B = A + C" by auto
1335   show ?thesis unfolding B by (induct C, auto)
1336 qed
1338 lemma mcard_filter_lesseq[simp]: "mcard (Multiset.filter f M) \<le> mcard M"
1339   by (rule mcard_mono[OF multiset_filter_subset])
1342 subsection {* Alternative representations *}
1344 subsubsection {* Lists *}
1346 context linorder
1347 begin
1349 lemma multiset_of_insort [simp]:
1350   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
1351   by (induct xs) (simp_all add: ac_simps)
1353 lemma multiset_of_sort [simp]:
1354   "multiset_of (sort_key k xs) = multiset_of xs"
1355   by (induct xs) (simp_all add: ac_simps)
1357 text {*
1358   This lemma shows which properties suffice to show that a function
1359   @{text "f"} with @{text "f xs = ys"} behaves like sort.
1360 *}
1362 lemma properties_for_sort_key:
1363   assumes "multiset_of ys = multiset_of xs"
1364   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
1365   and "sorted (map f ys)"
1366   shows "sort_key f xs = ys"
1367 using assms
1368 proof (induct xs arbitrary: ys)
1369   case Nil then show ?case by simp
1370 next
1371   case (Cons x xs)
1372   from Cons.prems(2) have
1373     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
1374     by (simp add: filter_remove1)
1375   with Cons.prems have "sort_key f xs = remove1 x ys"
1376     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
1377   moreover from Cons.prems have "x \<in> set ys"
1378     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
1379   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
1380 qed
1382 lemma properties_for_sort:
1383   assumes multiset: "multiset_of ys = multiset_of xs"
1384   and "sorted ys"
1385   shows "sort xs = ys"
1386 proof (rule properties_for_sort_key)
1387   from multiset show "multiset_of ys = multiset_of xs" .
1388   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
1389   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
1390     by (rule multiset_of_eq_length_filter)
1391   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
1392     by simp
1393   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
1394     by (simp add: replicate_length_filter)
1395 qed
1397 lemma sort_key_by_quicksort:
1398   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
1399     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
1400     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
1401 proof (rule properties_for_sort_key)
1402   show "multiset_of ?rhs = multiset_of ?lhs"
1403     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
1404 next
1405   show "sorted (map f ?rhs)"
1406     by (auto simp add: sorted_append intro: sorted_map_same)
1407 next
1408   fix l
1409   assume "l \<in> set ?rhs"
1410   let ?pivot = "f (xs ! (length xs div 2))"
1411   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
1412   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
1413     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
1414   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
1415   have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
1416   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
1417     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
1418   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
1419   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
1420   proof (cases "f l" ?pivot rule: linorder_cases)
1421     case less
1422     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
1423     with less show ?thesis
1424       by (simp add: filter_sort [symmetric] ** ***)
1425   next
1426     case equal then show ?thesis
1427       by (simp add: * less_le)
1428   next
1429     case greater
1430     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
1431     with greater show ?thesis
1432       by (simp add: filter_sort [symmetric] ** ***)
1433   qed
1434 qed
1436 lemma sort_by_quicksort:
1437   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
1438     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
1439     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
1440   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
1442 text {* A stable parametrized quicksort *}
1444 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
1445   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
1447 lemma part_code [code]:
1448   "part f pivot [] = ([], [], [])"
1449   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
1450      if x' < pivot then (x # lts, eqs, gts)
1451      else if x' > pivot then (lts, eqs, x # gts)
1452      else (lts, x # eqs, gts))"
1453   by (auto simp add: part_def Let_def split_def)
1455 lemma sort_key_by_quicksort_code [code]:
1456   "sort_key f xs = (case xs of [] \<Rightarrow> []
1457     | [x] \<Rightarrow> xs
1458     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
1459     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1460        in sort_key f lts @ eqs @ sort_key f gts))"
1461 proof (cases xs)
1462   case Nil then show ?thesis by simp
1463 next
1464   case (Cons _ ys) note hyps = Cons show ?thesis
1465   proof (cases ys)
1466     case Nil with hyps show ?thesis by simp
1467   next
1468     case (Cons _ zs) note hyps = hyps Cons show ?thesis
1469     proof (cases zs)
1470       case Nil with hyps show ?thesis by auto
1471     next
1472       case Cons
1473       from sort_key_by_quicksort [of f xs]
1474       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1475         in sort_key f lts @ eqs @ sort_key f gts)"
1476       by (simp only: split_def Let_def part_def fst_conv snd_conv)
1477       with hyps Cons show ?thesis by (simp only: list.cases)
1478     qed
1479   qed
1480 qed
1482 end
1484 hide_const (open) part
1486 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
1487   by (induct xs) (auto intro: order_trans)
1489 lemma multiset_of_update:
1490   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
1491 proof (induct ls arbitrary: i)
1492   case Nil then show ?case by simp
1493 next
1494   case (Cons x xs)
1495   show ?case
1496   proof (cases i)
1497     case 0 then show ?thesis by simp
1498   next
1499     case (Suc i')
1500     with Cons show ?thesis
1501       apply simp
1502       apply (subst add.assoc)
1503       apply (subst add.commute [of "{#v#}" "{#x#}"])
1504       apply (subst add.assoc [symmetric])
1505       apply simp
1506       apply (rule mset_le_multiset_union_diff_commute)
1507       apply (simp add: mset_le_single nth_mem_multiset_of)
1508       done
1509   qed
1510 qed
1512 lemma multiset_of_swap:
1513   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
1514     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
1515   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
1518 subsection {* The multiset order *}
1520 subsubsection {* Well-foundedness *}
1522 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1523   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1524       (\<forall>b. b :# K --> (b, a) \<in> r)}"
1526 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1527   "mult r = (mult1 r)\<^sup>+"
1529 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1530 by (simp add: mult1_def)
1532 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
1533     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1534     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
1535   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
1536 proof (unfold mult1_def)
1537   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
1538   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1539   let ?case1 = "?case1 {(N, M). ?R N M}"
1541   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
1542   then have "\<exists>a' M0' K.
