src/HOL/Library/Multiset.thy
 author wenzelm Fri Apr 10 11:52:55 2015 +0200 (2015-04-10) changeset 59997 90fb391a15c1 parent 59986 f38b94549dc8 child 59999 3fa68bacfa2b permissions -rw-r--r--
tuned proofs;
1 (*  Title:      HOL/Library/Multiset.thy
2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
3     Author:     Andrei Popescu, TU Muenchen
4     Author:     Jasmin Blanchette, Inria, LORIA, MPII
5     Author:     Dmitriy Traytel, TU Muenchen
6     Author:     Mathias Fleury, MPII
7 *)
9 section {* (Finite) multisets *}
11 theory Multiset
12 imports Main
13 begin
15 subsection {* The type of multisets *}
17 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
19 typedef 'a multiset = "multiset :: ('a => nat) set"
20   morphisms count Abs_multiset
21   unfolding multiset_def
22 proof
23   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
24 qed
26 setup_lifting type_definition_multiset
28 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
29   "a :# M == 0 < count M a"
31 notation (xsymbols)
32   Melem (infix "\<in>#" 50)
34 lemma multiset_eq_iff:
35   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
36   by (simp only: count_inject [symmetric] fun_eq_iff)
38 lemma multiset_eqI:
39   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
40   using multiset_eq_iff by auto
42 text {*
43  \medskip Preservation of the representing set @{term multiset}.
44 *}
46 lemma const0_in_multiset:
47   "(\<lambda>a. 0) \<in> multiset"
48   by (simp add: multiset_def)
50 lemma only1_in_multiset:
51   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
52   by (simp add: multiset_def)
54 lemma union_preserves_multiset:
55   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
56   by (simp add: multiset_def)
58 lemma diff_preserves_multiset:
59   assumes "M \<in> multiset"
60   shows "(\<lambda>a. M a - N a) \<in> multiset"
61 proof -
62   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
63     by auto
64   with assms show ?thesis
65     by (auto simp add: multiset_def intro: finite_subset)
66 qed
68 lemma filter_preserves_multiset:
69   assumes "M \<in> multiset"
70   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
71 proof -
72   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
73     by auto
74   with assms show ?thesis
75     by (auto simp add: multiset_def intro: finite_subset)
76 qed
78 lemmas in_multiset = const0_in_multiset only1_in_multiset
79   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
82 subsection {* Representing multisets *}
84 text {* Multiset enumeration *}
86 instantiation multiset :: (type) cancel_comm_monoid_add
87 begin
89 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
90 by (rule const0_in_multiset)
92 abbreviation Mempty :: "'a multiset" ("{#}") where
93   "Mempty \<equiv> 0"
95 lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
96 by (rule union_preserves_multiset)
98 lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
99 by (rule diff_preserves_multiset)
101 instance
102   by default (transfer, simp add: fun_eq_iff)+
104 end
106 lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
107 by (rule only1_in_multiset)
109 syntax
110   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
111 translations
112   "{#x, xs#}" == "{#x#} + {#xs#}"
113   "{#x#}" == "CONST single x"
115 lemma count_empty [simp]: "count {#} a = 0"
116   by (simp add: zero_multiset.rep_eq)
118 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
119   by (simp add: single.rep_eq)
122 subsection {* Basic operations *}
124 subsubsection {* Union *}
126 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
127   by (simp add: plus_multiset.rep_eq)
130 subsubsection {* Difference *}
132 instantiation multiset :: (type) comm_monoid_diff
133 begin
135 instance
136 by default (transfer, simp add: fun_eq_iff)+
138 end
140 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
141   by (simp add: minus_multiset.rep_eq)
143 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
144   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
146 lemma diff_cancel[simp]: "A - A = {#}"
147   by (fact Groups.diff_cancel)
149 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
150   by (fact add_diff_cancel_right')
152 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
153   by (fact add_diff_cancel_left')
155 lemma diff_right_commute:
156   "(M::'a multiset) - N - Q = M - Q - N"
157   by (fact diff_right_commute)
160   "(M::'a multiset) - (N + Q) = M - N - Q"
161   by (rule sym) (fact diff_diff_add)
163 lemma insert_DiffM:
164   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
165   by (clarsimp simp: multiset_eq_iff)
167 lemma insert_DiffM2 [simp]:
168   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
169   by (clarsimp simp: multiset_eq_iff)
171 lemma diff_union_swap:
172   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
173   by (auto simp add: multiset_eq_iff)
175 lemma diff_union_single_conv:
176   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
177   by (simp add: multiset_eq_iff)
180 subsubsection {* Equality of multisets *}
182 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
183   by (simp add: multiset_eq_iff)
185 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
186   by (auto simp add: multiset_eq_iff)
188 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
189   by (auto simp add: multiset_eq_iff)
191 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
192   by (auto simp add: multiset_eq_iff)
194 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
195   by (auto simp add: multiset_eq_iff)
197 lemma diff_single_trivial:
198   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
199   by (auto simp add: multiset_eq_iff)
201 lemma diff_single_eq_union:
202   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
203   by auto
205 lemma union_single_eq_diff:
206   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
207   by (auto dest: sym)
209 lemma union_single_eq_member:
210   "M + {#x#} = N \<Longrightarrow> x \<in># N"
211   by auto
213 lemma union_is_single:
214   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
215 proof
216   assume ?rhs then show ?lhs by auto
217 next
218   assume ?lhs then show ?rhs
219     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
220 qed
222 lemma single_is_union:
223   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
224   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
227   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
228 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
229 proof
230   assume ?rhs then show ?lhs
232     (drule sym, simp add: add.assoc [symmetric])
233 next
234   assume ?lhs
235   show ?rhs
236   proof (cases "a = b")
237     case True with `?lhs` show ?thesis by simp
238   next
239     case False
240     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
241     with False have "a \<in># N" by auto
242     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
243     moreover note False
244     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
245   qed
246 qed
248 lemma insert_noteq_member:
249   assumes BC: "B + {#b#} = C + {#c#}"
250    and bnotc: "b \<noteq> c"
251   shows "c \<in># B"
252 proof -
253   have "c \<in># C + {#c#}" by simp
254   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
255   then have "c \<in># B + {#b#}" using BC by simp
256   then show "c \<in># B" using nc by simp
257 qed
260   "(M + {#a#} = N + {#b#}) =
261     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
264 lemma multi_member_split:
265   "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
266   by (rule_tac x = "M - {#x#}" in exI, simp)
269   assumes "c \<in># B" and "b \<noteq> c"
270   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
271 proof -
272   from `c \<in># B` obtain A where B: "B = A + {#c#}"
273     by (blast dest: multi_member_split)
274   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
275   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
276     by (simp add: ac_simps)
277   then show ?thesis using B by simp
278 qed
281 subsubsection {* Pointwise ordering induced by count *}
283 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
284 begin
286 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)" .
