src/HOL/Library/Multiset.thy
 author blanchet Fri Jan 24 11:51:45 2014 +0100 (2014-01-24) changeset 55129 26bd1cba3ab5 parent 54868 bab6cade3cc5 child 55417 01fbfb60c33e permissions -rw-r--r--
killed 'More_BNFs' by moving its various bits where they (now) belong
1 (*  Title:      HOL/Library/Multiset.thy
2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
3     Author:     Andrei Popescu, TU Muenchen
4 *)
6 header {* (Finite) multisets *}
8 theory Multiset
9 imports Main
10 begin
12 subsection {* The type of multisets *}
14 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
16 typedef 'a multiset = "multiset :: ('a => nat) set"
17   morphisms count Abs_multiset
18   unfolding multiset_def
19 proof
20   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
21 qed
23 setup_lifting type_definition_multiset
25 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
26   "a :# M == 0 < count M a"
28 notation (xsymbols)
29   Melem (infix "\<in>#" 50)
31 lemma multiset_eq_iff:
32   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
33   by (simp only: count_inject [symmetric] fun_eq_iff)
35 lemma multiset_eqI:
36   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
37   using multiset_eq_iff by auto
39 text {*
40  \medskip Preservation of the representing set @{term multiset}.
41 *}
43 lemma const0_in_multiset:
44   "(\<lambda>a. 0) \<in> multiset"
45   by (simp add: multiset_def)
47 lemma only1_in_multiset:
48   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
49   by (simp add: multiset_def)
51 lemma union_preserves_multiset:
52   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
53   by (simp add: multiset_def)
55 lemma diff_preserves_multiset:
56   assumes "M \<in> multiset"
57   shows "(\<lambda>a. M a - N a) \<in> multiset"
58 proof -
59   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
60     by auto
61   with assms show ?thesis
62     by (auto simp add: multiset_def intro: finite_subset)
63 qed
65 lemma filter_preserves_multiset:
66   assumes "M \<in> multiset"
67   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
68 proof -
69   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
70     by auto
71   with assms show ?thesis
72     by (auto simp add: multiset_def intro: finite_subset)
73 qed
75 lemmas in_multiset = const0_in_multiset only1_in_multiset
76   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
79 subsection {* Representing multisets *}
81 text {* Multiset enumeration *}
83 instantiation multiset :: (type) cancel_comm_monoid_add
84 begin
86 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
87 by (rule const0_in_multiset)
89 abbreviation Mempty :: "'a multiset" ("{#}") where
90   "Mempty \<equiv> 0"
92 lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
93 by (rule union_preserves_multiset)
95 instance
96 by default (transfer, simp add: fun_eq_iff)+
98 end
100 lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
101 by (rule only1_in_multiset)
103 syntax
104   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
105 translations
106   "{#x, xs#}" == "{#x#} + {#xs#}"
107   "{#x#}" == "CONST single x"
109 lemma count_empty [simp]: "count {#} a = 0"
110   by (simp add: zero_multiset.rep_eq)
112 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
113   by (simp add: single.rep_eq)
116 subsection {* Basic operations *}
118 subsubsection {* Union *}
120 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
121   by (simp add: plus_multiset.rep_eq)
124 subsubsection {* Difference *}
126 instantiation multiset :: (type) comm_monoid_diff
127 begin
129 lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
130 by (rule diff_preserves_multiset)
132 instance
133 by default (transfer, simp add: fun_eq_iff)+
135 end
137 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
138   by (simp add: minus_multiset.rep_eq)
140 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
141   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
143 lemma diff_cancel[simp]: "A - A = {#}"
144   by (fact Groups.diff_cancel)
146 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
147   by (fact add_diff_cancel_right')
149 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
150   by (fact add_diff_cancel_left')
152 lemma diff_right_commute:
153   "(M::'a multiset) - N - Q = M - Q - N"
154   by (fact diff_right_commute)
157   "(M::'a multiset) - (N + Q) = M - N - Q"
158   by (rule sym) (fact diff_diff_add)
160 lemma insert_DiffM:
161   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
162   by (clarsimp simp: multiset_eq_iff)
164 lemma insert_DiffM2 [simp]:
165   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
166   by (clarsimp simp: multiset_eq_iff)
168 lemma diff_union_swap:
169   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
170   by (auto simp add: multiset_eq_iff)
172 lemma diff_union_single_conv:
173   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
174   by (simp add: multiset_eq_iff)
177 subsubsection {* Equality of multisets *}
179 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
180   by (simp add: multiset_eq_iff)
182 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
183   by (auto simp add: multiset_eq_iff)
185 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
186   by (auto simp add: multiset_eq_iff)
188 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
189   by (auto simp add: multiset_eq_iff)
191 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
192   by (auto simp add: multiset_eq_iff)
194 lemma diff_single_trivial:
195   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
196   by (auto simp add: multiset_eq_iff)
198 lemma diff_single_eq_union:
199   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
200   by auto
202 lemma union_single_eq_diff:
203   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
204   by (auto dest: sym)
206 lemma union_single_eq_member:
207   "M + {#x#} = N \<Longrightarrow> x \<in># N"
208   by auto
210 lemma union_is_single:
211   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
212 proof
213   assume ?rhs then show ?lhs by auto
214 next
215   assume ?lhs then show ?rhs
216     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
217 qed
219 lemma single_is_union:
220   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
221   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
224   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
225 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
226 proof
227   assume ?rhs then show ?lhs
229     (drule sym, simp add: add_assoc [symmetric])
230 next
231   assume ?lhs
232   show ?rhs
233   proof (cases "a = b")
234     case True with `?lhs` show ?thesis by simp
235   next
236     case False
237     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
238     with False have "a \<in># N" by auto
239     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
240     moreover note False
241     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
242   qed
243 qed
245 lemma insert_noteq_member:
246   assumes BC: "B + {#b#} = C + {#c#}"
247    and bnotc: "b \<noteq> c"
248   shows "c \<in># B"
249 proof -
250   have "c \<in># C + {#c#}" by simp
251   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
252   then have "c \<in># B + {#b#}" using BC by simp
253   then show "c \<in># B" using nc by simp
254 qed
257   "(M + {#a#} = N + {#b#}) =
258     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
261 lemma multi_member_split:
262   "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
263   by (rule_tac x = "M - {#x#}" in exI, simp)
266 subsubsection {* Pointwise ordering induced by count *}
268 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
269 begin
271 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)"
272 by simp
273 lemmas mset_le_def = less_eq_multiset_def
275 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
276   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
278 instance
279   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
281 end
283 lemma mset_less_eqI:
284   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
285   by (simp add: mset_le_def)
287 lemma mset_le_exists_conv:
288   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
289 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
290 apply (auto intro: multiset_eq_iff [THEN iffD2])
291 done
293 instance multiset :: (type) ordered_cancel_comm_monoid_diff
294   by default (simp, fact mset_le_exists_conv)
296 lemma mset_le_mono_add_right_cancel [simp]:
297   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
298   by (fact add_le_cancel_right)
300 lemma mset_le_mono_add_left_cancel [simp]:
301   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
302   by (fact add_le_cancel_left)
305   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
306   by (fact add_mono)
308 lemma mset_le_add_left [simp]:
309   "(A::'a multiset) \<le> A + B"
310   unfolding mset_le_def by auto
312 lemma mset_le_add_right [simp]:
313   "B \<le> (A::'a multiset) + B"
314   unfolding mset_le_def by auto
316 lemma mset_le_single:
317   "a :# B \<Longrightarrow> {#a#} \<le> B"
318   by (simp add: mset_le_def)
320 lemma multiset_diff_union_assoc:
321   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
322   by (simp add: multiset_eq_iff mset_le_def)
324 lemma mset_le_multiset_union_diff_commute:
325   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
326 by (simp add: multiset_eq_iff mset_le_def)
328 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
329 by(simp add: mset_le_def)
331 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
332 apply (clarsimp simp: mset_le_def mset_less_def)
333 apply (erule_tac x=x in allE)
334 apply auto
335 done
337 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
338 apply (clarsimp simp: mset_le_def mset_less_def)
339 apply (erule_tac x = x in allE)
340 apply auto
341 done
343 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
344 apply (rule conjI)
345  apply (simp add: mset_lessD)
346 apply (clarsimp simp: mset_le_def mset_less_def)
347 apply safe
348  apply (erule_tac x = a in allE)
349  apply (auto split: split_if_asm)
350 done
352 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
353 apply (rule conjI)
354  apply (simp add: mset_leD)
355 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
356 done
358 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
359   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
361 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
362   by (auto simp: mset_le_def mset_less_def)
364 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
365   by simp
368   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
369   by (fact add_less_imp_less_right)
371 lemma mset_less_empty_nonempty:
372   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
373   by (auto simp: mset_le_def mset_less_def)
375 lemma mset_less_diff_self:
376   "c \<in># B \<Longrightarrow> B - {#c#} < B"
377   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
380 subsubsection {* Intersection *}
382 instantiation multiset :: (type) semilattice_inf
383 begin
385 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
386   multiset_inter_def: "inf_multiset A B = A - (A - B)"
388 instance
389 proof -
390   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
391   show "OFCLASS('a multiset, semilattice_inf_class)"
392     by default (auto simp add: multiset_inter_def mset_le_def aux)
393 qed
395 end
397 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
398   "multiset_inter \<equiv> inf"
400 lemma multiset_inter_count [simp]:
401   "count (A #\<inter> B) x = min (count A x) (count B x)"
402   by (simp add: multiset_inter_def)
404 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
405   by (rule multiset_eqI) auto
407 lemma multiset_union_diff_commute:
408   assumes "B #\<inter> C = {#}"
409   shows "A + B - C = A - C + B"
410 proof (rule multiset_eqI)
411   fix x
412   from assms have "min (count B x) (count C x) = 0"
413     by (auto simp add: multiset_eq_iff)
414   then have "count B x = 0 \<or> count C x = 0"
415     by auto
416   then show "count (A + B - C) x = count (A - C + B) x"
417     by auto
418 qed
420 lemma empty_inter [simp]:
421   "{#} #\<inter> M = {#}"
422   by (simp add: multiset_eq_iff)
424 lemma inter_empty [simp]:
425   "M #\<inter> {#} = {#}"
426   by (simp add: multiset_eq_iff)
429   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
430   by (simp add: multiset_eq_iff)
433   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
434   by (simp add: multiset_eq_iff)
437   "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
438   by (simp add: multiset_eq_iff)
441   "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
442   by (simp add: multiset_eq_iff)
445 subsubsection {* Bounded union *}
447 instantiation multiset :: (type) semilattice_sup
448 begin
450 definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
451   "sup_multiset A B = A + (B - A)"
453 instance
454 proof -
455   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" by arith
456   show "OFCLASS('a multiset, semilattice_sup_class)"
457     by default (auto simp add: sup_multiset_def mset_le_def aux)
458 qed
460 end
462 abbreviation sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<union>" 70) where
463   "sup_multiset \<equiv> sup"
465 lemma sup_multiset_count [simp]:
466   "count (A #\<union> B) x = max (count A x) (count B x)"
467   by (simp add: sup_multiset_def)
469 lemma empty_sup [simp]:
470   "{#} #\<union> M = M"
471   by (simp add: multiset_eq_iff)
473 lemma sup_empty [simp]:
474   "M #\<union> {#} = M"
475   by (simp add: multiset_eq_iff)
478   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
479   by (simp add: multiset_eq_iff)
482   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
483   by (simp add: multiset_eq_iff)
486   "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
487   by (simp add: multiset_eq_iff)
490   "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
491   by (simp add: multiset_eq_iff)
494 subsubsection {* Filter (with comprehension syntax) *}
496 text {* Multiset comprehension *}
498 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"
499 by (rule filter_preserves_multiset)
501 hide_const (open) filter
503 lemma count_filter [simp]:
504   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
505   by (simp add: filter.rep_eq)
507 lemma filter_empty [simp]:
508   "Multiset.filter P {#} = {#}"
509   by (rule multiset_eqI) simp
511 lemma filter_single [simp]:
512   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
513   by (rule multiset_eqI) simp
515 lemma filter_union [simp]:
516   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
517   by (rule multiset_eqI) simp
519 lemma filter_diff [simp]:
520   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
521   by (rule multiset_eqI) simp
523 lemma filter_inter [simp]:
524   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
525   by (rule multiset_eqI) simp
527 syntax
528   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
529 syntax (xsymbol)
530   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
531 translations
532   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
535 subsubsection {* Set of elements *}
537 definition set_of :: "'a multiset => 'a set" where
538   "set_of M = {x. x :# M}"
540 lemma set_of_empty [simp]: "set_of {#} = {}"
541 by (simp add: set_of_def)
543 lemma set_of_single [simp]: "set_of {#b#} = {b}"
544 by (simp add: set_of_def)
546 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
547 by (auto simp add: set_of_def)
549 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
550 by (auto simp add: set_of_def multiset_eq_iff)
552 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
553 by (auto simp add: set_of_def)
555 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
556 by (auto simp add: set_of_def)
558 lemma finite_set_of [iff]: "finite (set_of M)"
559   using count [of M] by (simp add: multiset_def set_of_def)
561 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
562   unfolding set_of_def[symmetric] by simp
564 subsubsection {* Size *}
566 instantiation multiset :: (type) size
567 begin
569 definition size_def:
570   "size M = setsum (count M) (set_of M)"
572 instance ..