1543       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
1544   then show "?case1 \<or> ?case2"
1545   proof (elim exE conjE)
1546     fix a' M0' K
1547     assume N: "N = M0' + K" and r: "?r K a'"
1548     assume "M0 + {#a#} = M0' + {#a'#}"
1549     then have "M0 = M0' \<and> a = a' \<or>
1550         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
1551       by (simp only: add_eq_conv_ex)
1552     then show ?thesis
1553     proof (elim disjE conjE exE)
1554       assume "M0 = M0'" "a = a'"
1555       with N r have "?r K a \<and> N = M0 + K" by simp
1556       then have ?case2 .. then show ?thesis ..
1557     next
1558       fix K'
1559       assume "M0' = K' + {#a#}"
1560       with N have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
1562       assume "M0 = K' + {#a'#}"
1563       with r have "?R (K' + K) M0" by blast
1564       with n have ?case1 by simp then show ?thesis ..
1565     qed
1566   qed
1567 qed
1569 lemma all_accessible: "wf r ==> \<forall>M. M \<in> Wellfounded.acc (mult1 r)"
1570 proof
1571   let ?R = "mult1 r"
1572   let ?W = "Wellfounded.acc ?R"
1573   {
1574     fix M M0 a
1575     assume M0: "M0 \<in> ?W"
1576       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1577       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
1578     have "M0 + {#a#} \<in> ?W"
1579     proof (rule accI [of "M0 + {#a#}"])
1580       fix N
1581       assume "(N, M0 + {#a#}) \<in> ?R"
1582       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
1583           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
1584         by (rule less_add)
1585       then show "N \<in> ?W"
1586       proof (elim exE disjE conjE)
1587         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
1588         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
1589         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
1590         then show "N \<in> ?W" by (simp only: N)
1591       next
1592         fix K
1593         assume N: "N = M0 + K"
1594         assume "\<forall>b. b :# K --> (b, a) \<in> r"
1595         then have "M0 + K \<in> ?W"
1596         proof (induct K)
1597           case empty
1598           from M0 show "M0 + {#} \<in> ?W" by simp
1599         next
1600           case (add K x)
1601           from add.prems have "(x, a) \<in> r" by simp
1602           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
1603           moreover from add have "M0 + K \<in> ?W" by simp
1604           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
1605           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
1606         qed
1607         then show "N \<in> ?W" by (simp only: N)
1608       qed
1609     qed
1610   } note tedious_reasoning = this
1612   assume wf: "wf r"
1613   fix M
1614   show "M \<in> ?W"
1615   proof (induct M)
1616     show "{#} \<in> ?W"
1617     proof (rule accI)
1618       fix b assume "(b, {#}) \<in> ?R"
1619       with not_less_empty show "b \<in> ?W" by contradiction
1620     qed
1622     fix M a assume "M \<in> ?W"
1623     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1624     proof induct
1625       fix a
1626       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1627       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1628       proof
1629         fix M assume "M \<in> ?W"
1630         then show "M + {#a#} \<in> ?W"
1631           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
1632       qed
1633     qed
1634     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
1635   qed
1636 qed
1638 theorem wf_mult1: "wf r ==> wf (mult1 r)"
1639 by (rule acc_wfI) (rule all_accessible)
1641 theorem wf_mult: "wf r ==> wf (mult r)"
1642 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
1645 subsubsection {* Closure-free presentation *}
1647 text {* One direction. *}
1649 lemma mult_implies_one_step:
1650   "trans r ==> (M, N) \<in> mult r ==>
1651     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
1652     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
1653 apply (unfold mult_def mult1_def set_of_def)
1654 apply (erule converse_trancl_induct, clarify)
1655  apply (rule_tac x = M0 in exI, simp, clarify)
1656 apply (case_tac "a :# K")
1657  apply (rule_tac x = I in exI)
1658  apply (simp (no_asm))
1659  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1660  apply (simp (no_asm_simp) add: add.assoc [symmetric])
1661  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong)
1662  apply (simp add: diff_union_single_conv)
1663  apply (simp (no_asm_use) add: trans_def)
1664  apply blast
1665 apply (subgoal_tac "a :# I")
1666  apply (rule_tac x = "I - {#a#}" in exI)
1667  apply (rule_tac x = "J + {#a#}" in exI)
1668  apply (rule_tac x = "K + Ka" in exI)
1669  apply (rule conjI)
1670   apply (simp add: multiset_eq_iff split: nat_diff_split)
1671  apply (rule conjI)
1672   apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
1673   apply (simp add: multiset_eq_iff split: nat_diff_split)
1674  apply (simp (no_asm_use) add: trans_def)
1675  apply blast
1676 apply (subgoal_tac "a :# (M0 + {#a#})")
1677  apply simp
1678 apply (simp (no_asm))
1679 done
1681 lemma one_step_implies_mult_aux:
1682   "trans r ==>
1683     \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
1684       --> (I + K, I + J) \<in> mult r"
1685 apply (induct_tac n, auto)
1686 apply (frule size_eq_Suc_imp_eq_union, clarify)
1687 apply (rename_tac "J'", simp)
1688 apply (erule notE, auto)
1689 apply (case_tac "J' = {#}")
1690  apply (simp add: mult_def)
1691  apply (rule r_into_trancl)
1692  apply (simp add: mult1_def set_of_def, blast)
1693 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
1694 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
1695 apply (erule_tac P = "\<forall>k \<in> set_of K. P k" for P in rev_mp)
