src/HOL/Library/Multiset.thy
 author kuncar Tue Feb 18 23:03:50 2014 +0100 (2014-02-18) changeset 55565 f663fc1e653b parent 55467 a5c9002bc54d child 55808 488c3e8282c8 permissions -rw-r--r--
simplify proofs because of the stronger reflexivity prover
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)" .
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 functor 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 (rename_tac a b)
959 apply (rule conjI)
960 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
961 apply (erule_tac x = a in allE, simp, clarify)
962 apply (erule_tac x = aa in allE, simp)
963 done
965 lemma multiset_of_eq_setD:
966   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
967 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
969 lemma set_eq_iff_multiset_of_eq_distinct:
970   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
971     (set x = set y) = (multiset_of x = multiset_of y)"
972 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
974 lemma set_eq_iff_multiset_of_remdups_eq:
975    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
976 apply (rule iffI)
977 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
978 apply (drule distinct_remdups [THEN distinct_remdups
979       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
980 apply simp
981 done
983 lemma multiset_of_compl_union [simp]:
984   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
985   by (induct xs) (auto simp: add_ac)
987 lemma count_multiset_of_length_filter:
988   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
989   by (induct xs) auto
991 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
992 apply (induct ls arbitrary: i)
993  apply simp
994 apply (case_tac i)
995  apply auto
996 done
998 lemma multiset_of_remove1[simp]:
999   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
1000 by (induct xs) (auto simp add: multiset_eq_iff)
1002 lemma multiset_of_eq_length:
1003   assumes "multiset_of xs = multiset_of ys"
1004   shows "length xs = length ys"
1005   using assms by (metis size_multiset_of)
1007 lemma multiset_of_eq_length_filter:
1008   assumes "multiset_of xs = multiset_of ys"
1009   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
1010   using assms by (metis count_multiset_of)
1012 lemma fold_multiset_equiv:
1013   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"
1014     and equiv: "multiset_of xs = multiset_of ys"
1015   shows "List.fold f xs = List.fold f ys"
1016 using f equiv [symmetric]
1017 proof (induct xs arbitrary: ys)
1018   case Nil then show ?case by simp
1019 next
1020   case (Cons x xs)
1021   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
1022   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
1023     by (rule Cons.prems(1)) (simp_all add: *)
1024   moreover from * have "x \<in> set ys" by simp
1025   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
1026   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
1027   ultimately show ?case by simp
1028 qed
1030 lemma multiset_of_insort [simp]:
1031   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
1032   by (induct xs) (simp_all add: ac_simps)
1034 lemma in_multiset_of:
1035   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
1036   by (induct xs) simp_all
1038 lemma multiset_of_map:
1039   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
1040   by (induct xs) simp_all
1042 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
1043 where
1044   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
1046 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
1047 where
1048   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
1049 proof -
1050   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
1051   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
1052   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
1053 qed
1055 lemma count_multiset_of_set [simp]:
1056   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
1057   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
1058   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
1059 proof -
1060   { fix A
1061     assume "x \<notin> A"
1062     have "count (multiset_of_set A) x = 0"
1063     proof (cases "finite A")
1064       case False then show ?thesis by simp
1065     next
1066       case True from True `x \<notin> A` show ?thesis by (induct A) auto
1067     qed
1068   } note * = this
1069   then show "PROP ?P" "PROP ?Q" "PROP ?R"
1070   by (auto elim!: Set.set_insert)
1071 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
1073 context linorder
1074 begin
1076 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
1077 where
1078   "sorted_list_of_multiset M = fold insort [] M"
1080 lemma sorted_list_of_multiset_empty [simp]:
1081   "sorted_list_of_multiset {#} = []"
1082   by (simp add: sorted_list_of_multiset_def)
1084 lemma sorted_list_of_multiset_singleton [simp]:
1085   "sorted_list_of_multiset {#x#} = [x]"
1086 proof -
1087   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1088   show ?thesis by (simp add: sorted_list_of_multiset_def)
1089 qed
1091 lemma sorted_list_of_multiset_insert [simp]:
1092   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
1093 proof -
1094   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1095   show ?thesis by (simp add: sorted_list_of_multiset_def)
1096 qed
1098 end
1100 lemma multiset_of_sorted_list_of_multiset [simp]:
1101   "multiset_of (sorted_list_of_multiset M) = M"
1102   by (induct M) simp_all
1104 lemma sorted_list_of_multiset_multiset_of [simp]:
1105   "sorted_list_of_multiset (multiset_of xs) = sort xs"
1106   by (induct xs) simp_all
1108 lemma finite_set_of_multiset_of_set:
1109   assumes "finite A"
1110   shows "set_of (multiset_of_set A) = A"
1111   using assms by (induct A) simp_all
1113 lemma infinite_set_of_multiset_of_set:
1114   assumes "\<not> finite A"
1115   shows "set_of (multiset_of_set A) = {}"
1116   using assms by simp
1118 lemma set_sorted_list_of_multiset [simp]:
1119   "set (sorted_list_of_multiset M) = set_of M"
1120   by (induct M) (simp_all add: set_insort)
1122 lemma sorted_list_of_multiset_of_set [simp]:
1123   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
1124   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
1127 subsection {* Big operators *}
1129 no_notation times (infixl "*" 70)
1130 no_notation Groups.one ("1")
1132 locale comm_monoid_mset = comm_monoid
1133 begin
1135 definition F :: "'a multiset \<Rightarrow> 'a"
1136 where
1137   eq_fold: "F M = Multiset.fold f 1 M"
1139 lemma empty [simp]:
1140   "F {#} = 1"
1141   by (simp add: eq_fold)
1143 lemma singleton [simp]:
1144   "F {#x#} = x"
1145 proof -
1146   interpret comp_fun_commute
1147     by default (simp add: fun_eq_iff left_commute)
1148   show ?thesis by (simp add: eq_fold)
1149 qed
1151 lemma union [simp]:
1152   "F (M + N) = F M * F N"
1153 proof -
1154   interpret comp_fun_commute f
1155     by default (simp add: fun_eq_iff left_commute)
1156   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
1157 qed
1159 end
1161 notation times (infixl "*" 70)
1162 notation Groups.one ("1")
1165 begin
1167 definition msetsum :: "'a multiset \<Rightarrow> 'a"
1168 where
1169   "msetsum = comm_monoid_mset.F plus 0"
1171 sublocale msetsum!: comm_monoid_mset plus 0
1172 where
1173   "comm_monoid_mset.F plus 0 = msetsum"
1174 proof -
1175   show "comm_monoid_mset plus 0" ..