574 end
576 lemma size_empty [simp]: "size {#} = 0"
577 by (simp add: size_def)
579 lemma size_single [simp]: "size {#b#} = 1"
580 by (simp add: size_def)
582 lemma setsum_count_Int:
583   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
584 apply (induct rule: finite_induct)
585  apply simp
586 apply (simp add: Int_insert_left set_of_def)
587 done
589 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
590 apply (unfold size_def)
591 apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
592  prefer 2
593  apply (rule ext, simp)
594 apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
595 apply (subst Int_commute)
596 apply (simp (no_asm_simp) add: setsum_count_Int)
597 done
599 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
600 by (auto simp add: size_def multiset_eq_iff)
602 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
603 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
605 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
606 apply (unfold size_def)
607 apply (drule setsum_SucD)
608 apply auto
609 done
611 lemma size_eq_Suc_imp_eq_union:
612   assumes "size M = Suc n"
613   shows "\<exists>a N. M = N + {#a#}"
614 proof -
615   from assms obtain a where "a \<in># M"
616     by (erule size_eq_Suc_imp_elem [THEN exE])
617   then have "M = M - {#a#} + {#a#}" by simp
618   then show ?thesis by blast
619 qed
622 subsection {* Induction and case splits *}
624 theorem multiset_induct [case_names empty add, induct type: multiset]:
625   assumes empty: "P {#}"
626   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
627   shows "P M"
628 proof (induct n \<equiv> "size M" arbitrary: M)
629   case 0 thus "P M" by (simp add: empty)
630 next
631   case (Suc k)
632   obtain N x where "M = N + {#x#}"
633     using `Suc k = size M` [symmetric]
634     using size_eq_Suc_imp_eq_union by fast
635   with Suc add show "P M" by simp
636 qed
638 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
639 by (induct M) auto
641 lemma multiset_cases [cases type, case_names empty add]:
642 assumes em:  "M = {#} \<Longrightarrow> P"
643 assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
644 shows "P"
645 using assms by (induct M) simp_all
647 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
648 by (cases "B = {#}") (auto dest: multi_member_split)
650 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
651 apply (subst multiset_eq_iff)
652 apply auto
653 done
655 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
656 proof (induct A arbitrary: B)
657   case (empty M)
658   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
659   then obtain M' x where "M = M' + {#x#}"
660     by (blast dest: multi_nonempty_split)
661   then show ?case by simp
662 next
663   case (add S x T)
664   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
665   have SxsubT: "S + {#x#} < T" by fact
666   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
667   then obtain T' where T: "T = T' + {#x#}"
668     by (blast dest: multi_member_split)
669   then have "S < T'" using SxsubT
670     by (blast intro: mset_less_add_bothsides)
671   then have "size S < size T'" using IH by simp
672   then show ?case using T by simp
673 qed
676 subsubsection {* Strong induction and subset induction for multisets *}
678 text {* Well-foundedness of proper subset operator: *}
680 text {* proper multiset subset *}
682 definition
683   mset_less_rel :: "('a multiset * 'a multiset) set" where
684   "mset_less_rel = {(A,B). A < B}"
687   assumes "c \<in># B" and "b \<noteq> c"
688   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
689 proof -
690   from `c \<in># B` obtain A where B: "B = A + {#c#}"
691     by (blast dest: multi_member_split)
692   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
693   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
695   then show ?thesis using B by simp
696 qed
698 lemma wf_mset_less_rel: "wf mset_less_rel"
699 apply (unfold mset_less_rel_def)
700 apply (rule wf_measure [THEN wf_subset, where f1=size])
701 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
702 done
704 text {* The induction rules: *}
706 lemma full_multiset_induct [case_names less]:
707 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
708 shows "P B"
709 apply (rule wf_mset_less_rel [THEN wf_induct])
710 apply (rule ih, auto simp: mset_less_rel_def)
711 done
713 lemma multi_subset_induct [consumes 2, case_names empty add]:
714 assumes "F \<le> A"
715   and empty: "P {#}"
716   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
717 shows "P F"
718 proof -
719   from `F \<le> A`
720   show ?thesis
721   proof (induct F)
722     show "P {#}" by fact
723   next
724     fix x F
725     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
726     show "P (F + {#x#})"
727     proof (rule insert)
728       from i show "x \<in># A" by (auto dest: mset_le_insertD)
729       from i have "F \<le> A" by (auto dest: mset_le_insertD)
730       with P show "P F" .
731     qed
732   qed
733 qed
736 subsection {* The fold combinator *}
738 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
739 where
740   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
742 lemma fold_mset_empty [simp]:
743   "fold f s {#} = s"
744   by (simp add: fold_def)
746 context comp_fun_commute
747 begin
749 lemma fold_mset_insert:
750   "fold f s (M + {#x#}) = f x (fold f s M)"
751 proof -
752   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
753     by (fact comp_fun_commute_funpow)
754   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
755     by (fact comp_fun_commute_funpow)
756   show ?thesis
757   proof (cases "x \<in> set_of M")
758     case False
759     then have *: "count (M + {#x#}) x = 1" by simp
760     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
761       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
762       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
763     with False * show ?thesis
764       by (simp add: fold_def del: count_union)
765   next
766     case True
767     def N \<equiv> "set_of M - {x}"
768     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
769     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
770       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
771       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
772     with * show ?thesis by (simp add: fold_def del: count_union) simp
773   qed
774 qed
776 corollary fold_mset_single [simp]:
777   "fold f s {#x#} = f x s"
778 proof -
779   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
780   then show ?thesis by simp
781 qed
783 lemma fold_mset_fun_left_comm:
784   "f x (fold f s M) = fold f (f x s) M"
785   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
787 lemma fold_mset_union [simp]:
788   "fold f s (M + N) = fold f (fold f s M) N"
789 proof (induct M)
790   case empty then show ?case by simp
791 next
792   case (add M x)
793   have "M + {#x#} + N = (M + N) + {#x#}"
795   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
796 qed
798 lemma fold_mset_fusion:
799   assumes "comp_fun_commute g"
800   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")
801 proof -
802   interpret comp_fun_commute g by (fact assms)
803   show "PROP ?P" by (induct A) auto
804 qed
806 end
808 text {*
809   A note on code generation: When defining some function containing a
810   subterm @{term "fold F"}, code generation is not automatic. When
811   interpreting locale @{text left_commutative} with @{text F}, the
812   would be code thms for @{const fold} become thms like
813   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
814   contains defined symbols, i.e.\ is not a code thm. Hence a separate
815   constant with its own code thms needs to be introduced for @{text
816   F}. See the image operator below.
817 *}
820 subsection {* Image *}
822 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
823   "image_mset f = fold (plus o single o f) {#}"
825 lemma comp_fun_commute_mset_image:
826   "comp_fun_commute (plus o single o f)"
827 proof
830 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
831   by (simp add: image_mset_def)
833 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
834 proof -
835   interpret comp_fun_commute "plus o single o f"
836     by (fact comp_fun_commute_mset_image)
837   show ?thesis by (simp add: image_mset_def)
838 qed
840 lemma image_mset_union [simp]:
841   "image_mset f (M + N) = image_mset f M + image_mset f N"
842 proof -
843   interpret comp_fun_commute "plus o single o f"
844     by (fact comp_fun_commute_mset_image)
845   show ?thesis by (induct N) (simp_all add: image_mset_def add_ac)
846 qed
848 corollary image_mset_insert:
849   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
850   by simp
852 lemma set_of_image_mset [simp]:
853   "set_of (image_mset f M) = image f (set_of M)"
854   by (induct M) simp_all
856 lemma size_image_mset [simp]:
857   "size (image_mset f M) = size M"
858   by (induct M) simp_all
860 lemma image_mset_is_empty_iff [simp]:
861   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
862   by (cases M) auto
864 syntax
865   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
866       ("({#_/. _ :# _#})")
867 translations
868   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
870 syntax
871   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
872       ("({#_/ | _ :# _./ _#})")
873 translations
874   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
876 text {*
877   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
878   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
879   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
880   @{term "{#x+x|x:#M. x<c#}"}.
881 *}
883 enriched_type image_mset: image_mset
884 proof -
885   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
886   proof
887     fix A
888     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
889       by (induct A) simp_all
890   qed
891   show "image_mset id = id"
892   proof
893     fix A
894     show "image_mset id A = id A"
895       by (induct A) simp_all
896   qed
897 qed
899 declare image_mset.identity [simp]
902 subsection {* Further conversions *}
904 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
905   "multiset_of [] = {#}" |
906   "multiset_of (a # x) = multiset_of x + {# a #}"
908 lemma in_multiset_in_set:
909   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
910   by (induct xs) simp_all
912 lemma count_multiset_of:
913   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
914   by (induct xs) simp_all
916 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
917 by (induct x) auto
919 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
920 by (induct x) auto
922 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
923 by (induct x) auto
925 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
926 by (induct xs) auto
928 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
929   by (induct xs) simp_all
931 lemma multiset_of_append [simp]:
932   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
933   by (induct xs arbitrary: ys) (auto simp: add_ac)
935 lemma multiset_of_filter:
936   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
937   by (induct xs) simp_all
939 lemma multiset_of_rev [simp]:
940   "multiset_of (rev xs) = multiset_of xs"
941   by (induct xs) simp_all
943 lemma surj_multiset_of: "surj multiset_of"
944 apply (unfold surj_def)
945 apply (rule allI)
946 apply (rule_tac M = y in multiset_induct)
947  apply auto
948 apply (rule_tac x = "x # xa" in exI)
949 apply auto
950 done
952 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
953 by (induct x) auto
955 lemma distinct_count_atmost_1:
956   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
957 apply (induct x, simp, rule iffI, simp_all)
958 apply (rule conjI)
959 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
960 apply (erule_tac x = a in allE, simp, clarify)
961 apply (erule_tac x = aa in allE, simp)
962 done
964 lemma multiset_of_eq_setD:
965   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
966 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
968 lemma set_eq_iff_multiset_of_eq_distinct:
969   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
970     (set x = set y) = (multiset_of x = multiset_of y)"
971 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
973 lemma set_eq_iff_multiset_of_remdups_eq:
974    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
975 apply (rule iffI)
976 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
977 apply (drule distinct_remdups [THEN distinct_remdups
978       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
979 apply simp
980 done
982 lemma multiset_of_compl_union [simp]:
983   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
984   by (induct xs) (auto simp: add_ac)
986 lemma count_multiset_of_length_filter:
987   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
988   by (induct xs) auto
990 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
991 apply (induct ls arbitrary: i)
992  apply simp
993 apply (case_tac i)
994  apply auto
995 done
997 lemma multiset_of_remove1[simp]:
998   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
999 by (induct xs) (auto simp add: multiset_eq_iff)
1001 lemma multiset_of_eq_length:
1002   assumes "multiset_of xs = multiset_of ys"
1003   shows "length xs = length ys"
1004   using assms by (metis size_multiset_of)
1006 lemma multiset_of_eq_length_filter:
1007   assumes "multiset_of xs = multiset_of ys"
1008   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
1009   using assms by (metis count_multiset_of)
1011 lemma fold_multiset_equiv:
1012   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"
1013     and equiv: "multiset_of xs = multiset_of ys"
1014   shows "List.fold f xs = List.fold f ys"
1015 using f equiv [symmetric]
1016 proof (induct xs arbitrary: ys)
1017   case Nil then show ?case by simp
1018 next
1019   case (Cons x xs)
1020   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
1021   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
1022     by (rule Cons.prems(1)) (simp_all add: *)
1023   moreover from * have "x \<in> set ys" by simp
1024   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
1025   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
1026   ultimately show ?case by simp
1027 qed
1029 lemma multiset_of_insort [simp]:
1030   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
1031   by (induct xs) (simp_all add: ac_simps)
1033 lemma in_multiset_of:
1034   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
1035   by (induct xs) simp_all
1037 lemma multiset_of_map:
1038   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
1039   by (induct xs) simp_all
1041 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
1042 where
1043   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
1045 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
1046 where
1047   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
1048 proof -
1049   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
1050   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
1051   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
1052 qed
1054 lemma count_multiset_of_set [simp]:
1055   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
1056   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
1057   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
1058 proof -
1059   { fix A
1060     assume "x \<notin> A"
1061     have "count (multiset_of_set A) x = 0"
1062     proof (cases "finite A")
1063       case False then show ?thesis by simp
1064     next
1065       case True from True `x \<notin> A` show ?thesis by (induct A) auto
1066     qed
1067   } note * = this
1068   then show "PROP ?P" "PROP ?Q" "PROP ?R"
1069   by (auto elim!: Set.set_insert)
1070 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
1072 context linorder
1073 begin
1075 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
1076 where
1077   "sorted_list_of_multiset M = fold insort [] M"
1079 lemma sorted_list_of_multiset_empty [simp]:
1080   "sorted_list_of_multiset {#} = []"
1081   by (simp add: sorted_list_of_multiset_def)
1083 lemma sorted_list_of_multiset_singleton [simp]:
1084   "sorted_list_of_multiset {#x#} = [x]"
1085 proof -
1086   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1087   show ?thesis by (simp add: sorted_list_of_multiset_def)
1088 qed
1090 lemma sorted_list_of_multiset_insert [simp]:
1091   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
1092 proof -
1093   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1094   show ?thesis by (simp add: sorted_list_of_multiset_def)
1095 qed
1097 end
1099 lemma multiset_of_sorted_list_of_multiset [simp]:
1100   "multiset_of (sorted_list_of_multiset M) = M"
1101   by (induct M) simp_all
1103 lemma sorted_list_of_multiset_multiset_of [simp]:
1104   "sorted_list_of_multiset (multiset_of xs) = sort xs"
1105   by (induct xs) simp_all
1107 lemma finite_set_of_multiset_of_set:
1108   assumes "finite A"
1109   shows "set_of (multiset_of_set A) = A"
1110   using assms by (induct A) simp_all
1112 lemma infinite_set_of_multiset_of_set:
1113   assumes "\<not> finite A"
1114   shows "set_of (multiset_of_set A) = {}"
1115   using assms by simp
1117 lemma set_sorted_list_of_multiset [simp]:
1118   "set (sorted_list_of_multiset M) = set_of M"
1119   by (induct M) (simp_all add: set_insort)
1121 lemma sorted_list_of_multiset_of_set [simp]:
1122   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
1123   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
1126 subsection {* Big operators *}
1128 no_notation times (infixl "*" 70)
1129 no_notation Groups.one ("1")
1131 locale comm_monoid_mset = comm_monoid
1132 begin
1134 definition F :: "'a multiset \<Rightarrow> 'a"
1135 where
1136   eq_fold: "F M = Multiset.fold f 1 M"
1138 lemma empty [simp]:
1139   "F {#} = 1"
1140   by (simp add: eq_fold)
1142 lemma singleton [simp]:
1143   "F {#x#} = x"
1144 proof -
1145   interpret comp_fun_commute
1146     by default (simp add: fun_eq_iff left_commute)
1147   show ?thesis by (simp add: eq_fold)
1148 qed
1150 lemma union [simp]:
1151   "F (M + N) = F M * F N"
1152 proof -
1153   interpret comp_fun_commute f
1154     by default (simp add: fun_eq_iff left_commute)
1155   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
1156 qed
1158 end
1160 notation times (infixl "*" 70)
1161 notation Groups.one ("1")
1164 begin
1166 definition msetsum :: "'a multiset \<Rightarrow> 'a"
1167 where
1168   "msetsum = comm_monoid_mset.F plus 0"
1170 sublocale msetsum!: comm_monoid_mset plus 0
1171 where
1172   "comm_monoid_mset.F plus 0 = msetsum"
1173 proof -
1174   show "comm_monoid_mset plus 0" ..