1696 apply (erule ssubst)
1697 apply (simp add: Ball_def, auto)
1698 apply (subgoal_tac
1699   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
1700     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
1701  prefer 2
1702  apply force
1703 apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
1704 apply (erule trancl_trans)
1705 apply (rule r_into_trancl)
1706 apply (simp add: mult1_def set_of_def)
1707 apply (rule_tac x = a in exI)
1708 apply (rule_tac x = "I + J'" in exI)
1709 apply (simp add: ac_simps)
1710 done
1712 lemma one_step_implies_mult:
1713   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
1714     ==> (I + K, I + J) \<in> mult r"
1715 using one_step_implies_mult_aux by blast
1718 subsubsection {* Partial-order properties *}
1720 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
1721   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
1723 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
1724   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
1726 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
1727 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
1729 interpretation multiset_order: order le_multiset less_multiset
1730 proof -
1731   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
1732   proof
1733     fix M :: "'a multiset"
1734     assume "M \<subset># M"
1735     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
1736     have "trans {(x'::'a, x). x' < x}"
1737       by (rule transI) simp
1738     moreover note MM
1739     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
1740       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
1741       by (rule mult_implies_one_step)
1742     then obtain I J K where "M = I + J" and "M = I + K"
1743       and "J \<noteq> {#}" and "(\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})" by blast
1744     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
1745     have "finite (set_of K)" by simp
1746     moreover note aux2
1747     ultimately have "set_of K = {}"
1748       by (induct rule: finite_induct) (auto intro: order_less_trans)
1749     with aux1 show False by simp
1750   qed
1751   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
1752     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
1753   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
1754     by default (auto simp add: le_multiset_def irrefl dest: trans)
1755 qed
1757 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
1758   by simp
1761 subsubsection {* Monotonicity of multiset union *}
1763 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
1764 apply (unfold mult1_def)
1765 apply auto
1766 apply (rule_tac x = a in exI)
1767 apply (rule_tac x = "C + M0" in exI)
1769 done
1771 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
1772 apply (unfold less_multiset_def mult_def)
1773 apply (erule trancl_induct)
1774  apply (blast intro: mult1_union)
1775 apply (blast intro: mult1_union trancl_trans)
1776 done
1778 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
1779 apply (subst add.commute [of B C])
1780 apply (subst add.commute [of D C])
1781 apply (erule union_less_mono2)
1782 done
1784 lemma union_less_mono:
1785   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
1786   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
1788 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
1789 proof
1790 qed (auto simp add: le_multiset_def intro: union_less_mono2)
1793 subsection {* Termination proofs with multiset orders *}
1795 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
1796   and multi_member_this: "x \<in># {# x #} + XS"
1797   and multi_member_last: "x \<in># {# x #}"
1798   by auto
1800 definition "ms_strict = mult pair_less"
1801 definition "ms_weak = ms_strict \<union> Id"
1803 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
1804 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
1805 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
1807 lemma smsI:
1808   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
1809   unfolding ms_strict_def
1810 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
1812 lemma wmsI:
1813   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
1814   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
1815 unfolding ms_weak_def ms_strict_def
1816 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
1818 inductive pw_leq
1819 where
1820   pw_leq_empty: "pw_leq {#} {#}"
1821 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
1823 lemma pw_leq_lstep:
1824   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
1825 by (drule pw_leq_step) (rule pw_leq_empty, simp)
1827 lemma pw_leq_split:
1828   assumes "pw_leq X Y"
1829   shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1830   using assms
1831 proof (induct)
1832   case pw_leq_empty thus ?case by auto
1833 next
1834   case (pw_leq_step x y X Y)
1835   then obtain A B Z where
1836     [simp]: "X = A + Z" "Y = B + Z"
1837       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
1838     by auto
1839   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
1840     unfolding pair_leq_def by auto
1841   thus ?case
1842   proof
1843     assume [simp]: "x = y"
1844     have
1845       "{#x#} + X = A + ({#y#}+Z)
1846       \<and> {#y#} + Y = B + ({#y#}+Z)
1847       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1848       by (auto simp: ac_simps)
1849     thus ?case by (intro exI)
1850   next
1851     assume A: "(x, y) \<in> pair_less"
1852     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
1853     have "{#x#} + X = ?A' + Z"
1854       "{#y#} + Y = ?B' + Z"
1855       by (auto simp add: ac_simps)
1856     moreover have
1857       "(set_of ?A', set_of ?B') \<in> max_strict"
1858       using 1 A unfolding max_strict_def
1859       by (auto elim!: max_ext.cases)