288 lemmas mset_le_def = less_eq_multiset_def
290 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
291   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
293 instance
294   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
296 end
298 abbreviation less_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
299   "A <# B \<equiv> A < B"
300 abbreviation (xsymbols) subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subset>#" 50) where
301   "A \<subset># B \<equiv> A < B"
303 abbreviation less_eq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
304   "A <=# B \<equiv> A \<le> B"
305 abbreviation (xsymbols) leq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<le>#" 50) where
306   "A \<le># B \<equiv> A \<le> B"
307 abbreviation (xsymbols) subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subseteq>#" 50) where
308   "A \<subseteq># B \<equiv> A \<le> B"
310 lemma mset_less_eqI:
311   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
312   by (simp add: mset_le_def)
314 lemma mset_le_exists_conv:
315   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
316 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
317 apply (auto intro: multiset_eq_iff [THEN iffD2])
318 done
320 instance multiset :: (type) ordered_cancel_comm_monoid_diff
321   by default (simp, fact mset_le_exists_conv)
323 lemma mset_le_mono_add_right_cancel [simp]:
324   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
325   by (fact add_le_cancel_right)
327 lemma mset_le_mono_add_left_cancel [simp]:
328   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
329   by (fact add_le_cancel_left)
332   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
333   by (fact add_mono)
335 lemma mset_le_add_left [simp]:
336   "(A::'a multiset) \<le> A + B"
337   unfolding mset_le_def by auto
339 lemma mset_le_add_right [simp]:
340   "B \<le> (A::'a multiset) + B"
341   unfolding mset_le_def by auto
343 lemma mset_le_single:
344   "a :# B \<Longrightarrow> {#a#} \<le> B"
345   by (simp add: mset_le_def)
347 lemma multiset_diff_union_assoc:
348   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
349   by (simp add: multiset_eq_iff mset_le_def)
351 lemma mset_le_multiset_union_diff_commute:
352   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
353 by (simp add: multiset_eq_iff mset_le_def)
355 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
356 by(simp add: mset_le_def)
358 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
359 apply (clarsimp simp: mset_le_def mset_less_def)
360 apply (erule_tac x=x in allE)
361 apply auto
362 done
364 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
365 apply (clarsimp simp: mset_le_def mset_less_def)
366 apply (erule_tac x = x in allE)
367 apply auto
368 done
370 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
371 apply (rule conjI)
372  apply (simp add: mset_lessD)
373 apply (clarsimp simp: mset_le_def mset_less_def)
374 apply safe
375  apply (erule_tac x = a in allE)
376  apply (auto split: split_if_asm)
377 done
379 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
380 apply (rule conjI)
381  apply (simp add: mset_leD)
382 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
383 done
385 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
386   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
388 lemma empty_le[simp]: "{#} \<le> A"
389   unfolding mset_le_exists_conv by auto
391 lemma le_empty[simp]: "(M \<le> {#}) = (M = {#})"
392   unfolding mset_le_exists_conv by auto
394 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
395   by (auto simp: mset_le_def mset_less_def)
397 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
398   by simp
400 lemma mset_less_add_bothsides: "N + {#x#} < M + {#x#} \<Longrightarrow> N < M"
401   by (fact add_less_imp_less_right)
403 lemma mset_less_empty_nonempty:
404   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
405   by (auto simp: mset_le_def mset_less_def)
407 lemma mset_less_diff_self:
408   "c \<in># B \<Longrightarrow> B - {#c#} < B"
409   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
412 subsubsection {* Intersection *}
414 instantiation multiset :: (type) semilattice_inf
415 begin
417 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
418   multiset_inter_def: "inf_multiset A B = A - (A - B)"
420 instance
421 proof -
422   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
423   show "OFCLASS('a multiset, semilattice_inf_class)"
424     by default (auto simp add: multiset_inter_def mset_le_def aux)
425 qed
427 end
429 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
430   "multiset_inter \<equiv> inf"
432 lemma multiset_inter_count [simp]:
433   "count (A #\<inter> B) x = min (count A x) (count B x)"
434   by (simp add: multiset_inter_def)
436 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
437   by (rule multiset_eqI) auto
439 lemma multiset_union_diff_commute:
440   assumes "B #\<inter> C = {#}"
441   shows "A + B - C = A - C + B"
442 proof (rule multiset_eqI)
443   fix x
444   from assms have "min (count B x) (count C x) = 0"
445     by (auto simp add: multiset_eq_iff)
446   then have "count B x = 0 \<or> count C x = 0"
447     by auto
448   then show "count (A + B - C) x = count (A - C + B) x"
449     by auto
450 qed
452 lemma empty_inter [simp]:
453   "{#} #\<inter> M = {#}"
454   by (simp add: multiset_eq_iff)
456 lemma inter_empty [simp]:
457   "M #\<inter> {#} = {#}"
458   by (simp add: multiset_eq_iff)
461   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
462   by (simp add: multiset_eq_iff)
465   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
466   by (simp add: multiset_eq_iff)
469   "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
470   by (simp add: multiset_eq_iff)
473   "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
474   by (simp add: multiset_eq_iff)
477 subsubsection {* Bounded union *}
479 instantiation multiset :: (type) semilattice_sup
480 begin
482 definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
483   "sup_multiset A B = A + (B - A)"
485 instance
486 proof -
487   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" by arith
488   show "OFCLASS('a multiset, semilattice_sup_class)"
489     by default (auto simp add: sup_multiset_def mset_le_def aux)
490 qed
492 end
494 abbreviation sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<union>" 70) where
495   "sup_multiset \<equiv> sup"
497 lemma sup_multiset_count [simp]:
498   "count (A #\<union> B) x = max (count A x) (count B x)"
499   by (simp add: sup_multiset_def)
501 lemma empty_sup [simp]:
502   "{#} #\<union> M = M"
503   by (simp add: multiset_eq_iff)
505 lemma sup_empty [simp]:
506   "M #\<union> {#} = M"
507   by (simp add: multiset_eq_iff)
510   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
511   by (simp add: multiset_eq_iff)
514   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
515   by (simp add: multiset_eq_iff)
518   "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
519   by (simp add: multiset_eq_iff)
522   "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
523   by (simp add: multiset_eq_iff)
526 subsubsection {* Filter (with comprehension syntax) *}
528 text {* Multiset comprehension *}
530 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"
531 by (rule filter_preserves_multiset)
533 hide_const (open) filter
535 lemma count_filter [simp]:
536   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
537   by (simp add: filter.rep_eq)
539 lemma filter_empty [simp]:
540   "Multiset.filter P {#} = {#}"
541   by (rule multiset_eqI) simp
543 lemma filter_single [simp]:
544   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
545   by (rule multiset_eqI) simp
547 lemma filter_union [simp]:
548   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
549   by (rule multiset_eqI) simp
551 lemma filter_diff [simp]:
552   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
553   by (rule multiset_eqI) simp
555 lemma filter_inter [simp]:
556   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
557   by (rule multiset_eqI) simp
559 lemma multiset_filter_subset[simp]: "Multiset.filter f M \<le> M"
560   unfolding less_eq_multiset.rep_eq by auto
562 lemma multiset_filter_mono: assumes "A \<le> B"
563   shows "Multiset.filter f A \<le> Multiset.filter f B"
564 proof -
565   from assms[unfolded mset_le_exists_conv]
566   obtain C where B: "B = A + C" by auto
567   show ?thesis unfolding B by auto
568 qed
570 syntax
571   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
572 syntax (xsymbol)
573   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
574 translations
575   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
578 subsubsection {* Set of elements *}
580 definition set_of :: "'a multiset => 'a set" where
581   "set_of M = {x. x :# M}"
583 lemma set_of_empty [simp]: "set_of {#} = {}"
584 by (simp add: set_of_def)
586 lemma set_of_single [simp]: "set_of {#b#} = {b}"
587 by (simp add: set_of_def)
589 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
590 by (auto simp add: set_of_def)
592 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
593 by (auto simp add: set_of_def multiset_eq_iff)
595 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
596 by (auto simp add: set_of_def)
598 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
599 by (auto simp add: set_of_def)
601 lemma finite_set_of [iff]: "finite (set_of M)"
602   using count [of M] by (simp add: multiset_def set_of_def)
604 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
605   unfolding set_of_def[symmetric] by simp
607 lemma set_of_mono: "A \<le> B \<Longrightarrow> set_of A \<subseteq> set_of B"
608   by (metis mset_leD subsetI mem_set_of_iff)
610 lemma ball_set_of_iff: "(\<forall>x \<in> set_of M. P x) \<longleftrightarrow> (\<forall>x. x \<in># M \<longrightarrow> P x)"
611   by auto
614 subsubsection {* Size *}
616 definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
618 lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
619   by (auto simp: wcount_def add_mult_distrib)
621 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
622   "size_multiset f M = setsum (wcount f M) (set_of M)"
624 lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
626 instantiation multiset :: (type) size begin
627 definition size_multiset where
628   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
629 instance ..
630 end
632 lemmas size_multiset_overloaded_eq =
633   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
635 lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
636 by (simp add: size_multiset_def)
638 lemma size_empty [simp]: "size {#} = 0"
641 lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
642 by (simp add: size_multiset_eq)
644 lemma size_single [simp]: "size {#b#} = 1"
647 lemma setsum_wcount_Int:
648   "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_of N) = setsum (wcount f N) A"
649 apply (induct rule: finite_induct)
650  apply simp
651 apply (simp add: Int_insert_left set_of_def wcount_def)
652 done
654 lemma size_multiset_union [simp]:
655   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
656 apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
657 apply (subst Int_commute)
658 apply (simp add: setsum_wcount_Int)
659 done
661 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
664 lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
665 by (auto simp add: size_multiset_eq multiset_eq_iff)
667 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
670 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
671 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
673 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
674 apply (unfold size_multiset_overloaded_eq)
675 apply (drule setsum_SucD)
676 apply auto
677 done
679 lemma size_eq_Suc_imp_eq_union:
680   assumes "size M = Suc n"
681   shows "\<exists>a N. M = N + {#a#}"
682 proof -
683   from assms obtain a where "a \<in># M"
684     by (erule size_eq_Suc_imp_elem [THEN exE])
685   then have "M = M - {#a#} + {#a#}" by simp
686   then show ?thesis by blast
687 qed
689 lemma size_mset_mono: assumes "A \<le> B"
690   shows "size A \<le> size(B::_ multiset)"
691 proof -
692   from assms[unfolded mset_le_exists_conv]
693   obtain C where B: "B = A + C" by auto
694   show ?thesis unfolding B by (induct C, auto)
695 qed
697 lemma size_filter_mset_lesseq[simp]: "size (Multiset.filter f M) \<le> size M"
698 by (rule size_mset_mono[OF multiset_filter_subset])
700 lemma size_Diff_submset:
701   "M \<le> M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)"
702 by (metis add_diff_cancel_left' size_union mset_le_exists_conv)
704 subsection {* Induction and case splits *}
706 theorem multiset_induct [case_names empty add, induct type: multiset]:
707   assumes empty: "P {#}"
708   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
709   shows "P M"
710 proof (induct n \<equiv> "size M" arbitrary: M)
711   case 0 thus "P M" by (simp add: empty)
712 next
713   case (Suc k)
714   obtain N x where "M = N + {#x#}"
715     using `Suc k = size M` [symmetric]
716     using size_eq_Suc_imp_eq_union by fast
717   with Suc add show "P M" by simp
718 qed
720 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
721 by (induct M) auto
723 lemma multiset_cases [cases type]:
724   obtains (empty) "M = {#}"
725     | (add) N x where "M = N + {#x#}"
726   using assms by (induct M) simp_all
728 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
729 by (cases "B = {#}") (auto dest: multi_member_split)
731 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
732 apply (subst multiset_eq_iff)
733 apply auto
734 done
736 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
737 proof (induct A arbitrary: B)
738   case (empty M)
739   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
740   then obtain M' x where "M = M' + {#x#}"
741     by (blast dest: multi_nonempty_split)
742   then show ?case by simp
743 next
744   case (add S x T)
745   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
746   have SxsubT: "S + {#x#} < T" by fact
747   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
748   then obtain T' where T: "T = T' + {#x#}"
749     by (blast dest: multi_member_split)
750   then have "S < T'" using SxsubT
751     by (blast intro: mset_less_add_bothsides)
752   then have "size S < size T'" using IH by simp
753   then show ?case using T by simp
754 qed
757 lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}"
758 by (cases M) auto
760 subsubsection {* Strong induction and subset induction for multisets *}
762 text {* Well-foundedness of strict subset relation *}
764 lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M < N}"
765 apply (rule wf_measure [THEN wf_subset, where f1=size])
766 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
767 done
769 lemma full_multiset_induct [case_names less]:
770 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
771 shows "P B"
772 apply (rule wf_less_mset_rel [THEN wf_induct])
773 apply (rule ih, auto)
774 done
776 lemma multi_subset_induct [consumes 2, case_names empty add]:
777 assumes "F \<le> A"
778   and empty: "P {#}"
779   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
780 shows "P F"
781 proof -
782   from `F \<le> A`
783   show ?thesis
784   proof (induct F)
785     show "P {#}" by fact
786   next
787     fix x F
788     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
789     show "P (F + {#x#})"
790     proof (rule insert)
791       from i show "x \<in># A" by (auto dest: mset_le_insertD)
792       from i have "F \<le> A" by (auto dest: mset_le_insertD)
793       with P show "P F" .