1175   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
1176 qed
1178 lemma setsum_unfold_msetsum:
1179   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
1180   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1182 abbreviation msetsum_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1183 where
1184   "msetsum_image f M \<equiv> msetsum (image_mset f M)"
1186 end
1188 syntax
1189   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1190       ("(3SUM _:#_. _)" [0, 51, 10] 10)
1192 syntax (xsymbols)
1193   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1194       ("(3\<Sum>_:#_. _)" [0, 51, 10] 10)
1196 syntax (HTML output)
1197   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1198       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1200 translations
1201   "SUM i :# A. b" == "CONST msetsum_image (\<lambda>i. b) A"
1203 context comm_monoid_mult
1204 begin
1206 definition msetprod :: "'a multiset \<Rightarrow> 'a"
1207 where
1208   "msetprod = comm_monoid_mset.F times 1"
1210 sublocale msetprod!: comm_monoid_mset times 1
1211 where
1212   "comm_monoid_mset.F times 1 = msetprod"
1213 proof -
1214   show "comm_monoid_mset times 1" ..
1215   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
1216 qed
1218 lemma msetprod_empty:
1219   "msetprod {#} = 1"
1220   by (fact msetprod.empty)
1222 lemma msetprod_singleton:
1223   "msetprod {#x#} = x"
1224   by (fact msetprod.singleton)
1226 lemma msetprod_Un:
1227   "msetprod (A + B) = msetprod A * msetprod B"
1228   by (fact msetprod.union)
1230 lemma setprod_unfold_msetprod:
1231   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
1232   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1234 lemma msetprod_multiplicity:
1235   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
1236   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
1238 abbreviation msetprod_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1239 where
1240   "msetprod_image f M \<equiv> msetprod (image_mset f M)"
1242 end
1244 syntax
1245   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1246       ("(3PROD _:#_. _)" [0, 51, 10] 10)
1248 syntax (xsymbols)
1249   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1250       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1252 syntax (HTML output)
1253   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1254       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1256 translations
1257   "PROD i :# A. b" == "CONST msetprod_image (\<lambda>i. b) A"
1259 lemma (in comm_semiring_1) dvd_msetprod:
1260   assumes "x \<in># A"
1261   shows "x dvd msetprod A"
1262 proof -
1263   from assms have "A = (A - {#x#}) + {#x#}" by simp
1264   then obtain B where "A = B + {#x#}" ..
1265   then show ?thesis by simp
1266 qed
1269 subsection {* Cardinality *}
1271 definition mcard :: "'a multiset \<Rightarrow> nat"
1272 where
1273   "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
1275 lemma mcard_empty [simp]:
1276   "mcard {#} = 0"
1277   by (simp add: mcard_def)
1279 lemma mcard_singleton [simp]:
1280   "mcard {#a#} = Suc 0"
1281   by (simp add: mcard_def)
1283 lemma mcard_plus [simp]:
1284   "mcard (M + N) = mcard M + mcard N"
1285   by (simp add: mcard_def)
1287 lemma mcard_empty_iff [simp]:
1288   "mcard M = 0 \<longleftrightarrow> M = {#}"
1289   by (induct M) simp_all
1291 lemma mcard_unfold_setsum:
1292   "mcard M = setsum (count M) (set_of M)"
1293 proof (induct M)
1294   case empty then show ?case by simp
1295 next
1296   case (add M x) then show ?case
1297     by (cases "x \<in> set_of M")
1298       (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
1299 qed
1301 lemma size_eq_mcard:
1302   "size = mcard"
1303   by (simp add: fun_eq_iff size_def mcard_unfold_setsum)
1305 lemma mcard_multiset_of:
1306   "mcard (multiset_of xs) = length xs"
1307   by (induct xs) simp_all
1310 subsection {* Alternative representations *}
1312 subsubsection {* Lists *}
1314 context linorder
1315 begin
1317 lemma multiset_of_insort [simp]:
1318   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
1319   by (induct xs) (simp_all add: ac_simps)
1321 lemma multiset_of_sort [simp]:
1322   "multiset_of (sort_key k xs) = multiset_of xs"
1323   by (induct xs) (simp_all add: ac_simps)
1325 text {*
1326   This lemma shows which properties suffice to show that a function
1327   @{text "f"} with @{text "f xs = ys"} behaves like sort.
1328 *}
1330 lemma properties_for_sort_key:
1331   assumes "multiset_of ys = multiset_of xs"
1332   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
1333   and "sorted (map f ys)"
1334   shows "sort_key f xs = ys"
1335 using assms
1336 proof (induct xs arbitrary: ys)
1337   case Nil then show ?case by simp
1338 next
1339   case (Cons x xs)
1340   from Cons.prems(2) have
1341     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
1342     by (simp add: filter_remove1)
1343   with Cons.prems have "sort_key f xs = remove1 x ys"
1344     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
1345   moreover from Cons.prems have "x \<in> set ys"
1346     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
1347   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
1348 qed
1350 lemma properties_for_sort:
1351   assumes multiset: "multiset_of ys = multiset_of xs"
1352   and "sorted ys"
1353   shows "sort xs = ys"
1354 proof (rule properties_for_sort_key)
1355   from multiset show "multiset_of ys = multiset_of xs" .
1356   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
1357   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
1358     by (rule multiset_of_eq_length_filter)
1359   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
1360     by simp
1361   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
1362     by (simp add: replicate_length_filter)
1363 qed
1365 lemma sort_key_by_quicksort:
1366   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
1367     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
1368     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
1369 proof (rule properties_for_sort_key)
1370   show "multiset_of ?rhs = multiset_of ?lhs"
1371     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
1372 next
1373   show "sorted (map f ?rhs)"
1374     by (auto simp add: sorted_append intro: sorted_map_same)
1375 next
1376   fix l
1377   assume "l \<in> set ?rhs"
1378   let ?pivot = "f (xs ! (length xs div 2))"
1379   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
1380   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
1381     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
1382   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
1383   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
1384   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
1385     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
1386   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
1387   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
1388   proof (cases "f l" ?pivot rule: linorder_cases)
1389     case less
1390     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
1391     with less show ?thesis
1392       by (simp add: filter_sort [symmetric] ** ***)
1393   next
1394     case equal then show ?thesis
1395       by (simp add: * less_le)
1396   next
1397     case greater
1398     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
1399     with greater show ?thesis
1400       by (simp add: filter_sort [symmetric] ** ***)
1401   qed
1402 qed
1404 lemma sort_by_quicksort:
1405   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
1406     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
1407     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
1408   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
1410 text {* A stable parametrized quicksort *}
1412 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
1413   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
1415 lemma part_code [code]:
1416   "part f pivot [] = ([], [], [])"
1417   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
1418      if x' < pivot then (x # lts, eqs, gts)
1419      else if x' > pivot then (lts, eqs, x # gts)
1420      else (lts, x # eqs, gts))"
1421   by (auto simp add: part_def Let_def split_def)
1423 lemma sort_key_by_quicksort_code [code]:
1424   "sort_key f xs = (case xs of [] \<Rightarrow> []
1425     | [x] \<Rightarrow> xs
1426     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
1427     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1428        in sort_key f lts @ eqs @ sort_key f gts))"
1429 proof (cases xs)
1430   case Nil then show ?thesis by simp
1431 next
1432   case (Cons _ ys) note hyps = Cons show ?thesis
1433   proof (cases ys)
1434     case Nil with hyps show ?thesis by simp
1435   next
1436     case (Cons _ zs) note hyps = hyps Cons show ?thesis
1437     proof (cases zs)
1438       case Nil with hyps show ?thesis by auto
1439     next
1440       case Cons
1441       from sort_key_by_quicksort [of f xs]
1442       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1443         in sort_key f lts @ eqs @ sort_key f gts)"
1444       by (simp only: split_def Let_def part_def fst_conv snd_conv)
1445       with hyps Cons show ?thesis by (simp only: list.cases)
1446     qed
1447   qed
1448 qed
1450 end
1452 hide_const (open) part
1454 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
1455   by (induct xs) (auto intro: order_trans)
1457 lemma multiset_of_update:
1458   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
1459 proof (induct ls arbitrary: i)
1460   case Nil then show ?case by simp
1461 next
1462   case (Cons x xs)
1463   show ?case
1464   proof (cases i)
1465     case 0 then show ?thesis by simp
1466   next
1467     case (Suc i')
1468     with Cons show ?thesis
1469       apply simp
1470       apply (subst add_assoc)
1471       apply (subst add_commute [of "{#v#}" "{#x#}"])
1472       apply (subst add_assoc [symmetric])
1473       apply simp
1474       apply (rule mset_le_multiset_union_diff_commute)
1475       apply (simp add: mset_le_single nth_mem_multiset_of)
1476       done
1477   qed
1478 qed
1480 lemma multiset_of_swap:
1481   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
1482     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
1483   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
1486 subsection {* The multiset order *}
1488 subsubsection {* Well-foundedness *}
1490 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1491   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1492       (\<forall>b. b :# K --> (b, a) \<in> r)}"
1494 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1495   "mult r = (mult1 r)\<^sup>+"
1497 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1498 by (simp add: mult1_def)
1500 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
1501     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1502     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
1503   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
1504 proof (unfold mult1_def)
1505   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
1506   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1507   let ?case1 = "?case1 {(N, M). ?R N M}"
1509   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
1510   then have "\<exists>a' M0' K.