1860     ultimately show ?thesis by blast
1861   qed
1862 qed
1864 lemma
1865   assumes pwleq: "pw_leq Z Z'"
1866   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
1867   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
1868   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
1869 proof -
1870   from pw_leq_split[OF pwleq]
1871   obtain A' B' Z''
1872     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
1873     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
1874     by blast
1875   {
1876     assume max: "(set_of A, set_of B) \<in> max_strict"
1877     from mx_or_empty
1878     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
1879     proof
1880       assume max': "(set_of A', set_of B') \<in> max_strict"
1881       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
1882         by (auto simp: max_strict_def intro: max_ext_additive)
1883       thus ?thesis by (rule smsI)
1884     next
1885       assume [simp]: "A' = {#} \<and> B' = {#}"
1886       show ?thesis by (rule smsI) (auto intro: max)
1887     qed
1888     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:ac_simps)
1889     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
1890   }
1891   from mx_or_empty
1892   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
1893   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
1894 qed
1896 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
1897 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
1898 and nonempty_single: "{# x #} \<noteq> {#}"
1899 by auto
1901 setup {*
1902 let
1903   fun msetT T = Type (@{type_name multiset}, [T]);
1905   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
1906     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
1907     | mk_mset T (x :: xs) =
1908           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
1909                 mk_mset T [x] \$ mk_mset T xs
1911   fun mset_member_tac m i =
1912       (if m <= 0 then
1913            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
1914        else
1915            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
1917   val mset_nonempty_tac =
1918       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
1920   fun regroup_munion_conv ctxt =
1921     Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus}
1922       (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
1924   fun unfold_pwleq_tac i =
1925     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
1926       ORELSE (rtac @{thm pw_leq_lstep} i)
1927       ORELSE (rtac @{thm pw_leq_empty} i)
1929   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
1930                       @{thm Un_insert_left}, @{thm Un_empty_left}]
1931 in
1932   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
1933   {
1934     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
1935     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
1936     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
1937     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
1938     reduction_pair= @{thm ms_reduction_pair}
1939   })
1940 end
1941 *}
1944 subsection {* Legacy theorem bindings *}
1946 lemmas multi_count_eq = multiset_eq_iff [symmetric]
1948 lemma union_commute: "M + N = N + (M::'a multiset)"
1949   by (fact add.commute)
1951 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
1952   by (fact add.assoc)
1954 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
1955   by (fact add.left_commute)
1957 lemmas union_ac = union_assoc union_commute union_lcomm
1959 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
1960   by (fact add_right_cancel)
1962 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
1963   by (fact add_left_cancel)
1965 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
1966   by (fact add_left_imp_eq)
1968 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
1969   by (fact order_less_trans)
1971 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
1972   by (fact inf.commute)
1974 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
1975   by (fact inf.assoc [symmetric])
1977 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
1978   by (fact inf.left_commute)
1980 lemmas multiset_inter_ac =
1981   multiset_inter_commute
1982   multiset_inter_assoc
1983   multiset_inter_left_commute
1985 lemma mult_less_not_refl:
1986   "\<not> M \<subset># (M::'a::order multiset)"
1987   by (fact multiset_order.less_irrefl)
1989 lemma mult_less_trans:
1990   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
1991   by (fact multiset_order.less_trans)
1993 lemma mult_less_not_sym:
1994   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
1995   by (fact multiset_order.less_not_sym)
1997 lemma mult_less_asym:
1998   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
1999   by (fact multiset_order.less_asym)
2001 ML {*
2002 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
2003                       (Const _ \$ t') =
2004     let
2005       val (maybe_opt, ps) =
2006         Nitpick_Model.dest_plain_fun t' ||> op ~~
2007         ||> map (apsnd (snd o HOLogic.dest_number))
2008       fun elems_for t =
2009         case AList.lookup (op =) ps t of
2010           SOME n => replicate n t
2011         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
2012     in
2013       case maps elems_for (all_values elem_T) @
2014            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
2015             else []) of
2016         [] => Const (@{const_name zero_class.zero}, T)
2017       | ts => foldl1 (fn (t1, t2) =>
2018                          Const (@{const_name plus_class.plus}, T --> T --> T)
2019                          \$ t1 \$ t2)
2020                      (map (curry (op \$) (Const (@{const_name single},
2021                                                 elem_T --> T))) ts)