794     qed
795   qed
796 qed
799 subsection {* The fold combinator *}
801 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
802 where
803   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
805 lemma fold_mset_empty [simp]:
806   "fold f s {#} = s"
807   by (simp add: fold_def)
809 context comp_fun_commute
810 begin
812 lemma fold_mset_insert:
813   "fold f s (M + {#x#}) = f x (fold f s M)"
814 proof -
815   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
816     by (fact comp_fun_commute_funpow)
817   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
818     by (fact comp_fun_commute_funpow)
819   show ?thesis
820   proof (cases "x \<in> set_of M")
821     case False
822     then have *: "count (M + {#x#}) x = 1" by simp
823     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
824       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
825       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
826     with False * show ?thesis
827       by (simp add: fold_def del: count_union)
828   next
829     case True
830     def N \<equiv> "set_of M - {x}"
831     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
832     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
833       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
834       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
835     with * show ?thesis by (simp add: fold_def del: count_union) simp
836   qed
837 qed
839 corollary fold_mset_single [simp]:
840   "fold f s {#x#} = f x s"
841 proof -
842   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
843   then show ?thesis by simp
844 qed
846 lemma fold_mset_fun_left_comm:
847   "f x (fold f s M) = fold f (f x s) M"
848   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
850 lemma fold_mset_union [simp]:
851   "fold f s (M + N) = fold f (fold f s M) N"
852 proof (induct M)
853   case empty then show ?case by simp
854 next
855   case (add M x)
856   have "M + {#x#} + N = (M + N) + {#x#}"
857     by (simp add: ac_simps)
858   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
859 qed
861 lemma fold_mset_fusion:
862   assumes "comp_fun_commute g"
863   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")
864 proof -
865   interpret comp_fun_commute g by (fact assms)
866   show "PROP ?P" by (induct A) auto
867 qed
869 end
871 text {*
872   A note on code generation: When defining some function containing a
873   subterm @{term "fold F"}, code generation is not automatic. When
874   interpreting locale @{text left_commutative} with @{text F}, the
875   would be code thms for @{const fold} become thms like
876   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
877   contains defined symbols, i.e.\ is not a code thm. Hence a separate
878   constant with its own code thms needs to be introduced for @{text
879   F}. See the image operator below.
880 *}
883 subsection {* Image *}
885 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
886   "image_mset f = fold (plus o single o f) {#}"
888 lemma comp_fun_commute_mset_image:
889   "comp_fun_commute (plus o single o f)"
890 proof
891 qed (simp add: ac_simps fun_eq_iff)
893 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
894   by (simp add: image_mset_def)
896 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
897 proof -
898   interpret comp_fun_commute "plus o single o f"
899     by (fact comp_fun_commute_mset_image)
900   show ?thesis by (simp add: image_mset_def)
901 qed
903 lemma image_mset_union [simp]:
904   "image_mset f (M + N) = image_mset f M + image_mset f N"
905 proof -
906   interpret comp_fun_commute "plus o single o f"
907     by (fact comp_fun_commute_mset_image)
908   show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
909 qed
911 corollary image_mset_insert:
912   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
913   by simp
915 lemma set_of_image_mset [simp]:
916   "set_of (image_mset f M) = image f (set_of M)"
917   by (induct M) simp_all
919 lemma size_image_mset [simp]:
920   "size (image_mset f M) = size M"
921   by (induct M) simp_all
923 lemma image_mset_is_empty_iff [simp]:
924   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
925   by (cases M) auto
927 syntax
928   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
929       ("({#_/. _ :# _#})")
930 translations
931   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
933 syntax (xsymbols)
934   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
935       ("({#_/. _ \<in># _#})")
936 translations
937   "{#e. x \<in># M#}" == "CONST image_mset (\<lambda>x. e) M"
939 syntax
940   "_comprehension3_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
941       ("({#_/ | _ :# _./ _#})")
942 translations
943   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
945 syntax
946   "_comprehension4_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
947       ("({#_/ | _ \<in># _./ _#})")
948 translations
949   "{#e | x\<in>#M. P#}" => "{#e. x \<in># {# x\<in>#M. P#}#}"
951 text {*
952   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
953   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
954   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
955   @{term "{#x+x|x:#M. x<c#}"}.
956 *}
958 lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_of M"
959   by (metis mem_set_of_iff set_of_image_mset)
961 functor image_mset: image_mset
962 proof -
963   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
964   proof
965     fix A
966     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
967       by (induct A) simp_all
968   qed
969   show "image_mset id = id"
970   proof
971     fix A
972     show "image_mset id A = id A"
973       by (induct A) simp_all
974   qed
975 qed
977 declare
978   image_mset.id [simp]
979   image_mset.identity [simp]
981 lemma image_mset_id[simp]: "image_mset id x = x"
982   unfolding id_def by auto
984 lemma image_mset_cong: "(\<And>x. x \<in># M \<Longrightarrow> f x = g x) \<Longrightarrow> {#f x. x \<in># M#} = {#g x. x \<in># M#}"
985   by (induct M) auto
987 lemma image_mset_cong_pair:
988   "(\<forall>x y. (x, y) \<in># M \<longrightarrow> f x y = g x y) \<Longrightarrow> {#f x y. (x, y) \<in># M#} = {#g x y. (x, y) \<in># M#}"
989   by (metis image_mset_cong split_cong)
992 subsection {* Further conversions *}
994 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
995   "multiset_of [] = {#}" |
996   "multiset_of (a # x) = multiset_of x + {# a #}"
998 lemma in_multiset_in_set:
999   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
1000   by (induct xs) simp_all
1002 lemma count_multiset_of:
1003   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
1004   by (induct xs) simp_all
1006 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
1007   by (induct x) auto
1009 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
1010 by (induct x) auto
1012 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
1013 by (induct x) auto
1015 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
1016 by (induct xs) auto
1018 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
1019   by (induct xs) simp_all
1021 lemma multiset_of_append [simp]:
1022   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
1023   by (induct xs arbitrary: ys) (auto simp: ac_simps)
1025 lemma multiset_of_filter:
1026   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
1027   by (induct xs) simp_all
1029 lemma multiset_of_rev [simp]:
1030   "multiset_of (rev xs) = multiset_of xs"
1031   by (induct xs) simp_all
1033 lemma surj_multiset_of: "surj multiset_of"
1034 apply (unfold surj_def)
1035 apply (rule allI)
1036 apply (rule_tac M = y in multiset_induct)
1037  apply auto
1038 apply (rule_tac x = "x # xa" in exI)
1039 apply auto
1040 done
1042 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
1043 by (induct x) auto
1045 lemma distinct_count_atmost_1:
1046   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
1047 apply (induct x, simp, rule iffI, simp_all)
1048 apply (rename_tac a b)
1049 apply (rule conjI)
1050 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
1051 apply (erule_tac x = a in allE, simp, clarify)
1052 apply (erule_tac x = aa in allE, simp)
1053 done
1055 lemma multiset_of_eq_setD:
1056   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
1057 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
1059 lemma set_eq_iff_multiset_of_eq_distinct:
1060   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
1061     (set x = set y) = (multiset_of x = multiset_of y)"
1062 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
1064 lemma set_eq_iff_multiset_of_remdups_eq:
1065    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
1066 apply (rule iffI)
1067 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
1068 apply (drule distinct_remdups [THEN distinct_remdups
1069       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
1070 apply simp
1071 done
1073 lemma multiset_of_compl_union [simp]:
1074   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
1075   by (induct xs) (auto simp: ac_simps)
1077 lemma count_multiset_of_length_filter:
1078   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
1079   by (induct xs) auto
1081 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
1082 apply (induct ls arbitrary: i)
1083  apply simp
1084 apply (case_tac i)
1085  apply auto
1086 done
1088 lemma multiset_of_remove1[simp]:
1089   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
1090 by (induct xs) (auto simp add: multiset_eq_iff)
1092 lemma multiset_of_eq_length:
1093   assumes "multiset_of xs = multiset_of ys"
1094   shows "length xs = length ys"
1095   using assms by (metis size_multiset_of)
1097 lemma multiset_of_eq_length_filter:
1098   assumes "multiset_of xs = multiset_of ys"
1099   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
1100   using assms by (metis count_multiset_of)
1102 lemma fold_multiset_equiv:
1103   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"
1104     and equiv: "multiset_of xs = multiset_of ys"
1105   shows "List.fold f xs = List.fold f ys"
1106 using f equiv [symmetric]
1107 proof (induct xs arbitrary: ys)
1108   case Nil then show ?case by simp
1109 next
1110   case (Cons x xs)
1111   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
1112   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
1113     by (rule Cons.prems(1)) (simp_all add: *)