1511       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
1512   then show "?case1 \<or> ?case2"
1513   proof (elim exE conjE)
1514     fix a' M0' K
1515     assume N: "N = M0' + K" and r: "?r K a'"
1516     assume "M0 + {#a#} = M0' + {#a'#}"
1517     then have "M0 = M0' \<and> a = a' \<or>
1518         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
1519       by (simp only: add_eq_conv_ex)
1520     then show ?thesis
1521     proof (elim disjE conjE exE)
1522       assume "M0 = M0'" "a = a'"
1523       with N r have "?r K a \<and> N = M0 + K" by simp
1524       then have ?case2 .. then show ?thesis ..
1525     next
1526       fix K'
1527       assume "M0' = K' + {#a#}"
1528       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
1530       assume "M0 = K' + {#a'#}"
1531       with r have "?R (K' + K) M0" by blast
1532       with n have ?case1 by simp then show ?thesis ..
1533     qed
1534   qed
1535 qed
1537 lemma all_accessible: "wf r ==> \<forall>M. M \<in> Wellfounded.acc (mult1 r)"
1538 proof
1539   let ?R = "mult1 r"
1540   let ?W = "Wellfounded.acc ?R"
1541   {
1542     fix M M0 a
1543     assume M0: "M0 \<in> ?W"
1544       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1545       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
1546     have "M0 + {#a#} \<in> ?W"
1547     proof (rule accI [of "M0 + {#a#}"])
1548       fix N
1549       assume "(N, M0 + {#a#}) \<in> ?R"
1550       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
1551           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
1552         by (rule less_add)
1553       then show "N \<in> ?W"
1554       proof (elim exE disjE conjE)
1555         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
1556         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
1557         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
1558         then show "N \<in> ?W" by (simp only: N)
1559       next
1560         fix K
1561         assume N: "N = M0 + K"
1562         assume "\<forall>b. b :# K --> (b, a) \<in> r"
1563         then have "M0 + K \<in> ?W"
1564         proof (induct K)
1565           case empty
1566           from M0 show "M0 + {#} \<in> ?W" by simp
1567         next
1568           case (add K x)
1569           from add.prems have "(x, a) \<in> r" by simp
1570           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
1571           moreover from add have "M0 + K \<in> ?W" by simp
1572           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
1573           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
1574         qed
1575         then show "N \<in> ?W" by (simp only: N)
1576       qed
1577     qed
1578   } note tedious_reasoning = this
1580   assume wf: "wf r"
1581   fix M
1582   show "M \<in> ?W"
1583   proof (induct M)
1584     show "{#} \<in> ?W"
1585     proof (rule accI)
1586       fix b assume "(b, {#}) \<in> ?R"
1587       with not_less_empty show "b \<in> ?W" by contradiction
1588     qed
1590     fix M a assume "M \<in> ?W"
1591     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1592     proof induct
1593       fix a
1594       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1595       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1596       proof
1597         fix M assume "M \<in> ?W"
1598         then show "M + {#a#} \<in> ?W"
1599           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
1600       qed
1601     qed
1602     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
1603   qed
1604 qed
1606 theorem wf_mult1: "wf r ==> wf (mult1 r)"
1607 by (rule acc_wfI) (rule all_accessible)
1609 theorem wf_mult: "wf r ==> wf (mult r)"
1610 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
1613 subsubsection {* Closure-free presentation *}
1615 text {* One direction. *}
1617 lemma mult_implies_one_step:
1618   "trans r ==> (M, N) \<in> mult r ==>
1619     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
1620     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
1621 apply (unfold mult_def mult1_def set_of_def)
1622 apply (erule converse_trancl_induct, clarify)
1623  apply (rule_tac x = M0 in exI, simp, clarify)
1624 apply (case_tac "a :# K")
1625  apply (rule_tac x = I in exI)
1626  apply (simp (no_asm))
1627  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1628  apply (simp (no_asm_simp) add: add_assoc [symmetric])
1629  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
1630  apply (simp add: diff_union_single_conv)
1631  apply (simp (no_asm_use) add: trans_def)
1632  apply blast
1633 apply (subgoal_tac "a :# I")
1634  apply (rule_tac x = "I - {#a#}" in exI)
1635  apply (rule_tac x = "J + {#a#}" in exI)
1636  apply (rule_tac x = "K + Ka" in exI)
1637  apply (rule conjI)
1638   apply (simp add: multiset_eq_iff split: nat_diff_split)
1639  apply (rule conjI)
1640   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
1641   apply (simp add: multiset_eq_iff split: nat_diff_split)
1642  apply (simp (no_asm_use) add: trans_def)
1643  apply blast
1644 apply (subgoal_tac "a :# (M0 + {#a#})")
1645  apply simp
1646 apply (simp (no_asm))
1647 done
1649 lemma one_step_implies_mult_aux:
1650   "trans r ==>
1651     \<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))
1652       --> (I + K, I + J) \<in> mult r"
1653 apply (induct_tac n, auto)
1654 apply (frule size_eq_Suc_imp_eq_union, clarify)
1655 apply (rename_tac "J'", simp)
1656 apply (erule notE, auto)
1657 apply (case_tac "J' = {#}")
1658  apply (simp add: mult_def)
1659  apply (rule r_into_trancl)
1660  apply (simp add: mult1_def set_of_def, blast)
1661 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
1662 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
1663 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
1664 apply (erule ssubst)
1665 apply (simp add: Ball_def, auto)
1666 apply (subgoal_tac
1667   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
1668     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
1669  prefer 2
1670  apply force
1671 apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
1672 apply (erule trancl_trans)
1673 apply (rule r_into_trancl)
1674 apply (simp add: mult1_def set_of_def)
1675 apply (rule_tac x = a in exI)
1676 apply (rule_tac x = "I + J'" in exI)
1678 done
1680 lemma one_step_implies_mult:
1681   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
1682     ==> (I + K, I + J) \<in> mult r"
1683 using one_step_implies_mult_aux by blast
1686 subsubsection {* Partial-order properties *}
1688 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
1689   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
1691 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
1692   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
1694 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
1695 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
1697 interpretation multiset_order: order le_multiset less_multiset
1698 proof -
1699   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
1700   proof
1701     fix M :: "'a multiset"
1702     assume "M \<subset># M"
1703     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
1704     have "trans {(x'::'a, x). x' < x}"
1705       by (rule transI) simp
1706     moreover note MM
1707     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
1708       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
1709       by (rule mult_implies_one_step)
1710     then obtain I J K where "M = I + J" and "M = I + K"
1711       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
1712     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
1713     have "finite (set_of K)" by simp
1714     moreover note aux2
1715     ultimately have "set_of K = {}"
1716       by (induct rule: finite_induct) (auto intro: order_less_trans)
1717     with aux1 show False by simp
1718   qed
1719   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
1720     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
1721   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
1722     by default (auto simp add: le_multiset_def irrefl dest: trans)
1723 qed
1725 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
1726   by simp
1729 subsubsection {* Monotonicity of multiset union *}
1731 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
1732 apply (unfold mult1_def)
1733 apply auto
1734 apply (rule_tac x = a in exI)
1735 apply (rule_tac x = "C + M0" in exI)
1737 done
1739 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
1740 apply (unfold less_multiset_def mult_def)
1741 apply (erule trancl_induct)
1742  apply (blast intro: mult1_union)
1743 apply (blast intro: mult1_union trancl_trans)
1744 done
1746 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
1747 apply (subst add_commute [of B C])
1748 apply (subst add_commute [of D C])
1749 apply (erule union_less_mono2)
1750 done
1752 lemma union_less_mono:
1753   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
1754   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
1756 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
1757 proof
1758 qed (auto simp add: le_multiset_def intro: union_less_mono2)
1761 subsection {* Termination proofs with multiset orders *}
1763 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
1764   and multi_member_this: "x \<in># {# x #} + XS"
1765   and multi_member_last: "x \<in># {# x #}"
1766   by auto
1768 definition "ms_strict = mult pair_less"
1769 definition "ms_weak = ms_strict \<union> Id"
1771 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
1772 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
1773 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
1775 lemma smsI:
1776   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
1777   unfolding ms_strict_def
1778 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
1780 lemma wmsI:
1781   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
1782   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
1783 unfolding ms_weak_def ms_strict_def
1784 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
1786 inductive pw_leq
1787 where
1788   pw_leq_empty: "pw_leq {#} {#}"
1789 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
1791 lemma pw_leq_lstep:
1792   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
1793 by (drule pw_leq_step) (rule pw_leq_empty, simp)
1795 lemma pw_leq_split:
1796   assumes "pw_leq X Y"
1797   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 = {#}))"
1798   using assms
1799 proof (induct)
1800   case pw_leq_empty thus ?case by auto
1801 next
1802   case (pw_leq_step x y X Y)
1803   then obtain A B Z where
1804     [simp]: "X = A + Z" "Y = B + Z"
1805       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
1806     by auto
1807   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
1808     unfolding pair_leq_def by auto
1809   thus ?case
1810   proof
1811     assume [simp]: "x = y"
1812     have
1813       "{#x#} + X = A + ({#y#}+Z)
1814       \<and> {#y#} + Y = B + ({#y#}+Z)
1815       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1816       by (auto simp: add_ac)
1817     thus ?case by (intro exI)
1818   next
1819     assume A: "(x, y) \<in> pair_less"
1820     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
1821     have "{#x#} + X = ?A' + Z"
1822       "{#y#} + Y = ?B' + Z"
1824     moreover have
1825       "(set_of ?A', set_of ?B') \<in> max_strict"
1826       using 1 A unfolding max_strict_def
1827       by (auto elim!: max_ext.cases)
1828     ultimately show ?thesis by blast
1829   qed
1830 qed
1832 lemma
1833   assumes pwleq: "pw_leq Z Z'"
1834   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
1835   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
1836   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
1837 proof -
1838   from pw_leq_split[OF pwleq]
1839   obtain A' B' Z''
1840     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
1841     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
1842     by blast
1843   {
1844     assume max: "(set_of A, set_of B) \<in> max_strict"
1845     from mx_or_empty
1846     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
1847     proof
1848       assume max': "(set_of A', set_of B') \<in> max_strict"
1849       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
1850         by (auto simp: max_strict_def intro: max_ext_additive)
1851       thus ?thesis by (rule smsI)
1852     next
1853       assume [simp]: "A' = {#} \<and> B' = {#}"
1854       show ?thesis by (rule smsI) (auto intro: max)
1855     qed
1856     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
1857     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
1858   }
1859   from mx_or_empty
1860   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
1861   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
1862 qed
1864 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
1865 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
1866 and nonempty_single: "{# x #} \<noteq> {#}"
1867 by auto
1869 setup {*
1870 let
1871   fun msetT T = Type (@{type_name multiset}, [T]);
1873   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
1874     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
1875     | mk_mset T (x :: xs) =
1876           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
1877                 mk_mset T [x] \$ mk_mset T xs
1879   fun mset_member_tac m i =
1880       (if m <= 0 then
1881            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
1882        else
1883            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
1885   val mset_nonempty_tac =
1886       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
1888   val regroup_munion_conv =
1889       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
1890         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))
1892   fun unfold_pwleq_tac i =
1893     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
1894       ORELSE (rtac @{thm pw_leq_lstep} i)
1895       ORELSE (rtac @{thm pw_leq_empty} i)
1897   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
1898                       @{thm Un_insert_left}, @{thm Un_empty_left}]
1899 in
1900   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
1901   {
1902     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
1903     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
1904     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
1905     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
1906     reduction_pair= @{thm ms_reduction_pair}
1907   })
1908 end
1909 *}
1912 subsection {* Legacy theorem bindings *}
1914 lemmas multi_count_eq = multiset_eq_iff [symmetric]
1916 lemma union_commute: "M + N = N + (M::'a multiset)"
1917   by (fact add_commute)
1919 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
1920   by (fact add_assoc)
1922 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
1923   by (fact add_left_commute)
1925 lemmas union_ac = union_assoc union_commute union_lcomm
1927 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
1928   by (fact add_right_cancel)
1930 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
1931   by (fact add_left_cancel)
1933 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
1934   by (fact add_imp_eq)
1936 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
1937   by (fact order_less_trans)
1939 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
1940   by (fact inf.commute)
1942 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
1943   by (fact inf.assoc [symmetric])
1945 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
1946   by (fact inf.left_commute)
1948 lemmas multiset_inter_ac =
1949   multiset_inter_commute
1950   multiset_inter_assoc
1951   multiset_inter_left_commute
1953 lemma mult_less_not_refl:
1954   "\<not> M \<subset># (M::'a::order multiset)"
1955   by (fact multiset_order.less_irrefl)
1957 lemma mult_less_trans:
1958   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
1959   by (fact multiset_order.less_trans)
1961 lemma mult_less_not_sym:
1962   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
1963   by (fact multiset_order.less_not_sym)
1965 lemma mult_less_asym:
1966   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
1967   by (fact multiset_order.less_asym)
1969 ML {*
1970 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
1971                       (Const _ \$ t') =
1972     let
1973       val (maybe_opt, ps) =
1974         Nitpick_Model.dest_plain_fun t' ||> op ~~
1975         ||> map (apsnd (snd o HOLogic.dest_number))
1976       fun elems_for t =
1977         case AList.lookup (op =) ps t of
1978           SOME n => replicate n t
1979         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
1980     in
1981       case maps elems_for (all_values elem_T) @
1982            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
1983             else []) of
1984         [] => Const (@{const_name zero_class.zero}, T)
1985       | ts => foldl1 (fn (t1, t2) =>
1986                          Const (@{const_name plus_class.plus}, T --> T --> T)
1987                          \$ t1 \$ t2)
1988                      (map (curry (op \$) (Const (@{const_name single},
1989                                                 elem_T --> T))) ts)
1990     end
1991   | multiset_postproc _ _ _ _ t = t
1992 *}
1994 declaration {*
1995 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
1996     multiset_postproc
1997 *}
1999 hide_const (open) fold
2002 subsection {* Naive implementation using lists *}
2004 code_datatype multiset_of
2006 lemma [code]:
2007   "{#} = multiset_of []"
2008   by simp
2010 lemma [code]:
2011   "{#x#} = multiset_of [x]"
2012   by simp
2014 lemma union_code [code]:
2015   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
2016   by simp
2018 lemma [code]:
2019   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
2020   by (simp add: multiset_of_map)
2022 lemma [code]:
2023   "Multiset.filter f (multiset_of xs) = multiset_of (filter f xs)"
2024   by (simp add: multiset_of_filter)
2026 lemma [code]:
2027   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
2028   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
2030 lemma [code]:
2031   "multiset_of xs #\<inter> multiset_of ys =
2032     multiset_of (snd (fold (\<lambda>x (ys, zs).