2022     end
2023   | multiset_postproc _ _ _ _ t = t
2024 *}
2026 declaration {*
2027 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
2028     multiset_postproc
2029 *}
2031 hide_const (open) fold
2034 subsection {* Naive implementation using lists *}
2036 code_datatype multiset_of
2038 lemma [code]:
2039   "{#} = multiset_of []"
2040   by simp
2042 lemma [code]:
2043   "{#x#} = multiset_of [x]"
2044   by simp
2046 lemma union_code [code]:
2047   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
2048   by simp
2050 lemma [code]:
2051   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
2052   by (simp add: multiset_of_map)
2054 lemma [code]:
2055   "Multiset.filter f (multiset_of xs) = multiset_of (filter f xs)"
2056   by (simp add: multiset_of_filter)
2058 lemma [code]:
2059   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
2060   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
2062 lemma [code]:
2063   "multiset_of xs #\<inter> multiset_of ys =
2064     multiset_of (snd (fold (\<lambda>x (ys, zs).
2065       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
2066 proof -
2067   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
2068     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
2069       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
2070     by (induct xs arbitrary: ys)
2072   then show ?thesis by simp
2073 qed
2075 lemma [code]:
2076   "multiset_of xs #\<union> multiset_of ys =
2077     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
2078 proof -
2079   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
2080       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
2081     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
2082   then show ?thesis by simp
2083 qed
2085 lemma [code_unfold]:
2086   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
2087   by (simp add: in_multiset_of)
2089 lemma [code]:
2090   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
2091 proof -
2092   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
2093     by (induct xs) simp_all
2094   then show ?thesis by simp
2095 qed
2097 lemma [code]:
2098   "set_of (multiset_of xs) = set xs"
2099   by simp
2101 lemma [code]:
2102   "sorted_list_of_multiset (multiset_of xs) = sort xs"
2103   by (induct xs) simp_all
2105 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
2106   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
2107   apply (cases "finite A")
2108   apply simp_all
2109   apply (induct A rule: finite_induct)
2110   apply (simp_all add: union_commute)
2111   done
2113 lemma [code]:
2114   "mcard (multiset_of xs) = length xs"
2115   by (simp add: mcard_multiset_of)
2117 fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
2118   "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
2119 | "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
2120      None \<Rightarrow> None
2121    | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
2123 lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> multiset_of xs \<le> multiset_of ys) \<and>
2124   (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> multiset_of xs < multiset_of ys) \<and>
2125   (ms_lesseq_impl xs ys = Some False \<longrightarrow> multiset_of xs = multiset_of ys)"
2126 proof (induct xs arbitrary: ys)
2127   case (Nil ys)
2128   show ?case by (auto simp: mset_less_empty_nonempty)
2129 next
2130   case (Cons x xs ys)
2131   show ?case
2132   proof (cases "List.extract (op = x) ys")
2133     case None
2134     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
2135     {
2136       assume "multiset_of (x # xs) \<le> multiset_of ys"
2137       from set_of_mono[OF this] x have False by simp
2138     } note nle = this
2139     moreover
2140     {
2141       assume "multiset_of (x # xs) < multiset_of ys"
2142       hence "multiset_of (x # xs) \<le> multiset_of ys" by auto
2143       from nle[OF this] have False .
2144     }
2145     ultimately show ?thesis using None by auto
2146   next
2147     case (Some res)
2148     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
2149     note Some = Some[unfolded res]
2150     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
2151     hence id: "multiset_of ys = multiset_of (ys1 @ ys2) + {#x#}"
2152       by (auto simp: ac_simps)
2153     show ?thesis unfolding ms_lesseq_impl.simps
2154       unfolding Some option.simps split
2155       unfolding id
2156       using Cons[of "ys1 @ ys2"]
2157       unfolding mset_le_def mset_less_def by auto
2158   qed
2159 qed
2161 lemma [code]: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
2162   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2164 lemma [code]: "multiset_of xs < multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
2165   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2167 instantiation multiset :: (equal) equal
2168 begin
2170 definition
2171   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
2172 lemma [code]: "HOL.equal (multiset_of xs) (multiset_of ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
2173   unfolding equal_multiset_def
2174   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2176 instance
2177   by default (simp add: equal_multiset_def)
2178 end
2180 lemma [code]:
2181   "msetsum (multiset_of xs) = listsum xs"
2182   by (induct xs) (simp_all add: add.commute)
2184 lemma [code]:
2185   "msetprod (multiset_of xs) = fold times xs 1"
2186 proof -
2187   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
2188     by (induct xs) (simp_all add: mult.assoc)
2189   then show ?thesis by simp
2190 qed
2192 lemma [code]:
2193   "size = mcard"
2194   by (fact size_eq_mcard)
2196 text {*
2197   Exercise for the casual reader: add implementations for @{const le_multiset}
2198   and @{const less_multiset} (multiset order).