1114   moreover from * have "x \<in> set ys" by simp
1115   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
1116   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
1117   ultimately show ?case by simp
1118 qed
1120 lemma multiset_of_insort [simp]:
1121   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
1122   by (induct xs) (simp_all add: ac_simps)
1124 lemma multiset_of_map:
1125   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
1126   by (induct xs) simp_all
1128 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
1129 where
1130   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
1132 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
1133 where
1134   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
1135 proof -
1136   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
1137   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
1138   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
1139 qed
1141 lemma count_multiset_of_set [simp]:
1142   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
1143   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
1144   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
1145 proof -
1146   { fix A
1147     assume "x \<notin> A"
1148     have "count (multiset_of_set A) x = 0"
1149     proof (cases "finite A")
1150       case False then show ?thesis by simp
1151     next
1152       case True from True `x \<notin> A` show ?thesis by (induct A) auto
1153     qed
1154   } note * = this
1155   then show "PROP ?P" "PROP ?Q" "PROP ?R"
1156   by (auto elim!: Set.set_insert)
1157 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
1159 lemma elem_multiset_of_set[simp, intro]: "finite A \<Longrightarrow> x \<in># multiset_of_set A \<longleftrightarrow> x \<in> A"
1160   by (induct A rule: finite_induct) simp_all
1162 context linorder
1163 begin
1165 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
1166 where
1167   "sorted_list_of_multiset M = fold insort [] M"
1169 lemma sorted_list_of_multiset_empty [simp]:
1170   "sorted_list_of_multiset {#} = []"
1171   by (simp add: sorted_list_of_multiset_def)
1173 lemma sorted_list_of_multiset_singleton [simp]:
1174   "sorted_list_of_multiset {#x#} = [x]"
1175 proof -
1176   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1177   show ?thesis by (simp add: sorted_list_of_multiset_def)
1178 qed
1180 lemma sorted_list_of_multiset_insert [simp]:
1181   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
1182 proof -
1183   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1184   show ?thesis by (simp add: sorted_list_of_multiset_def)
1185 qed
1187 end
1189 lemma multiset_of_sorted_list_of_multiset [simp]:
1190   "multiset_of (sorted_list_of_multiset M) = M"
1191   by (induct M) simp_all
1193 lemma sorted_list_of_multiset_multiset_of [simp]:
1194   "sorted_list_of_multiset (multiset_of xs) = sort xs"
1195   by (induct xs) simp_all
1197 lemma finite_set_of_multiset_of_set:
1198   assumes "finite A"
1199   shows "set_of (multiset_of_set A) = A"
1200   using assms by (induct A) simp_all
1202 lemma infinite_set_of_multiset_of_set:
1203   assumes "\<not> finite A"
1204   shows "set_of (multiset_of_set A) = {}"
1205   using assms by simp
1207 lemma set_sorted_list_of_multiset [simp]:
1208   "set (sorted_list_of_multiset M) = set_of M"
1209   by (induct M) (simp_all add: set_insort)
1211 lemma sorted_list_of_multiset_of_set [simp]:
1212   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
1213   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
1216 subsection {* Big operators *}
1218 no_notation times (infixl "*" 70)
1219 no_notation Groups.one ("1")
1221 locale comm_monoid_mset = comm_monoid
1222 begin
1224 definition F :: "'a multiset \<Rightarrow> 'a"
1225 where
1226   eq_fold: "F M = Multiset.fold f 1 M"
1228 lemma empty [simp]:
1229   "F {#} = 1"
1230   by (simp add: eq_fold)
1232 lemma singleton [simp]:
1233   "F {#x#} = x"
1234 proof -
1235   interpret comp_fun_commute
1236     by default (simp add: fun_eq_iff left_commute)
1237   show ?thesis by (simp add: eq_fold)
1238 qed
1240 lemma union [simp]:
1241   "F (M + N) = F M * F N"
1242 proof -
1243   interpret comp_fun_commute f
1244     by default (simp add: fun_eq_iff left_commute)
1245   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
1246 qed
1248 end
1250 lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + \<Colon> 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
1251   by default (simp add: add_ac comp_def)
1253 declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp]
1255 lemma in_mset_fold_plus_iff[iff]: "x \<in># Multiset.fold (op +) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)"
1256   by (induct NN) auto
1258 notation times (infixl "*" 70)
1259 notation Groups.one ("1")
1262 begin
1264 definition msetsum :: "'a multiset \<Rightarrow> 'a"
1265 where
1266   "msetsum = comm_monoid_mset.F plus 0"
1268 sublocale msetsum!: comm_monoid_mset plus 0
1269 where
1270   "comm_monoid_mset.F plus 0 = msetsum"
1271 proof -
1272   show "comm_monoid_mset plus 0" ..
1273   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
1274 qed
1276 lemma setsum_unfold_msetsum:
1277   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
1278   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1280 end
1282 lemma msetsum_diff:
1283   fixes M N :: "('a \<Colon> ordered_cancel_comm_monoid_diff) multiset"
1284   shows "N \<le> M \<Longrightarrow> msetsum (M - N) = msetsum M - msetsum N"
1287 lemma size_eq_msetsum: "size M = msetsum (image_mset (\<lambda>_. 1) M)"
1288 proof (induct M)
1289   case empty then show ?case by simp
1290 next
1291   case (add M x) then show ?case
1292     by (cases "x \<in> set_of M")
1293       (simp_all del: mem_set_of_iff add: size_multiset_overloaded_eq setsum.distrib setsum.delta' insert_absorb, simp)
1294 qed
1297 abbreviation Union_mset :: "'a multiset multiset \<Rightarrow> 'a multiset" where
1298   "Union_mset MM \<equiv> msetsum MM"
1300 notation (xsymbols) Union_mset ("\<Union>#_"  900)
1302 lemma set_of_Union_mset[simp]: "set_of (\<Union># MM) = (\<Union>M \<in> set_of MM. set_of M)"
1303   by (induct MM) auto
1305 lemma in_Union_mset_iff[iff]: "x \<in># \<Union># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
1306   by (induct MM) auto
1308 syntax
1309   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1310       ("(3SUM _:#_. _)" [0, 51, 10] 10)
1312 syntax (xsymbols)
1313   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1314       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1316 syntax (HTML output)
1317   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1318       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1320 translations
1321   "SUM i :# A. b" == "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"
1323 context comm_monoid_mult
1324 begin
1326 definition msetprod :: "'a multiset \<Rightarrow> 'a"
1327 where
1328   "msetprod = comm_monoid_mset.F times 1"
1330 sublocale msetprod!: comm_monoid_mset times 1
1331 where
1332   "comm_monoid_mset.F times 1 = msetprod"
1333 proof -
1334   show "comm_monoid_mset times 1" ..
1335   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
1336 qed
1338 lemma msetprod_empty:
1339   "msetprod {#} = 1"
1340   by (fact msetprod.empty)
1342 lemma msetprod_singleton:
1343   "msetprod {#x#} = x"
1344   by (fact msetprod.singleton)
1346 lemma msetprod_Un:
1347   "msetprod (A + B) = msetprod A * msetprod B"
1348   by (fact msetprod.union)
1350 lemma setprod_unfold_msetprod:
1351   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
1352   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1354 lemma msetprod_multiplicity:
1355   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
1356   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
1358 end
1360 syntax
1361   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1362       ("(3PROD _:#_. _)" [0, 51, 10] 10)
1364 syntax (xsymbols)
1365   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1366       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1368 syntax (HTML output)
1369   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1370       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1372 translations
1373   "PROD i :# A. b" == "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"
1375 lemma (in comm_semiring_1) dvd_msetprod:
1376   assumes "x \<in># A"
1377   shows "x dvd msetprod A"
1378 proof -
1379   from assms have "A = (A - {#x#}) + {#x#}" by simp
1380   then obtain B where "A = B + {#x#}" ..
1381   then show ?thesis by simp
1382 qed
1385 subsection {* Replicate operation *}
1387 definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
1388   "replicate_mset n x = ((op + {#x#}) ^^ n) {#}"
1390 lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}"
1391   unfolding replicate_mset_def by simp
1393 lemma replicate_mset_Suc[simp]: "replicate_mset (Suc n) x = replicate_mset n x + {#x#}"
1394   unfolding replicate_mset_def by (induct n) (auto intro: add.commute)
1396 lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y"
1397   unfolding replicate_mset_def by (induct n) simp_all
1399 lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)"
1400   unfolding replicate_mset_def by (induct n) simp_all
1402 lemma set_of_replicate_mset_subset[simp]: "set_of (replicate_mset n x) = (if n = 0 then {} else {x})"
1403   by (auto split: if_splits)
1405 lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n"
1406   by (induct n, simp_all)
1408 lemma count_le_replicate_mset_le: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<le> M"
1409   by (auto simp add: assms mset_less_eqI) (metis count_replicate_mset less_eq_multiset.rep_eq)
1411 lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
1412   by (induct D) simp_all
1415 subsection {* Alternative representations *}
1417 subsubsection {* Lists *}
1419 context linorder
1420 begin
1422 lemma multiset_of_insort [simp]:
1423   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
1424   by (induct xs) (simp_all add: ac_simps)
1426 lemma multiset_of_sort [simp]:
1427   "multiset_of (sort_key k xs) = multiset_of xs"
1428   by (induct xs) (simp_all add: ac_simps)
1430 text {*
1431   This lemma shows which properties suffice to show that a function
1432   @{text "f"} with @{text "f xs = ys"} behaves like sort.