2033       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
2034 proof -
2035   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
2036     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
2037       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
2038     by (induct xs arbitrary: ys)
2040   then show ?thesis by simp
2041 qed
2043 lemma [code]:
2044   "multiset_of xs #\<union> multiset_of ys =
2045     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
2046 proof -
2047   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
2048       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
2049     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
2050   then show ?thesis by simp
2051 qed
2053 lemma [code_unfold]:
2054   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
2055   by (simp add: in_multiset_of)
2057 lemma [code]:
2058   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
2059 proof -
2060   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
2061     by (induct xs) simp_all
2062   then show ?thesis by simp
2063 qed
2065 lemma [code]:
2066   "set_of (multiset_of xs) = set xs"
2067   by simp
2069 lemma [code]:
2070   "sorted_list_of_multiset (multiset_of xs) = sort xs"
2071   by (induct xs) simp_all
2073 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
2074   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
2075   apply (cases "finite A")
2076   apply simp_all
2077   apply (induct A rule: finite_induct)
2078   apply (simp_all add: union_commute)
2079   done
2081 lemma [code]:
2082   "mcard (multiset_of xs) = length xs"
2083   by (simp add: mcard_multiset_of)
2085 lemma [code]:
2086   "A \<le> B \<longleftrightarrow> A #\<inter> B = A"
2087   by (auto simp add: inf.order_iff)
2089 instantiation multiset :: (equal) equal
2090 begin
2092 definition
2093   [code]: "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"
2095 instance
2096   by default (simp add: equal_multiset_def eq_iff)
2098 end
2100 lemma [code]:
2101   "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
2102   by auto
2104 lemma [code]:
2105   "msetsum (multiset_of xs) = listsum xs"
2106   by (induct xs) (simp_all add: add.commute)
2108 lemma [code]:
2109   "msetprod (multiset_of xs) = fold times xs 1"
2110 proof -
2111   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
2112     by (induct xs) (simp_all add: mult.assoc)
2113   then show ?thesis by simp
2114 qed
2116 lemma [code]:
2117   "size = mcard"
2118   by (fact size_eq_mcard)
2120 text {*
2121   Exercise for the casual reader: add implementations for @{const le_multiset}
2122   and @{const less_multiset} (multiset order).
2123 *}
2125 text {* Quickcheck generators *}
2127 definition (in term_syntax)
2128   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
2129     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
2130   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
2132 notation fcomp (infixl "\<circ>>" 60)
2133 notation scomp (infixl "\<circ>\<rightarrow>" 60)
2135 instantiation multiset :: (random) random
2136 begin
2138 definition
2139   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
2141 instance ..
2143 end
2145 no_notation fcomp (infixl "\<circ>>" 60)
2146 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
2148 instantiation multiset :: (full_exhaustive) full_exhaustive
2149 begin
2151 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
2152 where
2153   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
2155 instance ..
2157 end
2159 hide_const (open) msetify
2162 subsection {* BNF setup *}
2164 lemma setsum_gt_0_iff:
2165 fixes f :: "'a \<Rightarrow> nat" assumes "finite A"
2166 shows "setsum f A > 0 \<longleftrightarrow> (\<exists> a \<in> A. f a > 0)"
2167 (is "?L \<longleftrightarrow> ?R")
2168 proof-
2169   have "?L \<longleftrightarrow> \<not> setsum f A = 0" by fast
2170   also have "... \<longleftrightarrow> (\<exists> a \<in> A. f a \<noteq> 0)" using assms by simp
2171   also have "... \<longleftrightarrow> ?R" by simp
2172   finally show ?thesis .
2173 qed
2175 lift_definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" is
2176   "\<lambda>h f b. setsum f {a. h a = b \<and> f a > 0} :: nat"
2177 unfolding multiset_def proof safe
2178   fix h :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat"
2179   assume fin: "finite {a. 0 < f a}"  (is "finite ?A")
2180   show "finite {b. 0 < setsum f {a. h a = b \<and> 0 < f a}}"
2181   (is "finite {b. 0 < setsum f (?As b)}")
2182   proof- let ?B = "{b. 0 < setsum f (?As b)}"
2183     have "\<And> b. finite (?As b)" using fin by simp
2184     hence B: "?B = {b. ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
2185     hence "?B \<subseteq> h ` ?A" by auto
2186     thus ?thesis using finite_surj[OF fin] by auto
2187   qed
2188 qed
2190 lemma mmap_id0: "mmap id = id"
2191 proof (intro ext multiset_eqI)
2192   fix f a show "count (mmap id f) a = count (id f) a"
2193   proof (cases "count f a = 0")
2194     case False
2195     hence 1: "{aa. aa = a \<and> aa \<in># f} = {a}" by auto
2196     thus ?thesis by transfer auto
2197   qed (transfer, simp)
2198 qed
2200 lemma inj_on_setsum_inv:
2201 assumes 1: "(0::nat) < setsum (count f) {a. h a = b' \<and> a \<in># f}" (is "0 < setsum (count f) ?A'")
2202 and     2: "{a. h a = b \<and> a \<in># f} = {a. h a = b' \<and> a \<in># f}" (is "?A = ?A'")
2203 shows "b = b'"
2204 using assms by (auto simp add: setsum_gt_0_iff)
2206 lemma mmap_comp:
2207 fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
2208 shows "mmap (h2 o h1) = mmap h2 o mmap h1"
2209 proof (intro ext multiset_eqI)
2210   fix f :: "'a multiset" fix c :: 'c
2211   let ?A = "{a. h2 (h1 a) = c \<and> a \<in># f}"
2212   let ?As = "\<lambda> b. {a. h1 a = b \<and> a \<in># f}"
2213   let ?B = "{b. h2 b = c \<and> 0 < setsum (count f) (?As b)}"
2214   have 0: "{?As b | b.  b \<in> ?B} = ?As ` ?B" by auto
2215   have "\<And> b. finite (?As b)" by transfer (simp add: multiset_def)
2216   hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
2217   hence A: "?A = \<Union> {?As b | b.  b \<in> ?B}" by auto
2218   have "setsum (count f) ?A = setsum (setsum (count f)) {?As b | b.  b \<in> ?B}"
2219     unfolding A by transfer (intro setsum_Union_disjoint, auto simp: multiset_def)
2220   also have "... = setsum (setsum (count f)) (?As ` ?B)" unfolding 0 ..
2221   also have "... = setsum (setsum (count f) o ?As) ?B"
2222     by(intro setsum_reindex) (auto simp add: setsum_gt_0_iff inj_on_def)
2223   also have "... = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" unfolding comp_def ..
2224   finally have "setsum (count f) ?A = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" .
2225   thus "count (mmap (h2 \<circ> h1) f) c = count ((mmap h2 \<circ> mmap h1) f) c"
2226     by transfer (unfold comp_apply, blast)
2227 qed
2229 lemma mmap_cong:
2230 assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
2231 shows "mmap f M = mmap g M"
2232 using assms by transfer (auto intro!: setsum_cong)
2234 context
2235 begin
2236 interpretation lifting_syntax .
2238 lemma set_of_transfer[transfer_rule]: "(pcr_multiset op = ===> op =) (\<lambda>f. {a. 0 < f a}) set_of"
2239   unfolding set_of_def pcr_multiset_def cr_multiset_def fun_rel_def by auto
2241 end
2243 lemma set_of_mmap: "set_of o mmap h = image h o set_of"
2244 proof (rule ext, unfold comp_apply)
2245   fix M show "set_of (mmap h M) = h ` set_of M"
2246     by transfer (auto simp add: multiset_def setsum_gt_0_iff)
2247 qed
2249 lemma multiset_of_surj:
2250   "multiset_of ` {as. set as \<subseteq> A} = {M. set_of M \<subseteq> A}"
2251 proof safe
2252   fix M assume M: "set_of M \<subseteq> A"
2253   obtain as where eq: "M = multiset_of as" using surj_multiset_of unfolding surj_def by auto
2254   hence "set as \<subseteq> A" using M by auto
2255   thus "M \<in> multiset_of ` {as. set as \<subseteq> A}" using eq by auto
2256 next
2257   show "\<And>x xa xb. \<lbrakk>set xa \<subseteq> A; xb \<in> set_of (multiset_of xa)\<rbrakk> \<Longrightarrow> xb \<in> A"
2258   by (erule set_mp) (unfold set_of_multiset_of)
2259 qed
2261 lemma card_of_set_of:
2262 "(card_of {M. set_of M \<subseteq> A}, card_of {as. set as \<subseteq> A}) \<in> ordLeq"
2263 apply(rule card_of_ordLeqI2[of _ multiset_of]) using multiset_of_surj by auto
2265 lemma nat_sum_induct:
2266 assumes "\<And>n1 n2. (\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow> phi m1 m2) \<Longrightarrow> phi n1 n2"
2267 shows "phi (n1::nat) (n2::nat)"
2268 proof-
2269   let ?chi = "\<lambda> n1n2 :: nat * nat. phi (fst n1n2) (snd n1n2)"
2270   have "?chi (n1,n2)"
2271   apply(induct rule: measure_induct[of "\<lambda> n1n2. fst n1n2 + snd n1n2" ?chi])
2272   using assms by (metis fstI sndI)
2273   thus ?thesis by simp
2274 qed
2276 lemma matrix_count:
2277 fixes ct1 ct2 :: "nat \<Rightarrow> nat"
2278 assumes "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
2279 shows
2280 "\<exists> ct. (\<forall> i1 \<le> n1. setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ct1 i1) \<and>
2281        (\<forall> i2 \<le> n2. setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ct2 i2)"
2282 (is "?phi ct1 ct2 n1 n2")
2283 proof-
2284   have "\<forall> ct1 ct2 :: nat \<Rightarrow> nat.
2285         setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"
2286   proof(induct rule: nat_sum_induct[of
2287 "\<lambda> n1 n2. \<forall> ct1 ct2 :: nat \<Rightarrow> nat.
2288      setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"],
2289       clarify)
2290   fix n1 n2 :: nat and ct1 ct2 :: "nat \<Rightarrow> nat"
2291   assume IH: "\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow>
2292                 \<forall> dt1 dt2 :: nat \<Rightarrow> nat.