2199 *}
2201 text {* Quickcheck generators *}
2203 definition (in term_syntax)
2204   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
2205     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
2206   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
2208 notation fcomp (infixl "\<circ>>" 60)
2209 notation scomp (infixl "\<circ>\<rightarrow>" 60)
2211 instantiation multiset :: (random) random
2212 begin
2214 definition
2215   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
2217 instance ..
2219 end
2221 no_notation fcomp (infixl "\<circ>>" 60)
2222 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
2224 instantiation multiset :: (full_exhaustive) full_exhaustive
2225 begin
2227 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
2228 where
2229   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
2231 instance ..
2233 end
2235 hide_const (open) msetify
2238 subsection {* BNF setup *}
2240 definition rel_mset where
2241   "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. multiset_of xs = X \<and> multiset_of ys = Y \<and> list_all2 R xs ys)"
2243 lemma multiset_of_zip_take_Cons_drop_twice:
2244   assumes "length xs = length ys" "j \<le> length xs"
2245   shows "multiset_of (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
2246     multiset_of (zip xs ys) + {#(x, y)#}"
2247 using assms
2248 proof (induct xs ys arbitrary: x y j rule: list_induct2)
2249   case Nil
2250   thus ?case
2251     by simp
2252 next
2253   case (Cons x xs y ys)
2254   thus ?case
2255   proof (cases "j = 0")
2256     case True
2257     thus ?thesis
2258       by simp
2259   next
2260     case False
2261     then obtain k where k: "j = Suc k"
2262       by (case_tac j) simp
2263     hence "k \<le> length xs"
2264       using Cons.prems by auto
2265     hence "multiset_of (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
2266       multiset_of (zip xs ys) + {#(x, y)#}"
2267       by (rule Cons.hyps(2))
2268     thus ?thesis
2269       unfolding k by (auto simp: add.commute union_lcomm)
2270   qed
2271 qed
2273 lemma ex_multiset_of_zip_left:
2274   assumes "length xs = length ys" "multiset_of xs' = multiset_of xs"
2275   shows "\<exists>ys'. length ys' = length xs' \<and> multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
2276 using assms
2277 proof (induct xs ys arbitrary: xs' rule: list_induct2)
2278   case Nil
2279   thus ?case
2280     by auto
2281 next
2282   case (Cons x xs y ys xs')
2283   obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
2284     by (metis Cons.prems in_set_conv_nth list.set_intros(1) multiset_of_eq_setD)
2286   def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
2287   have "multiset_of xs' = {#x#} + multiset_of xsa"
2288     unfolding xsa_def using j_len nth_j
2289     by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
2290       multiset_of.simps(2) union_code union_commute)
2291   hence ms_x: "multiset_of xsa = multiset_of xs"
2292     by (metis Cons.prems add.commute add_right_imp_eq multiset_of.simps(2))
2293   then obtain ysa where
2294     len_a: "length ysa = length xsa" and ms_a: "multiset_of (zip xsa ysa) = multiset_of (zip xs ys)"
2295     using Cons.hyps(2) by blast
2297   def ys' \<equiv> "take j ysa @ y # drop j ysa"
2298   have xs': "xs' = take j xsa @ x # drop j xsa"
2299     using ms_x j_len nth_j Cons.prems xsa_def
2300     by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
2301       length_drop mcard_multiset_of)
2302   have j_len': "j \<le> length xsa"
2303     using j_len xs' xsa_def
2304     by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
2305   have "length ys' = length xs'"
2306     unfolding ys'_def using Cons.prems len_a ms_x
2307     by (metis add_Suc_right append_take_drop_id length_Cons length_append multiset_of_eq_length)
2308   moreover have "multiset_of (zip xs' ys') = multiset_of (zip (x # xs) (y # ys))"
2309     unfolding xs' ys'_def
2310     by (rule trans[OF multiset_of_zip_take_Cons_drop_twice])
2311       (auto simp: len_a ms_a j_len' add.commute)
2312   ultimately show ?case
2313     by blast
2314 qed
2316 lemma list_all2_reorder_left_invariance:
2317   assumes rel: "list_all2 R xs ys" and ms_x: "multiset_of xs' = multiset_of xs"
2318   shows "\<exists>ys'. list_all2 R xs' ys' \<and> multiset_of ys' = multiset_of ys"
2319 proof -
2320   have len: "length xs = length ys"
2321     using rel list_all2_conv_all_nth by auto
2322   obtain ys' where
2323     len': "length xs' = length ys'" and ms_xy: "multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
2324     using len ms_x by (metis ex_multiset_of_zip_left)
2325   have "list_all2 R xs' ys'"
2326     using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: multiset_of_eq_setD)