1433 *}
1435 lemma properties_for_sort_key:
1436   assumes "multiset_of ys = multiset_of xs"
1437   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
1438   and "sorted (map f ys)"
1439   shows "sort_key f xs = ys"
1440 using assms
1441 proof (induct xs arbitrary: ys)
1442   case Nil then show ?case by simp
1443 next
1444   case (Cons x xs)
1445   from Cons.prems(2) have
1446     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
1447     by (simp add: filter_remove1)
1448   with Cons.prems have "sort_key f xs = remove1 x ys"
1449     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
1450   moreover from Cons.prems have "x \<in> set ys"
1451     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
1452   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
1453 qed
1455 lemma properties_for_sort:
1456   assumes multiset: "multiset_of ys = multiset_of xs"
1457   and "sorted ys"
1458   shows "sort xs = ys"
1459 proof (rule properties_for_sort_key)
1460   from multiset show "multiset_of ys = multiset_of xs" .
1461   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
1462   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
1463     by (rule multiset_of_eq_length_filter)
1464   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
1465     by simp
1466   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
1467     by (simp add: replicate_length_filter)
1468 qed
1470 lemma sort_key_by_quicksort:
1471   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
1472     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
1473     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
1474 proof (rule properties_for_sort_key)
1475   show "multiset_of ?rhs = multiset_of ?lhs"
1476     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
1477 next
1478   show "sorted (map f ?rhs)"
1479     by (auto simp add: sorted_append intro: sorted_map_same)
1480 next
1481   fix l
1482   assume "l \<in> set ?rhs"
1483   let ?pivot = "f (xs ! (length xs div 2))"
1484   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
1485   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
1486     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
1487   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
1488   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
1489   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
1490     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
1491   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
1492   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
1493   proof (cases "f l" ?pivot rule: linorder_cases)
1494     case less
1495     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
1496     with less show ?thesis
1497       by (simp add: filter_sort [symmetric] ** ***)
1498   next
1499     case equal then show ?thesis
1500       by (simp add: * less_le)
1501   next
1502     case greater
1503     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
1504     with greater show ?thesis
1505       by (simp add: filter_sort [symmetric] ** ***)
1506   qed
1507 qed
1509 lemma sort_by_quicksort:
1510   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
1511     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
1512     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
1513   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
1515 text {* A stable parametrized quicksort *}
1517 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
1518   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
1520 lemma part_code [code]:
1521   "part f pivot [] = ([], [], [])"
1522   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
1523      if x' < pivot then (x # lts, eqs, gts)
1524      else if x' > pivot then (lts, eqs, x # gts)
1525      else (lts, x # eqs, gts))"
1526   by (auto simp add: part_def Let_def split_def)
1528 lemma sort_key_by_quicksort_code [code]:
1529   "sort_key f xs = (case xs of [] \<Rightarrow> []
1530     | [x] \<Rightarrow> xs
1531     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
1532     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1533        in sort_key f lts @ eqs @ sort_key f gts))"
1534 proof (cases xs)
1535   case Nil then show ?thesis by simp
1536 next
1537   case (Cons _ ys) note hyps = Cons show ?thesis
1538   proof (cases ys)
1539     case Nil with hyps show ?thesis by simp
1540   next
1541     case (Cons _ zs) note hyps = hyps Cons show ?thesis
1542     proof (cases zs)
1543       case Nil with hyps show ?thesis by auto
1544     next
1545       case Cons
1546       from sort_key_by_quicksort [of f xs]
1547       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1548         in sort_key f lts @ eqs @ sort_key f gts)"
1549       by (simp only: split_def Let_def part_def fst_conv snd_conv)
1550       with hyps Cons show ?thesis by (simp only: list.cases)
1551     qed
1552   qed
1553 qed
1555 end
1557 hide_const (open) part
1559 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
1560   by (induct xs) (auto intro: order_trans)
1562 lemma multiset_of_update:
1563   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
1564 proof (induct ls arbitrary: i)
1565   case Nil then show ?case by simp
1566 next
1567   case (Cons x xs)
1568   show ?case
1569   proof (cases i)
1570     case 0 then show ?thesis by simp
1571   next
1572     case (Suc i')
1573     with Cons show ?thesis
1574       apply simp
1575       apply (subst add.assoc)
1576       apply (subst add.commute [of "{#v#}" "{#x#}"])
1577       apply (subst add.assoc [symmetric])
1578       apply simp
1579       apply (rule mset_le_multiset_union_diff_commute)
1580       apply (simp add: mset_le_single nth_mem_multiset_of)
1581       done
1582   qed
1583 qed
1585 lemma multiset_of_swap:
1586   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
1587     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
1588   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
1591 subsection {* The multiset order *}
1593 subsubsection {* Well-foundedness *}
1595 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1596   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1597       (\<forall>b. b :# K --> (b, a) \<in> r)}"
1599 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1600   "mult r = (mult1 r)\<^sup>+"
1602 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1603 by (simp add: mult1_def)
1605 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
1606     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1607     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
1608   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
1609 proof (unfold mult1_def)
1610   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
1611   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1612   let ?case1 = "?case1 {(N, M). ?R N M}"
1614   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
1615   then have "\<exists>a' M0' K.
1616       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
1617   then show "?case1 \<or> ?case2"
1618   proof (elim exE conjE)
1619     fix a' M0' K
1620     assume N: "N = M0' + K" and r: "?r K a'"
1621     assume "M0 + {#a#} = M0' + {#a'#}"
1622     then have "M0 = M0' \<and> a = a' \<or>
1623         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
1624       by (simp only: add_eq_conv_ex)
1625     then show ?thesis
1626     proof (elim disjE conjE exE)
1627       assume "M0 = M0'" "a = a'"
1628       with N r have "?r K a \<and> N = M0 + K" by simp
1629       then have ?case2 .. then show ?thesis ..
1630     next
1631       fix K'
1632       assume "M0' = K' + {#a#}"
1633       with N have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
1635       assume "M0 = K' + {#a'#}"
1636       with r have "?R (K' + K) M0" by blast
1637       with n have ?case1 by simp then show ?thesis ..
1638     qed
1639   qed
1640 qed
1642 lemma all_accessible: "wf r ==> \<forall>M. M \<in> Wellfounded.acc (mult1 r)"
1643 proof
1644   let ?R = "mult1 r"
1645   let ?W = "Wellfounded.acc ?R"
1646   {
1647     fix M M0 a
1648     assume M0: "M0 \<in> ?W"
1649       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1650       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
1651     have "M0 + {#a#} \<in> ?W"
1652     proof (rule accI [of "M0 + {#a#}"])
1653       fix N
1654       assume "(N, M0 + {#a#}) \<in> ?R"
1655       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
1656           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
1657         by (rule less_add)
1658       then show "N \<in> ?W"
1659       proof (elim exE disjE conjE)
1660         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
1661         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
1662         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
1663         then show "N \<in> ?W" by (simp only: N)
1664       next
1665         fix K
1666         assume N: "N = M0 + K"
1667         assume "\<forall>b. b :# K --> (b, a) \<in> r"
1668         then have "M0 + K \<in> ?W"
1669         proof (induct K)
1670           case empty
1671           from M0 show "M0 + {#} \<in> ?W" by simp
1672         next
1673           case (add K x)
1674           from add.prems have "(x, a) \<in> r" by simp
1675           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
1676           moreover from add have "M0 + K \<in> ?W" by simp
1677           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
1678           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
1679         qed
1680         then show "N \<in> ?W" by (simp only: N)
1681       qed
1682     qed
1683   } note tedious_reasoning = this
1685   assume wf: "wf r"
1686   fix M
1687   show "M \<in> ?W"
1688   proof (induct M)
1689     show "{#} \<in> ?W"
1690     proof (rule accI)
1691       fix b assume "(b, {#}) \<in> ?R"
1692       with not_less_empty show "b \<in> ?W" by contradiction
1693     qed
1695     fix M a assume "M \<in> ?W"
1696     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1697     proof induct
1698       fix a
1699       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1700       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1701       proof
1702         fix M assume "M \<in> ?W"
1703         then show "M + {#a#} \<in> ?W"
1704           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
1705       qed
1706     qed
1707     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
1708   qed
1709 qed
1711 theorem wf_mult1: "wf r ==> wf (mult1 r)"
1712 by (rule acc_wfI) (rule all_accessible)
1714 theorem wf_mult: "wf r ==> wf (mult r)"
1715 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
1718 subsubsection {* Closure-free presentation *}
1720 text {* One direction. *}
1722 lemma mult_implies_one_step:
1723   "trans r ==> (M, N) \<in> mult r ==>
1724     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
1725     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
1726 apply (unfold mult_def mult1_def set_of_def)
1727 apply (erule converse_trancl_induct, clarify)
1728  apply (rule_tac x = M0 in exI, simp, clarify)
1729 apply (case_tac "a :# K")
1730  apply (rule_tac x = I in exI)
1731  apply (simp (no_asm))
1732  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1733  apply (simp (no_asm_simp) add: add.assoc [symmetric])
1734  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong)
1735  apply (simp add: diff_union_single_conv)
1736  apply (simp (no_asm_use) add: trans_def)
1737  apply blast
1738 apply (subgoal_tac "a :# I")
1739  apply (rule_tac x = "I - {#a#}" in exI)
1740  apply (rule_tac x = "J + {#a#}" in exI)
1741  apply (rule_tac x = "K + Ka" in exI)
1742  apply (rule conjI)
1743   apply (simp add: multiset_eq_iff split: nat_diff_split)
1744  apply (rule conjI)
1745   apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
1746   apply (simp add: multiset_eq_iff split: nat_diff_split)
1747  apply (simp (no_asm_use) add: trans_def)
1748  apply blast
1749 apply (subgoal_tac "a :# (M0 + {#a#})")
1750  apply simp
1751 apply (simp (no_asm))
1752 done
1754 lemma one_step_implies_mult_aux:
1755   "trans r ==>
1756     \<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))
1757       --> (I + K, I + J) \<in> mult r"
1758 apply (induct_tac n, auto)
1759 apply (frule size_eq_Suc_imp_eq_union, clarify)
1760 apply (rename_tac "J'", simp)
1761 apply (erule notE, auto)
1762 apply (case_tac "J' = {#}")
1763  apply (simp add: mult_def)
1764  apply (rule r_into_trancl)
1765  apply (simp add: mult1_def set_of_def, blast)
1766 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
1767 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
1768 apply (erule_tac P = "\<forall>k \<in> set_of K. P k" for P in rev_mp)
1769 apply (erule ssubst)
1770 apply (simp add: Ball_def, auto)
1771 apply (subgoal_tac
1772   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