2293                 setsum dt1 {..<Suc m1} = setsum dt2 {..<Suc m2} \<longrightarrow> ?phi dt1 dt2 m1 m2"
2294   and ss: "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
2295   show "?phi ct1 ct2 n1 n2"
2296   proof(cases n1)
2297     case 0 note n1 = 0
2298     show ?thesis
2299     proof(cases n2)
2300       case 0 note n2 = 0
2301       let ?ct = "\<lambda> i1 i2. ct2 0"
2302       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by simp
2303     next
2304       case (Suc m2) note n2 = Suc
2305       let ?ct = "\<lambda> i1 i2. ct2 i2"
2306       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
2307     qed
2308   next
2309     case (Suc m1) note n1 = Suc
2310     show ?thesis
2311     proof(cases n2)
2312       case 0 note n2 = 0
2313       let ?ct = "\<lambda> i1 i2. ct1 i1"
2314       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
2315     next
2316       case (Suc m2) note n2 = Suc
2317       show ?thesis
2318       proof(cases "ct1 n1 \<le> ct2 n2")
2319         case True
2320         def dt2 \<equiv> "\<lambda> i2. if i2 = n2 then ct2 i2 - ct1 n1 else ct2 i2"
2321         have "setsum ct1 {..<Suc m1} = setsum dt2 {..<Suc n2}"
2322         unfolding dt2_def using ss n1 True by auto
2323         hence "?phi ct1 dt2 m1 n2" using IH[of m1 n2] n1 by simp
2324         then obtain dt where
2325         1: "\<And> i1. i1 \<le> m1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc n2} = ct1 i1" and
2326         2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc m1} = dt2 i2" by auto
2327         let ?ct = "\<lambda> i1 i2. if i1 = n1 then (if i2 = n2 then ct1 n1 else 0)
2328                                        else dt i1 i2"
2329         show ?thesis apply(rule exI[of _ ?ct])
2330         using n1 n2 1 2 True unfolding dt2_def by simp
2331       next
2332         case False
2333         hence False: "ct2 n2 < ct1 n1" by simp
2334         def dt1 \<equiv> "\<lambda> i1. if i1 = n1 then ct1 i1 - ct2 n2 else ct1 i1"
2335         have "setsum dt1 {..<Suc n1} = setsum ct2 {..<Suc m2}"
2336         unfolding dt1_def using ss n2 False by auto
2337         hence "?phi dt1 ct2 n1 m2" using IH[of n1 m2] n2 by simp
2338         then obtain dt where
2339         1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc m2} = dt1 i1" and
2340         2: "\<And> i2. i2 \<le> m2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc n1} = ct2 i2" by force
2341         let ?ct = "\<lambda> i1 i2. if i2 = n2 then (if i1 = n1 then ct2 n2 else 0)
2342                                        else dt i1 i2"
2343         show ?thesis apply(rule exI[of _ ?ct])
2344         using n1 n2 1 2 False unfolding dt1_def by simp
2345       qed
2346     qed
2347   qed
2348   qed
2349   thus ?thesis using assms by auto
2350 qed
2352 definition
2353 "inj2 u B1 B2 \<equiv>
2354  \<forall> b1 b1' b2 b2'. {b1,b1'} \<subseteq> B1 \<and> {b2,b2'} \<subseteq> B2 \<and> u b1 b2 = u b1' b2'
2355                   \<longrightarrow> b1 = b1' \<and> b2 = b2'"
2357 lemma matrix_setsum_finite:
2358 assumes B1: "B1 \<noteq> {}" "finite B1" and B2: "B2 \<noteq> {}" "finite B2" and u: "inj2 u B1 B2"
2359 and ss: "setsum N1 B1 = setsum N2 B2"
2360 shows "\<exists> M :: 'a \<Rightarrow> nat.
2361             (\<forall> b1 \<in> B1. setsum (\<lambda> b2. M (u b1 b2)) B2 = N1 b1) \<and>
2362             (\<forall> b2 \<in> B2. setsum (\<lambda> b1. M (u b1 b2)) B1 = N2 b2)"
2363 proof-
2364   obtain n1 where "card B1 = Suc n1" using B1 by (metis card_insert finite.simps)
2365   then obtain e1 where e1: "bij_betw e1 {..<Suc n1} B1"
2366   using ex_bij_betw_finite_nat[OF B1(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
2367   hence e1_inj: "inj_on e1 {..<Suc n1}" and e1_surj: "e1 ` {..<Suc n1} = B1"
2368   unfolding bij_betw_def by auto
2369   def f1 \<equiv> "inv_into {..<Suc n1} e1"
2370   have f1: "bij_betw f1 B1 {..<Suc n1}"
2371   and f1e1[simp]: "\<And> i1. i1 < Suc n1 \<Longrightarrow> f1 (e1 i1) = i1"
2372   and e1f1[simp]: "\<And> b1. b1 \<in> B1 \<Longrightarrow> e1 (f1 b1) = b1" unfolding f1_def
2373   apply (metis bij_betw_inv_into e1, metis bij_betw_inv_into_left e1 lessThan_iff)
2374   by (metis e1_surj f_inv_into_f)
2375   (*  *)
2376   obtain n2 where "card B2 = Suc n2" using B2 by (metis card_insert finite.simps)
2377   then obtain e2 where e2: "bij_betw e2 {..<Suc n2} B2"
2378   using ex_bij_betw_finite_nat[OF B2(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
2379   hence e2_inj: "inj_on e2 {..<Suc n2}" and e2_surj: "e2 ` {..<Suc n2} = B2"
2380   unfolding bij_betw_def by auto
2381   def f2 \<equiv> "inv_into {..<Suc n2} e2"
2382   have f2: "bij_betw f2 B2 {..<Suc n2}"
2383   and f2e2[simp]: "\<And> i2. i2 < Suc n2 \<Longrightarrow> f2 (e2 i2) = i2"
2384   and e2f2[simp]: "\<And> b2. b2 \<in> B2 \<Longrightarrow> e2 (f2 b2) = b2" unfolding f2_def
2385   apply (metis bij_betw_inv_into e2, metis bij_betw_inv_into_left e2 lessThan_iff)
2386   by (metis e2_surj f_inv_into_f)
2387   (*  *)
2388   let ?ct1 = "N1 o e1"  let ?ct2 = "N2 o e2"
2389   have ss: "setsum ?ct1 {..<Suc n1} = setsum ?ct2 {..<Suc n2}"
2390   unfolding setsum_reindex[OF e1_inj, symmetric] setsum_reindex[OF e2_inj, symmetric]
2391   e1_surj e2_surj using ss .
2392   obtain ct where
2393   ct1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ?ct1 i1" and
2394   ct2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ?ct2 i2"
2395   using matrix_count[OF ss] by blast
2396   (*  *)
2397   def A \<equiv> "{u b1 b2 | b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"
2398   have "\<forall> a \<in> A. \<exists> b1b2 \<in> B1 <*> B2. u (fst b1b2) (snd b1b2) = a"
2399   unfolding A_def Ball_def mem_Collect_eq by auto
2400   then obtain h1h2 where h12:
2401   "\<And>a. a \<in> A \<Longrightarrow> u (fst (h1h2 a)) (snd (h1h2 a)) = a \<and> h1h2 a \<in> B1 <*> B2" by metis
2402   def h1 \<equiv> "fst o h1h2"  def h2 \<equiv> "snd o h1h2"
2403   have h12[simp]: "\<And>a. a \<in> A \<Longrightarrow> u (h1 a) (h2 a) = a"
2404                   "\<And> a. a \<in> A \<Longrightarrow> h1 a \<in> B1"  "\<And> a. a \<in> A \<Longrightarrow> h2 a \<in> B2"
2405   using h12 unfolding h1_def h2_def by force+
2406   {fix b1 b2 assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2"
2407    hence inA: "u b1 b2 \<in> A" unfolding A_def by auto
2408    hence "u b1 b2 = u (h1 (u b1 b2)) (h2 (u b1 b2))" by auto
2409    moreover have "h1 (u b1 b2) \<in> B1" "h2 (u b1 b2) \<in> B2" using inA by auto
2410    ultimately have "h1 (u b1 b2) = b1 \<and> h2 (u b1 b2) = b2"
2411    using u b1 b2 unfolding inj2_def by fastforce
2412   }
2413   hence h1[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h1 (u b1 b2) = b1" and
2414         h2[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h2 (u b1 b2) = b2" by auto
2415   def M \<equiv> "\<lambda> a. ct (f1 (h1 a)) (f2 (h2 a))"
2416   show ?thesis
2417   apply(rule exI[of _ M]) proof safe
2418     fix b1 assume b1: "b1 \<in> B1"
2419     hence f1b1: "f1 b1 \<le> n1" using f1 unfolding bij_betw_def
2420     by (metis image_eqI lessThan_iff less_Suc_eq_le)
2421     have "(\<Sum>b2\<in>B2. M (u b1 b2)) = (\<Sum>i2<Suc n2. ct (f1 b1) (f2 (e2 i2)))"
2422     unfolding e2_surj[symmetric] setsum_reindex[OF e2_inj]
2423     unfolding M_def comp_def apply(intro setsum_cong) apply force
2424     by (metis e2_surj b1 h1 h2 imageI)
2425     also have "... = N1 b1" using b1 ct1[OF f1b1] by simp
2426     finally show "(\<Sum>b2\<in>B2. M (u b1 b2)) = N1 b1" .
2427   next
2428     fix b2 assume b2: "b2 \<in> B2"
2429     hence f2b2: "f2 b2 \<le> n2" using f2 unfolding bij_betw_def
2430     by (metis image_eqI lessThan_iff less_Suc_eq_le)
2431     have "(\<Sum>b1\<in>B1. M (u b1 b2)) = (\<Sum>i1<Suc n1. ct (f1 (e1 i1)) (f2 b2))"
2432     unfolding e1_surj[symmetric] setsum_reindex[OF e1_inj]
2433     unfolding M_def comp_def apply(intro setsum_cong) apply force
2434     by (metis e1_surj b2 h1 h2 imageI)
2435     also have "... = N2 b2" using b2 ct2[OF f2b2] by simp
2436     finally show "(\<Sum>b1\<in>B1. M (u b1 b2)) = N2 b2" .