2327   moreover have "multiset_of ys' = multiset_of ys"
2328     using len len' ms_xy map_snd_zip multiset_of_map by metis
2329   ultimately show ?thesis
2330     by blast
2331 qed
2333 lemma ex_multiset_of: "\<exists>xs. multiset_of xs = X"
2334   by (induct X) (simp, metis multiset_of.simps(2))
2336 bnf "'a multiset"
2337   map: image_mset
2338   sets: set_of
2339   bd: natLeq
2340   wits: "{#}"
2341   rel: rel_mset
2342 proof -
2343   show "image_mset id = id"
2344     by (rule image_mset.id)
2345 next
2346   show "\<And>f g. image_mset (g \<circ> f) = image_mset g \<circ> image_mset f"
2347     unfolding comp_def by (rule ext) (simp add: image_mset.compositionality comp_def)
2348 next
2349   fix X :: "'a multiset"
2350   show "\<And>f g. (\<And>z. z \<in> set_of X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X"
2351     by (induct X, (simp (no_asm))+,
2352       metis One_nat_def Un_iff count_single mem_set_of_iff set_of_union zero_less_Suc)
2353 next
2354   show "\<And>f. set_of \<circ> image_mset f = op ` f \<circ> set_of"
2355     by auto
2356 next
2357   show "card_order natLeq"
2358     by (rule natLeq_card_order)
2359 next
2360   show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
2361     by (rule natLeq_cinfinite)
2362 next
2363   show "\<And>X. ordLeq3 (card_of (set_of X)) natLeq"
2364     by transfer
2365       (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
2366 next
2367   show "\<And>R S. rel_mset R OO rel_mset S \<le> rel_mset (R OO S)"
2368     unfolding rel_mset_def[abs_def] OO_def
2369     apply clarify
2370     apply (rename_tac X Z Y xs ys' ys zs)
2371     apply (drule_tac xs = ys' and ys = zs and xs' = ys in list_all2_reorder_left_invariance)
2372     by (auto intro: list_all2_trans)
2373 next
2374   show "\<And>R. rel_mset R =
2375     (BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
2376     BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset snd)"
2377     unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
2378     apply (rule ext)+
2379     apply auto
2380      apply (rule_tac x = "multiset_of (zip xs ys)" in exI)
2381      apply auto
2382         apply (metis list_all2_lengthD map_fst_zip multiset_of_map)
2383        apply (auto simp: list_all2_iff)
2384       apply (metis list_all2_lengthD map_snd_zip multiset_of_map)
2385      apply (auto simp: list_all2_iff)
2386     apply (rename_tac XY)
2387     apply (cut_tac X = XY in ex_multiset_of)
2388     apply (erule exE)
2389     apply (rename_tac xys)
2390     apply (rule_tac x = "map fst xys" in exI)
2391     apply (auto simp: multiset_of_map)
2392     apply (rule_tac x = "map snd xys" in exI)
2393     by (auto simp: multiset_of_map list_all2I subset_eq zip_map_fst_snd)
2394 next
2395   show "\<And>z. z \<in> set_of {#} \<Longrightarrow> False"
2396     by auto
2397 qed
2399 inductive rel_mset' where
2400   Zero[intro]: "rel_mset' R {#} {#}"
2401 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
2403 lemma rel_mset_Zero: "rel_mset R {#} {#}"
2404 unfolding rel_mset_def Grp_def by auto
2406 declare multiset.count[simp]
2407 declare Abs_multiset_inverse[simp]
2408 declare multiset.count_inverse[simp]
2409 declare union_preserves_multiset[simp]
2411 lemma rel_mset_Plus:
2412 assumes ab: "R a b" and MN: "rel_mset R M N"
2413 shows "rel_mset R (M + {#a#}) (N + {#b#})"
2414 proof-
2415   {fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
2416    hence "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
2417                image_mset snd y + {#b#} = image_mset snd ya \<and>
2418                set_of ya \<subseteq> {(x, y). R x y}"
2419    apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
2420   }
2421   thus ?thesis
2422   using assms
2423   unfolding multiset.rel_compp_Grp Grp_def by blast
2424 qed
2426 lemma rel_mset'_imp_rel_mset:
2427 "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
2428 apply(induct rule: rel_mset'.induct)
2429 using rel_mset_Zero rel_mset_Plus by auto
2431 lemma mcard_image_mset[simp]: "mcard (image_mset f M) = mcard M"
2432   unfolding size_eq_mcard[symmetric] by (rule size_image_mset)
2434 lemma rel_mset_mcard:
2435   assumes "rel_mset R M N"
2436   shows "mcard M = mcard N"
2437 using assms unfolding multiset.rel_compp_Grp Grp_def by auto
2440 assumes empty: "P {#} {#}"
2441 and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
2442 and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
2443 shows "P M N"
2444 apply(induct N rule: multiset_induct)
2445   apply(induct M rule: multiset_induct, rule empty, erule addL)
2446   apply(induct M rule: multiset_induct, erule addR, erule addR)
2447 done
2449 lemma multiset_induct2_mcard[consumes 1, case_names empty add]:
2450 assumes c: "mcard M = mcard N"