1773     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
1774  prefer 2
1775  apply force
1776 apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
1777 apply (erule trancl_trans)
1778 apply (rule r_into_trancl)
1779 apply (simp add: mult1_def set_of_def)
1780 apply (rule_tac x = a in exI)
1781 apply (rule_tac x = "I + J'" in exI)
1782 apply (simp add: ac_simps)
1783 done
1785 lemma one_step_implies_mult:
1786   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
1787     ==> (I + K, I + J) \<in> mult r"
1788 using one_step_implies_mult_aux by blast
1791 subsubsection {* Partial-order properties *}
1793 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "#<#" 50) where
1794   "M' #<# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
1796 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "#<=#" 50) where
1797   "M' #<=# M \<longleftrightarrow> M' #<# M \<or> M' = M"
1799 notation (xsymbols) less_multiset (infix "#\<subset>#" 50)
1800 notation (xsymbols) le_multiset (infix "#\<subseteq>#" 50)
1802 interpretation multiset_order: order le_multiset less_multiset
1803 proof -
1804   have irrefl: "\<And>M :: 'a multiset. \<not> M #\<subset># M"
1805   proof
1806     fix M :: "'a multiset"
1807     assume "M #\<subset># M"
1808     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
1809     have "trans {(x'::'a, x). x' < x}"
1810       by (rule transI) simp
1811     moreover note MM
1812     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
1813       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
1814       by (rule mult_implies_one_step)
1815     then obtain I J K where "M = I + J" and "M = I + K"
1816       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
1817     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
1818     have "finite (set_of K)" by simp
1819     moreover note aux2
1820     ultimately have "set_of K = {}"
1821       by (induct rule: finite_induct) (auto intro: order_less_trans)
1822     with aux1 show False by simp
1823   qed
1824   have trans: "\<And>K M N :: 'a multiset. K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># N"
1825     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
1826   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
1827     by default (auto simp add: le_multiset_def irrefl dest: trans)
1828 qed
1830 lemma mult_less_irrefl [elim!]: "M #\<subset># (M::'a::order multiset) ==> R"
1831   by simp
1834 subsubsection {* Monotonicity of multiset union *}
1836 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
1837 apply (unfold mult1_def)
1838 apply auto
1839 apply (rule_tac x = a in exI)
1840 apply (rule_tac x = "C + M0" in exI)
1842 done
1844 lemma union_less_mono2: "B #\<subset># D ==> C + B #\<subset># C + (D::'a::order multiset)"
1845 apply (unfold less_multiset_def mult_def)
1846 apply (erule trancl_induct)
1847  apply (blast intro: mult1_union)
1848 apply (blast intro: mult1_union trancl_trans)
1849 done
1851 lemma union_less_mono1: "B #\<subset># D ==> B + C #\<subset># D + (C::'a::order multiset)"
1852 apply (subst add.commute [of B C])
1853 apply (subst add.commute [of D C])
1854 apply (erule union_less_mono2)
1855 done
1857 lemma union_less_mono:
1858   "A #\<subset># C ==> B #\<subset># D ==> A + B #\<subset># C + (D::'a::order multiset)"
1859   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
1861 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
1862 proof
1863 qed (auto simp add: le_multiset_def intro: union_less_mono2)
1866 subsubsection {* Termination proofs with multiset orders *}
1868 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
1869   and multi_member_this: "x \<in># {# x #} + XS"
1870   and multi_member_last: "x \<in># {# x #}"
1871   by auto
1873 definition "ms_strict = mult pair_less"
1874 definition "ms_weak = ms_strict \<union> Id"
1876 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
1877 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
1878 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
1880 lemma smsI:
1881   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
1882   unfolding ms_strict_def
1883 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
1885 lemma wmsI:
1886   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
1887   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
1888 unfolding ms_weak_def ms_strict_def
1889 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
1891 inductive pw_leq
1892 where
1893   pw_leq_empty: "pw_leq {#} {#}"
1894 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
1896 lemma pw_leq_lstep:
1897   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
1898 by (drule pw_leq_step) (rule pw_leq_empty, simp)
1900 lemma pw_leq_split:
1901   assumes "pw_leq X Y"
1902   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 = {#}))"
1903   using assms
1904 proof (induct)
1905   case pw_leq_empty thus ?case by auto
1906 next
1907   case (pw_leq_step x y X Y)
1908   then obtain A B Z where
1909     [simp]: "X = A + Z" "Y = B + Z"
1910       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
1911     by auto
1912   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
1913     unfolding pair_leq_def by auto
1914   thus ?case
1915   proof
1916     assume [simp]: "x = y"
1917     have
1918       "{#x#} + X = A + ({#y#}+Z)
1919       \<and> {#y#} + Y = B + ({#y#}+Z)
1920       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1921       by (auto simp: ac_simps)
1922     thus ?case by (intro exI)
1923   next
1924     assume A: "(x, y) \<in> pair_less"
1925     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
1926     have "{#x#} + X = ?A' + Z"
1927       "{#y#} + Y = ?B' + Z"
1928       by (auto simp add: ac_simps)
1929     moreover have
1930       "(set_of ?A', set_of ?B') \<in> max_strict"
1931       using 1 A unfolding max_strict_def
1932       by (auto elim!: max_ext.cases)
1933     ultimately show ?thesis by blast
1934   qed
1935 qed
1937 lemma
1938   assumes pwleq: "pw_leq Z Z'"
1939   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
1940   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
1941   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
1942 proof -
1943   from pw_leq_split[OF pwleq]
1944   obtain A' B' Z''
1945     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
1946     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
1947     by blast
1948   {
1949     assume max: "(set_of A, set_of B) \<in> max_strict"
1950     from mx_or_empty
1951     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
1952     proof
1953       assume max': "(set_of A', set_of B') \<in> max_strict"
1954       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
1955         by (auto simp: max_strict_def intro: max_ext_additive)
1956       thus ?thesis by (rule smsI)
1957     next
1958       assume [simp]: "A' = {#} \<and> B' = {#}"
1959       show ?thesis by (rule smsI) (auto intro: max)
1960     qed
1961     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:ac_simps)
1962     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
1963   }
1964   from mx_or_empty
1965   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
1966   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
1967 qed
1969 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
1970 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
1971 and nonempty_single: "{# x #} \<noteq> {#}"
1972 by auto
1974 setup {*
1975 let
1976   fun msetT T = Type (@{type_name multiset}, [T]);
1978   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
1979     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
1980     | mk_mset T (x :: xs) =
1981           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
1982                 mk_mset T [x] \$ mk_mset T xs
1984   fun mset_member_tac m i =
1985       (if m <= 0 then
1986            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
1987        else
1988            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
1990   val mset_nonempty_tac =
1991       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
1993   fun regroup_munion_conv ctxt =
1994     Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus}
1995       (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
1997   fun unfold_pwleq_tac i =
1998     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
1999       ORELSE (rtac @{thm pw_leq_lstep} i)
2000       ORELSE (rtac @{thm pw_leq_empty} i)
2002   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
2003                       @{thm Un_insert_left}, @{thm Un_empty_left}]
2004 in
2005   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
2006   {
2007     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
2008     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
2009     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
2010     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
2011     reduction_pair= @{thm ms_reduction_pair}
2012   })
2013 end
2014 *}
2017 subsection {* Legacy theorem bindings *}
2019 lemmas multi_count_eq = multiset_eq_iff [symmetric]
2021 lemma union_commute: "M + N = N + (M::'a multiset)"
2022   by (fact add.commute)
2024 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
2025   by (fact add.assoc)
2027 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
2028   by (fact add.left_commute)
2030 lemmas union_ac = union_assoc union_commute union_lcomm
2032 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
2033   by (fact add_right_cancel)
2035 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
2036   by (fact add_left_cancel)
2038 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
2039   by (fact add_left_imp_eq)
2041 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
2042   by (fact order_less_trans)
2044 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
2045   by (fact inf.commute)
2047 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
2048   by (fact inf.assoc [symmetric])
2050 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
2051   by (fact inf.left_commute)
2053 lemmas multiset_inter_ac =
2054   multiset_inter_commute
2055   multiset_inter_assoc
2056   multiset_inter_left_commute
2058 lemma mult_less_not_refl:
2059   "\<not> M #\<subset># (M::'a::order multiset)"
2060   by (fact multiset_order.less_irrefl)
2062 lemma mult_less_trans:
2063   "K #\<subset># M ==> M #\<subset># N ==> K #\<subset># (N::'a::order multiset)"
2064   by (fact multiset_order.less_trans)
2066 lemma mult_less_not_sym:
2067   "M #\<subset># N ==> \<not> N #\<subset># (M::'a::order multiset)"
2068   by (fact multiset_order.less_not_sym)
2070 lemma mult_less_asym:
2071   "M #\<subset># N ==> (\<not> P ==> N #\<subset># (M::'a::order multiset)) ==> P"
2072   by (fact multiset_order.less_asym)
2074 ML {*
2075 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
2076                       (Const _ \$ t') =
2077     let
2078       val (maybe_opt, ps) =
2079         Nitpick_Model.dest_plain_fun t' ||> op ~~
2080         ||> map (apsnd (snd o HOLogic.dest_number))
2081       fun elems_for t =
2082         case AList.lookup (op =) ps t of
2083           SOME n => replicate n t
2084         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
2085     in
2086       case maps elems_for (all_values elem_T) @
2087            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
2088             else []) of
2089         [] => Const (@{const_name zero_class.zero}, T)
2090       | ts => foldl1 (fn (t1, t2) =>
2091                          Const (@{const_name plus_class.plus}, T --> T --> T)
2092                          \$ t1 \$ t2)
2093                      (map (curry (op \$) (Const (@{const_name single},
2094                                                 elem_T --> T))) ts)
2095     end
2096   | multiset_postproc _ _ _ _ t = t
2097 *}
2099 declaration {*
2100 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
2101     multiset_postproc
2102 *}
2104 hide_const (open) fold
2107 subsection {* Naive implementation using lists *}
2109 code_datatype multiset_of
2111 lemma [code]:
2112   "{#} = multiset_of []"
2113   by simp
2115 lemma [code]:
2116   "{#x#} = multiset_of [x]"
2117   by simp
2119 lemma union_code [code]:
2120   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
2121   by simp
2123 lemma [code]:
2124   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
2125   by (simp add: multiset_of_map)
2127 lemma [code]:
2128   "Multiset.filter f (multiset_of xs) = multiset_of (filter f xs)"
2129   by (simp add: multiset_of_filter)
2131 lemma [code]:
2132   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
2133   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
2135 lemma [code]:
2136   "multiset_of xs #\<inter> multiset_of ys =
2137     multiset_of (snd (fold (\<lambda>x (ys, zs).