2437   qed
2438 qed
2440 lemma supp_vimage_mmap: "set_of M \<subseteq> f -` (set_of (mmap f M))"
2441   by transfer (auto simp: multiset_def setsum_gt_0_iff)
2443 lemma mmap_ge_0: "b \<in># mmap f M \<longleftrightarrow> (\<exists>a. a \<in># M \<and> f a = b)"
2444   by transfer (auto simp: multiset_def setsum_gt_0_iff)
2446 lemma finite_twosets:
2447 assumes "finite B1" and "finite B2"
2448 shows "finite {u b1 b2 |b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"  (is "finite ?A")
2449 proof-
2450   have A: "?A = (\<lambda> b1b2. u (fst b1b2) (snd b1b2)) ` (B1 <*> B2)" by force
2451   show ?thesis unfolding A using finite_cartesian_product[OF assms] by auto
2452 qed
2454 (* Weak pullbacks: *)
2455 definition wpull where
2456 "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow>
2457  (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow> (\<exists> a \<in> A. p1 a = b1 \<and> p2 a = b2))"
2459 (* Weak pseudo-pullbacks *)
2460 definition wppull where
2461 "wppull A B1 B2 f1 f2 e1 e2 p1 p2 \<longleftrightarrow>
2462  (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow>
2463            (\<exists> a \<in> A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2))"
2466 (* The pullback of sets *)
2467 definition thePull where
2468 "thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
2470 lemma wpull_thePull:
2471 "wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
2472 unfolding wpull_def thePull_def by auto
2474 lemma wppull_thePull:
2475 assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
2476 shows
2477 "\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
2478    j a' \<in> A \<and>
2479    e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
2480 (is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
2481 proof(rule bchoice[of ?A' ?phi], default)
2482   fix a' assume a': "a' \<in> ?A'"
2483   hence "fst a' \<in> B1" unfolding thePull_def by auto
2484   moreover
2485   from a' have "snd a' \<in> B2" unfolding thePull_def by auto
2486   moreover have "f1 (fst a') = f2 (snd a')"
2487   using a' unfolding csquare_def thePull_def by auto
2488   ultimately show "\<exists> ja'. ?phi a' ja'"
2489   using assms unfolding wppull_def by blast
2490 qed
2492 lemma wpull_wppull:
2493 assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
2494 1: "\<forall> a' \<in> A'. j a' \<in> A \<and> e1 (p1 (j a')) = e1 (p1' a') \<and> e2 (p2 (j a')) = e2 (p2' a')"
2495 shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
2496 unfolding wppull_def proof safe
2497   fix b1 b2
2498   assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
2499   then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
2500   using wp unfolding wpull_def by blast
2501   show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
2502   apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
2503 qed
2505 lemma wppull_fstOp_sndOp:
2506 shows "wppull (Collect (split (P OO Q))) (Collect (split P)) (Collect (split Q))
2507   snd fst fst snd (BNF_Def.fstOp P Q) (BNF_Def.sndOp P Q)"
2508 using pick_middlep unfolding wppull_def fstOp_def sndOp_def relcompp.simps by auto
2510 lemma wpull_mmap:
2511 fixes A :: "'a set" and B1 :: "'b1 set" and B2 :: "'b2 set"
2512 assumes wp: "wpull A B1 B2 f1 f2 p1 p2"
2513 shows
2514 "wpull {M. set_of M \<subseteq> A}
2515        {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
2516        (mmap f1) (mmap f2) (mmap p1) (mmap p2)"
2517 unfolding wpull_def proof (safe, unfold Bex_def mem_Collect_eq)
2518   fix N1 :: "'b1 multiset" and N2 :: "'b2 multiset"
2519   assume mmap': "mmap f1 N1 = mmap f2 N2"
2520   and N1[simp]: "set_of N1 \<subseteq> B1"
2521   and N2[simp]: "set_of N2 \<subseteq> B2"
2522   def P \<equiv> "mmap f1 N1"
2523   have P1: "P = mmap f1 N1" and P2: "P = mmap f2 N2" unfolding P_def using mmap' by auto
2524   note P = P1 P2
2525   have fin_N1[simp]: "finite (set_of N1)"
2526    and fin_N2[simp]: "finite (set_of N2)"
2527    and fin_P[simp]: "finite (set_of P)" by auto
2529   def set1 \<equiv> "\<lambda> c. {b1 \<in> set_of N1. f1 b1 = c}"
2530   have set1[simp]: "\<And> c b1. b1 \<in> set1 c \<Longrightarrow> f1 b1 = c" unfolding set1_def by auto
2531   have fin_set1: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set1 c)"
2532     using N1(1) unfolding set1_def multiset_def by auto
2533   have set1_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<noteq> {}"
2534    unfolding set1_def set_of_def P mmap_ge_0 by auto
2535   have supp_N1_set1: "set_of N1 = (\<Union> c \<in> set_of P. set1 c)"
2536     using supp_vimage_mmap[of N1 f1] unfolding set1_def P1 by auto
2537   hence set1_inclN1: "\<And>c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> set_of N1" by auto
2538   hence set1_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> B1" using N1 by blast
2539   have set1_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set1 c \<inter> set1 c' = {}"
2540     unfolding set1_def by auto
2541   have setsum_set1: "\<And> c. setsum (count N1) (set1 c) = count P c"
2542     unfolding P1 set1_def by transfer (auto intro: setsum_cong)
2544   def set2 \<equiv> "\<lambda> c. {b2 \<in> set_of N2. f2 b2 = c}"
2545   have set2[simp]: "\<And> c b2. b2 \<in> set2 c \<Longrightarrow> f2 b2 = c" unfolding set2_def by auto
2546   have fin_set2: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set2 c)"
2547   using N2(1) unfolding set2_def multiset_def by auto
2548   have set2_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<noteq> {}"
2549     unfolding set2_def P2 mmap_ge_0 set_of_def by auto
2550   have supp_N2_set2: "set_of N2 = (\<Union> c \<in> set_of P. set2 c)"
2551     using supp_vimage_mmap[of N2 f2] unfolding set2_def P2 by auto
2552   hence set2_inclN2: "\<And>c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> set_of N2" by auto
2553   hence set2_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> B2" using N2 by blast
2554   have set2_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set2 c \<inter> set2 c' = {}"
2555     unfolding set2_def by auto
2556   have setsum_set2: "\<And> c. setsum (count N2) (set2 c) = count P c"
2557     unfolding P2 set2_def by transfer (auto intro: setsum_cong)
2559   have ss: "\<And> c. c \<in> set_of P \<Longrightarrow> setsum (count N1) (set1 c) = setsum (count N2) (set2 c)"
2560     unfolding setsum_set1 setsum_set2 ..
2561   have "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
2562           \<exists> a \<in> A. p1 a = fst b1b2 \<and> p2 a = snd b1b2"
2563     using wp set1_incl set2_incl unfolding wpull_def Ball_def mem_Collect_eq
2564     by simp (metis set1 set2 set_rev_mp)
2565   then obtain uu where uu:
2566   "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
2567      uu c b1b2 \<in> A \<and> p1 (uu c b1b2) = fst b1b2 \<and> p2 (uu c b1b2) = snd b1b2" by metis
2568   def u \<equiv> "\<lambda> c b1 b2. uu c (b1,b2)"
2569   have u[simp]:
2570   "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> A"
2571   "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p1 (u c b1 b2) = b1"
2572   "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p2 (u c b1 b2) = b2"
2573     using uu unfolding u_def by auto
2574   {fix c assume c: "c \<in> set_of P"
2575    have "inj2 (u c) (set1 c) (set2 c)" unfolding inj2_def proof clarify
2576      fix b1 b1' b2 b2'
2577      assume "{b1, b1'} \<subseteq> set1 c" "{b2, b2'} \<subseteq> set2 c" and 0: "u c b1 b2 = u c b1' b2'"
2578      hence "p1 (u c b1 b2) = b1 \<and> p2 (u c b1 b2) = b2 \<and>
2579             p1 (u c b1' b2') = b1' \<and> p2 (u c b1' b2') = b2'"
2580      using u(2)[OF c] u(3)[OF c] by simp metis
2581      thus "b1 = b1' \<and> b2 = b2'" using 0 by auto
2582    qed
2583   } note inj = this
2584   def sset \<equiv> "\<lambda> c. {u c b1 b2 | b1 b2. b1 \<in> set1 c \<and> b2 \<in> set2 c}"
2585   have fin_sset[simp]: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (sset c)" unfolding sset_def
2586     using fin_set1 fin_set2 finite_twosets by blast
2587   have sset_A: "\<And> c. c \<in> set_of P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
2588   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
2589    then obtain b1 b2 where b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
2590    and a: "a = u c b1 b2" unfolding sset_def by auto
2591    have "p1 a \<in> set1 c" and p2a: "p2 a \<in> set2 c"
2592    using ac a b1 b2 c u(2) u(3) by simp+
2593    hence "u c (p1 a) (p2 a) = a" unfolding a using b1 b2 inj[OF c]
2594    unfolding inj2_def by (metis c u(2) u(3))
2595   } note u_p12[simp] = this
2596   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
2597    hence "p1 a \<in> set1 c" unfolding sset_def by auto
2598   }note p1[simp] = this
2599   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
2600    hence "p2 a \<in> set2 c" unfolding sset_def by auto
2601   }note p2[simp] = this
2603   {fix c assume c: "c \<in> set_of P"
2604    hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = count N1 b1) \<and>
2605                (\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = count N2 b2)"
2606    unfolding sset_def
2607    using matrix_setsum_finite[OF set1_NE[OF c] fin_set1[OF c]
2608                                  set2_NE[OF c] fin_set2[OF c] inj[OF c] ss[OF c]] by auto
2609   }
2610   then obtain Ms where
2611   ss1: "\<And> c b1. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
2612                    setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = count N1 b1" and
2613   ss2: "\<And> c b2. \<lbrakk>c \<in> set_of P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
2614                    setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = count N2 b2"
2615   by metis
2616   def SET \<equiv> "\<Union> c \<in> set_of P. sset c"
2617   have fin_SET[simp]: "finite SET" unfolding SET_def apply(rule finite_UN_I) by auto
2618   have SET_A: "SET \<subseteq> A" unfolding SET_def using sset_A by blast
2619   have u_SET[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> SET"
2620     unfolding SET_def sset_def by blast
2621   {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
2622    then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
2623     unfolding SET_def by auto
2624    hence "p1 a \<in> set1 c'" unfolding sset_def by auto
2625    hence eq: "c = c'" using p1a c c' set1_disj by auto
2626    hence "a \<in> sset c" using ac' by simp
2627   } note p1_rev = this
2628   {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
2629    then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
2630    unfolding SET_def by auto
2631    hence "p2 a \<in> set2 c'" unfolding sset_def by auto
2632    hence eq: "c = c'" using p2a c c' set2_disj by auto
2633    hence "a \<in> sset c" using ac' by simp
2634   } note p2_rev = this
2636   have "\<forall> a \<in> SET. \<exists> c \<in> set_of P. a \<in> sset c" unfolding SET_def by auto
2637   then obtain h where h: "\<forall> a \<in> SET. h a \<in> set_of P \<and> a \<in> sset (h a)" by metis
2638   have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
2639                       \<Longrightarrow> h (u c b1 b2) = c"
2640   by (metis h p2 set2 u(3) u_SET)
2641   have h_u1: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
2642                       \<Longrightarrow> h (u c b1 b2) = f1 b1"
2643   using h unfolding sset_def by auto
2644   have h_u2: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
2645                       \<Longrightarrow> h (u c b1 b2) = f2 b2"
2646   using h unfolding sset_def by auto
2647   def M \<equiv>
2648     "Abs_multiset (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0)"
2649   have "(\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) \<in> multiset"
2650     unfolding multiset_def by auto
2651   hence [transfer_rule]: "pcr_multiset op = (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) M"
2652     unfolding M_def pcr_multiset_def cr_multiset_def by (auto simp: Abs_multiset_inverse)
2653   have sM: "set_of M \<subseteq> SET" "set_of M \<subseteq> p1 -` (set_of N1)" "set_of M \<subseteq> p2 -` set_of N2"
2654     by (transfer, auto split: split_if_asm)+
2655   show "\<exists>M. set_of M \<subseteq> A \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
2656   proof(rule exI[of _ M], safe)
2657     fix a assume *: "a \<in> set_of M"
2658     from SET_A show "a \<in> A"
2659     proof (cases "a \<in> SET")
2660       case False thus ?thesis using * by transfer' auto
2661     qed blast
2662   next
2663     show "mmap p1 M = N1"
2664     proof(intro multiset_eqI)
2665       fix b1
2666       let ?K = "{a. p1 a = b1 \<and> a \<in># M}"
2667       have "setsum (count M) ?K = count N1 b1"
2668       proof(cases "b1 \<in> set_of N1")
2669         case False
2670         hence "?K = {}" using sM(2) by auto
2671         thus ?thesis using False by auto
2672       next
2673         case True
2674         def c \<equiv> "f1 b1"
2675         have c: "c \<in> set_of P" and b1: "b1 \<in> set1 c"
2676           unfolding set1_def c_def P1 using True by (auto simp: comp_eq_dest[OF set_of_mmap])
2677         with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p1 a = b1 \<and> a \<in> SET}"
2678           by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
2679         also have "... = setsum (count M) ((\<lambda> b2. u c b1 b2) ` (set2 c))"
2680           apply(rule setsum_cong) using c b1 proof safe
2681           fix a assume p1a: "p1 a \<in> set1 c" and "c \<in> set_of P" and "a \<in> SET"
2682           hence ac: "a \<in> sset c" using p1_rev by auto
2683           hence "a = u c (p1 a) (p2 a)" using c by auto
2684           moreover have "p2 a \<in> set2 c" using ac c by auto
2685           ultimately show "a \<in> u c (p1 a) ` set2 c" by auto
2686         qed auto
2687         also have "... = setsum (\<lambda> b2. count M (u c b1 b2)) (set2 c)"
2688           unfolding comp_def[symmetric] apply(rule setsum_reindex)
2689           using inj unfolding inj_on_def inj2_def using b1 c u(3) by blast
2690         also have "... = count N1 b1" unfolding ss1[OF c b1, symmetric]
2691           apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b1 set2)
2692           using True h_u[OF c b1] set2_def u(2,3)[OF c b1] u_SET[OF c b1] by fastforce
2693         finally show ?thesis .
2694       qed
2695       thus "count (mmap p1 M) b1 = count N1 b1" by transfer
2696     qed
2697   next
2698     show "mmap p2 M = N2"
2699     proof(intro multiset_eqI)
2700       fix b2
2701       let ?K = "{a. p2 a = b2 \<and> a \<in># M}"
2702       have "setsum (count M) ?K = count N2 b2"
2703       proof(cases "b2 \<in> set_of N2")
2704         case False
2705         hence "?K = {}" using sM(3) by auto
2706         thus ?thesis using False by auto
2707       next
2708         case True
2709         def c \<equiv> "f2 b2"
2710         have c: "c \<in> set_of P" and b2: "b2 \<in> set2 c"
2711           unfolding set2_def c_def P2 using True by (auto simp: comp_eq_dest[OF set_of_mmap])
2712         with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p2 a = b2 \<and> a \<in> SET}"
2713           by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
2714         also have "... = setsum (count M) ((\<lambda> b1. u c b1 b2) ` (set1 c))"
2715           apply(rule setsum_cong) using c b2 proof safe
2716           fix a assume p2a: "p2 a \<in> set2 c" and "c \<in> set_of P" and "a \<in> SET"
2717           hence ac: "a \<in> sset c" using p2_rev by auto
2718           hence "a = u c (p1 a) (p2 a)" using c by auto
2719           moreover have "p1 a \<in> set1 c" using ac c by auto
2720           ultimately show "a \<in> (\<lambda>x. u c x (p2 a)) ` set1 c" by auto
2721         qed auto
2722         also have "... = setsum (count M o (\<lambda> b1. u c b1 b2)) (set1 c)"
2723           apply(rule setsum_reindex)
2724           using inj unfolding inj_on_def inj2_def using b2 c u(2) by blast
2725         also have "... = setsum (\<lambda> b1. count M (u c b1 b2)) (set1 c)" by simp
2726         also have "... = count N2 b2" unfolding ss2[OF c b2, symmetric] comp_def
2727           apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b2 set1)
2728           using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2] set1_def by fastforce
2729         finally show ?thesis .