2451 and empty: "P {#} {#}"
2452 and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
2453 shows "P M N"
2454 using c proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
2455   case (less M)  show ?case
2456   proof(cases "M = {#}")
2457     case True hence "N = {#}" using less.prems by auto
2458     thus ?thesis using True empty by auto
2459   next
2460     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2461     have "N \<noteq> {#}" using False less.prems by auto
2462     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
2463     have "mcard M1 = mcard N1" using less.prems unfolding M N by auto
2464     thus ?thesis using M N less.hyps add by auto
2465   qed
2466 qed
2468 lemma msed_map_invL:
2469 assumes "image_mset f (M + {#a#}) = N"
2470 shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1"
2471 proof-
2472   have "f a \<in># N"
2473   using assms multiset.set_map[of f "M + {#a#}"] by auto
2474   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
2475   have "image_mset f M = N1" using assms unfolding N by simp
2476   thus ?thesis using N by blast
2477 qed
2479 lemma msed_map_invR:
2480 assumes "image_mset f M = N + {#b#}"
2481 shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N"
2482 proof-
2483   obtain a where a: "a \<in># M" and fa: "f a = b"
2484   using multiset.set_map[of f M] unfolding assms
2485   by (metis image_iff mem_set_of_iff union_single_eq_member)
2486   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
2487   have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
2488   thus ?thesis using M fa by blast
2489 qed
2491 lemma msed_rel_invL:
2492 assumes "rel_mset R (M + {#a#}) N"
2493 shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1"
2494 proof-
2495   obtain K where KM: "image_mset fst K = M + {#a#}"
2496   and KN: "image_mset snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2497   using assms
2498   unfolding multiset.rel_compp_Grp Grp_def by auto
2499   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
2500   and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
2501   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1"
2502   using msed_map_invL[OF KN[unfolded K]] by auto
2503   have Rab: "R a (snd ab)" using sK a unfolding K by auto
2504   have "rel_mset R M N1" using sK K1M K1N1
2505   unfolding K multiset.rel_compp_Grp Grp_def by auto
2506   thus ?thesis using N Rab by auto
2507 qed
2509 lemma msed_rel_invR:
2510 assumes "rel_mset R M (N + {#b#})"
2511 shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N"
2512 proof-
2513   obtain K where KN: "image_mset snd K = N + {#b#}"
2514   and KM: "image_mset fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2515   using assms
2516   unfolding multiset.rel_compp_Grp Grp_def by auto
2517   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
2518   and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
2519   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1"
2520   using msed_map_invL[OF KM[unfolded K]] by auto
2521   have Rab: "R (fst ab) b" using sK b unfolding K by auto
2522   have "rel_mset R M1 N" using sK K1N K1M1
2523   unfolding K multiset.rel_compp_Grp Grp_def by auto
2524   thus ?thesis using M Rab by auto
2525 qed
2527 lemma rel_mset_imp_rel_mset':
2528 assumes "rel_mset R M N"
2529 shows "rel_mset' R M N"
2530 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
2531   case (less M)
2532   have c: "mcard M = mcard N" using rel_mset_mcard[OF less.prems] .
2533   show ?case
2534   proof(cases "M = {#}")
2535     case True hence "N = {#}" using c by simp
2536     thus ?thesis using True rel_mset'.Zero by auto
2537   next
2538     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2539     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1"
2540     using msed_rel_invL[OF less.prems[unfolded M]] by auto
2541     have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
2542     thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
2543   qed
2544 qed
2546 lemma rel_mset_rel_mset':
2547 "rel_mset R M N = rel_mset' R M N"
2548 using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
2550 (* The main end product for rel_mset: inductive characterization *)
2551 theorems rel_mset_induct[case_names empty add, induct pred: rel_mset] =
2552          rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
2555 subsection {* Size setup *}
2557 lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
2558   unfolding o_apply by (rule ext) (induct_tac, auto)
2560 setup {*
2561 BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
2562   @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
2563     size_union}
2564   @{thms multiset_size_o_map}
2565 *}
2567 hide_const (open) wcount
2569 end