2138       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
2139 proof -
2140   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
2141     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
2142       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
2143     by (induct xs arbitrary: ys)
2145   then show ?thesis by simp
2146 qed
2148 lemma [code]:
2149   "multiset_of xs #\<union> multiset_of ys =
2150     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
2151 proof -
2152   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
2153       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
2154     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
2155   then show ?thesis by simp
2156 qed
2158 declare in_multiset_in_set [code_unfold]
2160 lemma [code]:
2161   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
2162 proof -
2163   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
2164     by (induct xs) simp_all
2165   then show ?thesis by simp
2166 qed
2168 declare set_of_multiset_of [code]
2170 declare sorted_list_of_multiset_multiset_of [code]
2172 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
2173   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
2174   apply (cases "finite A")
2175   apply simp_all
2176   apply (induct A rule: finite_induct)
2178   done
2180 declare size_multiset_of [code]
2182 fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
2183   "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
2184 | "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
2185      None \<Rightarrow> None
2186    | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
2188 lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> multiset_of xs \<le> multiset_of ys) \<and>
2189   (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> multiset_of xs < multiset_of ys) \<and>
2190   (ms_lesseq_impl xs ys = Some False \<longrightarrow> multiset_of xs = multiset_of ys)"
2191 proof (induct xs arbitrary: ys)
2192   case (Nil ys)
2193   show ?case by (auto simp: mset_less_empty_nonempty)
2194 next
2195   case (Cons x xs ys)
2196   show ?case
2197   proof (cases "List.extract (op = x) ys")
2198     case None
2199     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
2200     {
2201       assume "multiset_of (x # xs) \<le> multiset_of ys"
2202       from set_of_mono[OF this] x have False by simp
2203     } note nle = this
2204     moreover
2205     {
2206       assume "multiset_of (x # xs) < multiset_of ys"
2207       hence "multiset_of (x # xs) \<le> multiset_of ys" by auto
2208       from nle[OF this] have False .
2209     }
2210     ultimately show ?thesis using None by auto
2211   next
2212     case (Some res)
2213     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
2214     note Some = Some[unfolded res]
2215     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
2216     hence id: "multiset_of ys = multiset_of (ys1 @ ys2) + {#x#}"
2217       by (auto simp: ac_simps)
2218     show ?thesis unfolding ms_lesseq_impl.simps
2219       unfolding Some option.simps split
2220       unfolding id
2221       using Cons[of "ys1 @ ys2"]
2222       unfolding mset_le_def mset_less_def by auto
2223   qed
2224 qed
2226 lemma [code]: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
2227   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2229 lemma [code]: "multiset_of xs < multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
2230   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2232 instantiation multiset :: (equal) equal
2233 begin
2235 definition
2236   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
2237 lemma [code]: "HOL.equal (multiset_of xs) (multiset_of ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
2238   unfolding equal_multiset_def
2239   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2241 instance
2242   by default (simp add: equal_multiset_def)
2243 end
2245 lemma [code]:
2246   "msetsum (multiset_of xs) = listsum xs"
2247   by (induct xs) (simp_all add: add.commute)
2249 lemma [code]:
2250   "msetprod (multiset_of xs) = fold times xs 1"
2251 proof -
2252   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
2253     by (induct xs) (simp_all add: mult.assoc)
2254   then show ?thesis by simp
2255 qed
2257 text {*
2258   Exercise for the casual reader: add implementations for @{const le_multiset}
2259   and @{const less_multiset} (multiset order).
2260 *}
2262 text {* Quickcheck generators *}
2264 definition (in term_syntax)
2265   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
2266     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
2267   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
2269 notation fcomp (infixl "\<circ>>" 60)
2270 notation scomp (infixl "\<circ>\<rightarrow>" 60)
2272 instantiation multiset :: (random) random
2273 begin
2275 definition
2276   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
2278 instance ..
2280 end
2282 no_notation fcomp (infixl "\<circ>>" 60)
2283 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
2285 instantiation multiset :: (full_exhaustive) full_exhaustive
2286 begin
2288 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
2289 where
2290   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
2292 instance ..
2294 end
2296 hide_const (open) msetify
2299 subsection {* BNF setup *}
2301 definition rel_mset where
2302   "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. multiset_of xs = X \<and> multiset_of ys = Y \<and> list_all2 R xs ys)"
2304 lemma multiset_of_zip_take_Cons_drop_twice:
2305   assumes "length xs = length ys" "j \<le> length xs"
2306   shows "multiset_of (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
2307     multiset_of (zip xs ys) + {#(x, y)#}"
2308 using assms
2309 proof (induct xs ys arbitrary: x y j rule: list_induct2)
2310   case Nil
2311   thus ?case
2312     by simp
2313 next
2314   case (Cons x xs y ys)
2315   thus ?case
2316   proof (cases "j = 0")
2317     case True
2318     thus ?thesis
2319       by simp
2320   next
2321     case False
2322     then obtain k where k: "j = Suc k"
2323       by (case_tac j) simp
2324     hence "k \<le> length xs"
2325       using Cons.prems by auto
2326     hence "multiset_of (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
2327       multiset_of (zip xs ys) + {#(x, y)#}"
2328       by (rule Cons.hyps(2))
2329     thus ?thesis
2330       unfolding k by (auto simp: add.commute union_lcomm)
2331   qed
2332 qed
2334 lemma ex_multiset_of_zip_left:
2335   assumes "length xs = length ys" "multiset_of xs' = multiset_of xs"
2336   shows "\<exists>ys'. length ys' = length xs' \<and> multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
2337 using assms
2338 proof (induct xs ys arbitrary: xs' rule: list_induct2)
2339   case Nil
2340   thus ?case
2341     by auto
2342 next
2343   case (Cons x xs y ys xs')
2344   obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
2345     by (metis Cons.prems in_set_conv_nth list.set_intros(1) multiset_of_eq_setD)
2347   def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
2348   have "multiset_of xs' = {#x#} + multiset_of xsa"
2349     unfolding xsa_def using j_len nth_j
2350     by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
2351       multiset_of.simps(2) union_code add.commute)
2352   hence ms_x: "multiset_of xsa = multiset_of xs"
2353     by (metis Cons.prems add.commute add_right_imp_eq multiset_of.simps(2))
2354   then obtain ysa where
2355     len_a: "length ysa = length xsa" and ms_a: "multiset_of (zip xsa ysa) = multiset_of (zip xs ys)"
2356     using Cons.hyps(2) by blast
2358   def ys' \<equiv> "take j ysa @ y # drop j ysa"
2359   have xs': "xs' = take j xsa @ x # drop j xsa"
2360     using ms_x j_len nth_j Cons.prems xsa_def
2361     by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
2362       length_drop size_multiset_of)
2363   have j_len': "j \<le> length xsa"
2364     using j_len xs' xsa_def
2365     by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
2366   have "length ys' = length xs'"
2367     unfolding ys'_def using Cons.prems len_a ms_x
2368     by (metis add_Suc_right append_take_drop_id length_Cons length_append multiset_of_eq_length)
2369   moreover have "multiset_of (zip xs' ys') = multiset_of (zip (x # xs) (y # ys))"
2370     unfolding xs' ys'_def
2371     by (rule trans[OF multiset_of_zip_take_Cons_drop_twice])
2372       (auto simp: len_a ms_a j_len' add.commute)
2373   ultimately show ?case
2374     by blast
2375 qed
2377 lemma list_all2_reorder_left_invariance:
2378   assumes rel: "list_all2 R xs ys" and ms_x: "multiset_of xs' = multiset_of xs"
2379   shows "\<exists>ys'. list_all2 R xs' ys' \<and> multiset_of ys' = multiset_of ys"
2380 proof -
2381   have len: "length xs = length ys"
2382     using rel list_all2_conv_all_nth by auto
2383   obtain ys' where
2384     len': "length xs' = length ys'" and ms_xy: "multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
2385     using len ms_x by (metis ex_multiset_of_zip_left)
2386   have "list_all2 R xs' ys'"
2387     using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: multiset_of_eq_setD)
2388   moreover have "multiset_of ys' = multiset_of ys"