2730       qed
2731       thus "count (mmap p2 M) b2 = count N2 b2" by transfer
2732     qed
2733   qed
2734 qed
2736 lemma set_of_bd: "(card_of (set_of x), natLeq) \<in> ordLeq"
2737   by transfer
2738     (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
2740 lemma wppull_mmap:
2741   assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
2742   shows "wppull {M. set_of M \<subseteq> A} {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
2743     (mmap f1) (mmap f2) (mmap e1) (mmap e2) (mmap p1) (mmap p2)"
2744 proof -
2745   from assms obtain j where j: "\<forall>a'\<in>thePull B1 B2 f1 f2.
2746     j a' \<in> A \<and> e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
2747     by (blast dest: wppull_thePull)
2748   then show ?thesis
2749     by (intro wpull_wppull[OF wpull_mmap[OF wpull_thePull], of _ _ _ _ "mmap j"])
2750       (auto simp: comp_eq_dest_lhs[OF mmap_comp[symmetric]] comp_eq_dest[OF set_of_mmap]
2751         intro!: mmap_cong simp del: mem_set_of_iff simp: mem_set_of_iff[symmetric])
2752 qed
2754 bnf "'a multiset"
2755   map: mmap
2756   sets: set_of
2757   bd: natLeq
2758   wits: "{#}"
2759 by (auto simp add: mmap_id0 mmap_comp set_of_mmap natLeq_card_order natLeq_cinfinite set_of_bd
2760   Grp_def relcompp.simps intro: mmap_cong)
2761   (metis wppull_mmap[OF wppull_fstOp_sndOp, unfolded wppull_def
2762     o_eq_dest_lhs[OF mmap_comp[symmetric]] fstOp_def sndOp_def comp_def, simplified])
2764 inductive rel_multiset' where
2765   Zero[intro]: "rel_multiset' R {#} {#}"
2766 | Plus[intro]: "\<lbrakk>R a b; rel_multiset' R M N\<rbrakk> \<Longrightarrow> rel_multiset' R (M + {#a#}) (N + {#b#})"
2768 lemma map_multiset_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"
2769 by (metis image_is_empty multiset.set_map set_of_eq_empty_iff)
2771 lemma map_multiset_Zero[simp]: "mmap f {#} = {#}" by simp
2773 lemma rel_multiset_Zero: "rel_multiset R {#} {#}"
2774 unfolding rel_multiset_def Grp_def by auto
2776 declare multiset.count[simp]
2777 declare Abs_multiset_inverse[simp]
2778 declare multiset.count_inverse[simp]
2779 declare union_preserves_multiset[simp]
2781 lemma map_multiset_Plus[simp]: "mmap f (M1 + M2) = mmap f M1 + mmap f M2"
2782 proof (intro multiset_eqI, transfer fixing: f)
2783   fix x :: 'a and M1 M2 :: "'b \<Rightarrow> nat"
2784   assume "M1 \<in> multiset" "M2 \<in> multiset"
2785   hence "setsum M1 {a. f a = x \<and> 0 < M1 a} = setsum M1 {a. f a = x \<and> 0 < M1 a + M2 a}"
2786         "setsum M2 {a. f a = x \<and> 0 < M2 a} = setsum M2 {a. f a = x \<and> 0 < M1 a + M2 a}"
2787     by (auto simp: multiset_def intro!: setsum_mono_zero_cong_left)
2788   then show "(\<Sum>a | f a = x \<and> 0 < M1 a + M2 a. M1 a + M2 a) =
2789        setsum M1 {a. f a = x \<and> 0 < M1 a} +
2790        setsum M2 {a. f a = x \<and> 0 < M2 a}"
2791     by (auto simp: setsum.distrib[symmetric])
2792 qed
2794 lemma map_multiset_single[simp]: "mmap f {#a#} = {#f a#}"
2795   by transfer auto
2797 lemma rel_multiset_Plus:
2798 assumes ab: "R a b" and MN: "rel_multiset R M N"
2799 shows "rel_multiset R (M + {#a#}) (N + {#b#})"
2800 proof-
2801   {fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
2802    hence "\<exists>ya. mmap fst y + {#a#} = mmap fst ya \<and>
2803                mmap snd y + {#b#} = mmap snd ya \<and>
2804                set_of ya \<subseteq> {(x, y). R x y}"
2805    apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
2806   }
2807   thus ?thesis
2808   using assms
2809   unfolding rel_multiset_def Grp_def by force
2810 qed
2812 lemma rel_multiset'_imp_rel_multiset:
2813 "rel_multiset' R M N \<Longrightarrow> rel_multiset R M N"
2814 apply(induct rule: rel_multiset'.induct)
2815 using rel_multiset_Zero rel_multiset_Plus by auto
2817 lemma mcard_mmap[simp]: "mcard (mmap f M) = mcard M"
2818 proof -
2819   def A \<equiv> "\<lambda> b. {a. f a = b \<and> a \<in># M}"
2820   let ?B = "{b. 0 < setsum (count M) (A b)}"
2821   have "{b. \<exists>a. f a = b \<and> a \<in># M} \<subseteq> f ` {a. a \<in># M}" by auto
2822   moreover have "finite (f ` {a. a \<in># M})" apply(rule finite_imageI)
2823   using finite_Collect_mem .
2824   ultimately have fin: "finite {b. \<exists>a. f a = b \<and> a \<in># M}" by(rule finite_subset)
2825   have i: "inj_on A ?B" unfolding inj_on_def A_def apply clarsimp
2826     by (metis (lifting, full_types) mem_Collect_eq neq0_conv setsum.neutral)
2827   have 0: "\<And> b. 0 < setsum (count M) (A b) \<longleftrightarrow> (\<exists> a \<in> A b. count M a > 0)"
2828   apply safe
2829     apply (metis less_not_refl setsum_gt_0_iff setsum_infinite)
2830     by (metis A_def finite_Collect_conjI finite_Collect_mem setsum_gt_0_iff)
2831   hence AB: "A ` ?B = {A b | b. \<exists> a \<in> A b. count M a > 0}" by auto
2833   have "setsum (\<lambda> x. setsum (count M) (A x)) ?B = setsum (setsum (count M) o A) ?B"
2834   unfolding comp_def ..
2835   also have "... = (\<Sum>x\<in> A ` ?B. setsum (count M) x)"
2836   unfolding setsum.reindex [OF i, symmetric] ..
2837   also have "... = setsum (count M) (\<Union>x\<in>A ` {b. 0 < setsum (count M) (A b)}. x)"
2838   (is "_ = setsum (count M) ?J")
2839   apply(rule setsum.UNION_disjoint[symmetric])
2840   using 0 fin unfolding A_def by auto
2841   also have "?J = {a. a \<in># M}" unfolding AB unfolding A_def by auto
2842   finally have "setsum (\<lambda> x. setsum (count M) (A x)) ?B =
2843                 setsum (count M) {a. a \<in># M}" .
2844   then show ?thesis unfolding mcard_unfold_setsum A_def by transfer
2845 qed
2847 lemma rel_multiset_mcard:
2848 assumes "rel_multiset R M N"
2849 shows "mcard M = mcard N"
2850 using assms unfolding rel_multiset_def Grp_def by auto
2853 assumes empty: "P {#} {#}"
2854 and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
2855 and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
2856 shows "P M N"
2857 apply(induct N rule: multiset_induct)
2858   apply(induct M rule: multiset_induct, rule empty, erule addL)
2859   apply(induct M rule: multiset_induct, erule addR, erule addR)
2860 done
2862 lemma multiset_induct2_mcard[consumes 1, case_names empty add]:
2863 assumes c: "mcard M = mcard N"
2864 and empty: "P {#} {#}"
2865 and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
2866 shows "P M N"
2867 using c proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
2868   case (less M)  show ?case
2869   proof(cases "M = {#}")
2870     case True hence "N = {#}" using less.prems by auto
2871     thus ?thesis using True empty by auto
2872   next
2873     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2874     have "N \<noteq> {#}" using False less.prems by auto
2875     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
2876     have "mcard M1 = mcard N1" using less.prems unfolding M N by auto
2877     thus ?thesis using M N less.hyps add by auto
2878   qed
2879 qed
2881 lemma msed_map_invL:
2882 assumes "mmap f (M + {#a#}) = N"
2883 shows "\<exists> N1. N = N1 + {#f a#} \<and> mmap f M = N1"
2884 proof-
2885   have "f a \<in># N"
2886   using assms multiset.set_map[of f "M + {#a#}"] by auto
2887   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
2888   have "mmap f M = N1" using assms unfolding N by simp
2889   thus ?thesis using N by blast
2890 qed
2892 lemma msed_map_invR:
2893 assumes "mmap f M = N + {#b#}"
2894 shows "\<exists> M1 a. M = M1 + {#a#} \<and> f a = b \<and> mmap f M1 = N"
2895 proof-
2896   obtain a where a: "a \<in># M" and fa: "f a = b"
2897   using multiset.set_map[of f M] unfolding assms
2898   by (metis image_iff mem_set_of_iff union_single_eq_member)
2899   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
2900   have "mmap f M1 = N" using assms unfolding M fa[symmetric] by simp
2901   thus ?thesis using M fa by blast
2902 qed
2904 lemma msed_rel_invL:
2905 assumes "rel_multiset R (M + {#a#}) N"
2906 shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_multiset R M N1"
2907 proof-
2908   obtain K where KM: "mmap fst K = M + {#a#}"
2909   and KN: "mmap snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2910   using assms
2911   unfolding rel_multiset_def Grp_def by auto
2912   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
2913   and K1M: "mmap fst K1 = M" using msed_map_invR[OF KM] by auto
2914   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "mmap snd K1 = N1"
2915   using msed_map_invL[OF KN[unfolded K]] by auto
2916   have Rab: "R a (snd ab)" using sK a unfolding K by auto
2917   have "rel_multiset R M N1" using sK K1M K1N1
2918   unfolding K rel_multiset_def Grp_def by auto
2919   thus ?thesis using N Rab by auto
2920 qed
2922 lemma msed_rel_invR:
2923 assumes "rel_multiset R M (N + {#b#})"
2924 shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_multiset R M1 N"
2925 proof-
2926   obtain K where KN: "mmap snd K = N + {#b#}"
2927   and KM: "mmap fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2928   using assms
2929   unfolding rel_multiset_def Grp_def by auto
2930   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
2931   and K1N: "mmap snd K1 = N" using msed_map_invR[OF KN] by auto
2932   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "mmap fst K1 = M1"
2933   using msed_map_invL[OF KM[unfolded K]] by auto
2934   have Rab: "R (fst ab) b" using sK b unfolding K by auto
2935   have "rel_multiset R M1 N" using sK K1N K1M1
2936   unfolding K rel_multiset_def Grp_def by auto
2937   thus ?thesis using M Rab by auto
2938 qed
2940 lemma rel_multiset_imp_rel_multiset':
2941 assumes "rel_multiset R M N"
2942 shows "rel_multiset' R M N"
2943 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
2944   case (less M)
2945   have c: "mcard M = mcard N" using rel_multiset_mcard[OF less.prems] .
2946   show ?case
2947   proof(cases "M = {#}")
2948     case True hence "N = {#}" using c by simp
2949     thus ?thesis using True rel_multiset'.Zero by auto
2950   next
2951     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2952     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_multiset R M1 N1"
2953     using msed_rel_invL[OF less.prems[unfolded M]] by auto
2954     have "rel_multiset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
2955     thus ?thesis using rel_multiset'.Plus[of R a b, OF R] unfolding M N by simp
2956   qed
2957 qed
2959 lemma rel_multiset_rel_multiset':
2960 "rel_multiset R M N = rel_multiset' R M N"
2961 using  rel_multiset_imp_rel_multiset' rel_multiset'_imp_rel_multiset by auto
2963 (* The main end product for rel_multiset: inductive characterization *)
2964 theorems rel_multiset_induct[case_names empty add, induct pred: rel_multiset] =
2965          rel_multiset'.induct[unfolded rel_multiset_rel_multiset'[symmetric]]
2967 end