2389     using len len' ms_xy map_snd_zip multiset_of_map by metis
2390   ultimately show ?thesis
2391     by blast
2392 qed
2394 lemma ex_multiset_of: "\<exists>xs. multiset_of xs = X"
2395   by (induct X) (simp, metis multiset_of.simps(2))
2397 bnf "'a multiset"
2398   map: image_mset
2399   sets: set_of
2400   bd: natLeq
2401   wits: "{#}"
2402   rel: rel_mset
2403 proof -
2404   show "image_mset id = id"
2405     by (rule image_mset.id)
2406 next
2407   show "\<And>f g. image_mset (g \<circ> f) = image_mset g \<circ> image_mset f"
2408     unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality)
2409 next
2410   fix X :: "'a multiset"
2411   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"
2412     by (induct X, (simp (no_asm))+,
2413       metis One_nat_def Un_iff count_single mem_set_of_iff set_of_union zero_less_Suc)
2414 next
2415   show "\<And>f. set_of \<circ> image_mset f = op ` f \<circ> set_of"
2416     by auto
2417 next
2418   show "card_order natLeq"
2419     by (rule natLeq_card_order)
2420 next
2421   show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
2422     by (rule natLeq_cinfinite)
2423 next
2424   show "\<And>X. ordLeq3 (card_of (set_of X)) natLeq"
2425     by transfer
2426       (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
2427 next
2428   show "\<And>R S. rel_mset R OO rel_mset S \<le> rel_mset (R OO S)"
2429     unfolding rel_mset_def[abs_def] OO_def
2430     apply clarify
2431     apply (rename_tac X Z Y xs ys' ys zs)
2432     apply (drule_tac xs = ys' and ys = zs and xs' = ys in list_all2_reorder_left_invariance)
2433     by (auto intro: list_all2_trans)
2434 next
2435   show "\<And>R. rel_mset R =
2436     (BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
2437     BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset snd)"
2438     unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
2439     apply (rule ext)+
2440     apply auto
2441      apply (rule_tac x = "multiset_of (zip xs ys)" in exI; auto)
2442         apply (metis list_all2_lengthD map_fst_zip multiset_of_map)
2443        apply (auto simp: list_all2_iff)
2444       apply (metis list_all2_lengthD map_snd_zip multiset_of_map)
2445      apply (auto simp: list_all2_iff)
2446     apply (rename_tac XY)
2447     apply (cut_tac X = XY in ex_multiset_of)
2448     apply (erule exE)
2449     apply (rename_tac xys)
2450     apply (rule_tac x = "map fst xys" in exI)
2451     apply (auto simp: multiset_of_map)
2452     apply (rule_tac x = "map snd xys" in exI)
2453     apply (auto simp: multiset_of_map list_all2I subset_eq zip_map_fst_snd)
2454     done
2455 next
2456   show "\<And>z. z \<in> set_of {#} \<Longrightarrow> False"
2457     by auto
2458 qed
2460 inductive rel_mset' where
2461   Zero[intro]: "rel_mset' R {#} {#}"
2462 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
2464 lemma rel_mset_Zero: "rel_mset R {#} {#}"
2465 unfolding rel_mset_def Grp_def by auto
2467 declare multiset.count[simp]
2468 declare Abs_multiset_inverse[simp]
2469 declare multiset.count_inverse[simp]
2470 declare union_preserves_multiset[simp]
2472 lemma rel_mset_Plus:
2473 assumes ab: "R a b" and MN: "rel_mset R M N"
2474 shows "rel_mset R (M + {#a#}) (N + {#b#})"
2475 proof-
2476   {fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
2477    hence "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
2478                image_mset snd y + {#b#} = image_mset snd ya \<and>
2479                set_of ya \<subseteq> {(x, y). R x y}"
2480    apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
2481   }
2482   thus ?thesis
2483   using assms
2484   unfolding multiset.rel_compp_Grp Grp_def by blast
2485 qed
2487 lemma rel_mset'_imp_rel_mset:
2488   "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
2489 apply(induct rule: rel_mset'.induct)
2490 using rel_mset_Zero rel_mset_Plus by auto
2492 lemma rel_mset_size:
2493   "rel_mset R M N \<Longrightarrow> size M = size N"
2494 unfolding multiset.rel_compp_Grp Grp_def by auto
2497 assumes empty: "P {#} {#}"
2498 and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
2499 and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
2500 shows "P M N"
2501 apply(induct N rule: multiset_induct)
2502   apply(induct M rule: multiset_induct, rule empty, erule addL)
2503   apply(induct M rule: multiset_induct, erule addR, erule addR)
2504 done
2506 lemma multiset_induct2_size[consumes 1, case_names empty add]:
2507 assumes c: "size M = size N"
2508 and empty: "P {#} {#}"
2509 and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
2510 shows "P M N"
2511 using c proof(induct M arbitrary: N rule: measure_induct_rule[of size])
2512   case (less M)  show ?case
2513   proof(cases "M = {#}")
2514     case True hence "N = {#}" using less.prems by auto
2515     thus ?thesis using True empty by auto
2516   next
2517     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2518     have "N \<noteq> {#}" using False less.prems by auto
2519     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
2520     have "size M1 = size N1" using less.prems unfolding M N by auto
2521     thus ?thesis using M N less.hyps add by auto
2522   qed
2523 qed
2525 lemma msed_map_invL:
2526 assumes "image_mset f (M + {#a#}) = N"
2527 shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1"
2528 proof-
2529   have "f a \<in># N"
2530   using assms multiset.set_map[of f "M + {#a#}"] by auto
2531   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
2532   have "image_mset f M = N1" using assms unfolding N by simp
2533   thus ?thesis using N by blast
2534 qed
2536 lemma msed_map_invR:
2537 assumes "image_mset f M = N + {#b#}"
2538 shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N"
2539 proof-
2540   obtain a where a: "a \<in># M" and fa: "f a = b"
2541   using multiset.set_map[of f M] unfolding assms
2542   by (metis image_iff mem_set_of_iff union_single_eq_member)
2543   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
2544   have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
2545   thus ?thesis using M fa by blast
2546 qed
2548 lemma msed_rel_invL:
2549 assumes "rel_mset R (M + {#a#}) N"
2550 shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1"
2551 proof-
2552   obtain K where KM: "image_mset fst K = M + {#a#}"
2553   and KN: "image_mset snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2554   using assms
2555   unfolding multiset.rel_compp_Grp Grp_def by auto
2556   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
2557   and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
2558   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1"
2559   using msed_map_invL[OF KN[unfolded K]] by auto
2560   have Rab: "R a (snd ab)" using sK a unfolding K by auto
2561   have "rel_mset R M N1" using sK K1M K1N1
2562   unfolding K multiset.rel_compp_Grp Grp_def by auto
2563   thus ?thesis using N Rab by auto
2564 qed
2566 lemma msed_rel_invR:
2567 assumes "rel_mset R M (N + {#b#})"
2568 shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N"
2569 proof-
2570   obtain K where KN: "image_mset snd K = N + {#b#}"
2571   and KM: "image_mset fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2572   using assms
2573   unfolding multiset.rel_compp_Grp Grp_def by auto
2574   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
2575   and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
2576   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1"
2577   using msed_map_invL[OF KM[unfolded K]] by auto
2578   have Rab: "R (fst ab) b" using sK b unfolding K by auto
2579   have "rel_mset R M1 N" using sK K1N K1M1
2580   unfolding K multiset.rel_compp_Grp Grp_def by auto
2581   thus ?thesis using M Rab by auto
2582 qed
2584 lemma rel_mset_imp_rel_mset':
2585 assumes "rel_mset R M N"
2586 shows "rel_mset' R M N"
2587 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of size])
2588   case (less M)
2589   have c: "size M = size N" using rel_mset_size[OF less.prems] .
2590   show ?case
2591   proof(cases "M = {#}")
2592     case True hence "N = {#}" using c by simp
2593     thus ?thesis using True rel_mset'.Zero by auto
2594   next
2595     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2596     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1"
2597     using msed_rel_invL[OF less.prems[unfolded M]] by auto
2598     have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
2599     thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
2600   qed
2601 qed
2603 lemma rel_mset_rel_mset':
2604 "rel_mset R M N = rel_mset' R M N"
2605 using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
2607 (* The main end product for rel_mset: inductive characterization *)
2608 theorems rel_mset_induct[case_names empty add, induct pred: rel_mset] =
2609          rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
2612 subsection {* Size setup *}
2614 lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
2615   unfolding o_apply by (rule ext) (induct_tac, auto)
2617 setup {*
2618 BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
2619   @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
2620     size_union}
2621   @{thms multiset_size_o_map}
2622 *}
2624 hide_const (open) wcount
2626 end