1176   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
1177 qed
1179 lemma setsum_unfold_msetsum:
1180   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
1181   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1183 abbreviation msetsum_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1184 where
1185   "msetsum_image f M \<equiv> msetsum (image_mset f M)"
1187 end
1189 syntax
1190   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1191       ("(3SUM _:#_. _)" [0, 51, 10] 10)
1193 syntax (xsymbols)
1194   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1195       ("(3\<Sum>_:#_. _)" [0, 51, 10] 10)
1197 syntax (HTML output)
1198   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1199       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1201 translations
1202   "SUM i :# A. b" == "CONST msetsum_image (\<lambda>i. b) A"
1204 context comm_monoid_mult
1205 begin
1207 definition msetprod :: "'a multiset \<Rightarrow> 'a"
1208 where
1209   "msetprod = comm_monoid_mset.F times 1"
1211 sublocale msetprod!: comm_monoid_mset times 1
1212 where
1213   "comm_monoid_mset.F times 1 = msetprod"
1214 proof -
1215   show "comm_monoid_mset times 1" ..
1216   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
1217 qed
1219 lemma msetprod_empty:
1220   "msetprod {#} = 1"
1221   by (fact msetprod.empty)
1223 lemma msetprod_singleton:
1224   "msetprod {#x#} = x"
1225   by (fact msetprod.singleton)
1227 lemma msetprod_Un:
1228   "msetprod (A + B) = msetprod A * msetprod B"
1229   by (fact msetprod.union)
1231 lemma setprod_unfold_msetprod:
1232   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
1233   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1235 lemma msetprod_multiplicity:
1236   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
1237   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
1239 abbreviation msetprod_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1240 where
1241   "msetprod_image f M \<equiv> msetprod (image_mset f M)"
1243 end
1245 syntax
1246   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1247       ("(3PROD _:#_. _)" [0, 51, 10] 10)
1249 syntax (xsymbols)
1250   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1251       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1253 syntax (HTML output)
1254   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1255       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1257 translations
1258   "PROD i :# A. b" == "CONST msetprod_image (\<lambda>i. b) A"
1260 lemma (in comm_semiring_1) dvd_msetprod:
1261   assumes "x \<in># A"
1262   shows "x dvd msetprod A"
1263 proof -
1264   from assms have "A = (A - {#x#}) + {#x#}" by simp
1265   then obtain B where "A = B + {#x#}" ..
1266   then show ?thesis by simp
1267 qed
1270 subsection {* Cardinality *}
1272 definition mcard :: "'a multiset \<Rightarrow> nat"
1273 where
1274   "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
1276 lemma mcard_empty [simp]:
1277   "mcard {#} = 0"
1278   by (simp add: mcard_def)
1280 lemma mcard_singleton [simp]:
1281   "mcard {#a#} = Suc 0"
1282   by (simp add: mcard_def)
1284 lemma mcard_plus [simp]:
1285   "mcard (M + N) = mcard M + mcard N"
1286   by (simp add: mcard_def)
1288 lemma mcard_empty_iff [simp]:
1289   "mcard M = 0 \<longleftrightarrow> M = {#}"
1290   by (induct M) simp_all
1292 lemma mcard_unfold_setsum:
1293   "mcard M = setsum (count M) (set_of M)"
1294 proof (induct M)
1295   case empty then show ?case by simp
1296 next
1297   case (add M x) then show ?case
1298     by (cases "x \<in> set_of M")
1299       (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
1300 qed
1302 lemma size_eq_mcard:
1303   "size = mcard"
1304   by (simp add: fun_eq_iff size_def mcard_unfold_setsum)
1306 lemma mcard_multiset_of:
1307   "mcard (multiset_of xs) = length xs"
1308   by (induct xs) simp_all
1311 subsection {* Alternative representations *}
1313 subsubsection {* Lists *}
1315 context linorder
1316 begin
1318 lemma multiset_of_insort [simp]:
1319   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
1320   by (induct xs) (simp_all add: ac_simps)
1322 lemma multiset_of_sort [simp]:
1323   "multiset_of (sort_key k xs) = multiset_of xs"
1324   by (induct xs) (simp_all add: ac_simps)
1326 text {*
1327   This lemma shows which properties suffice to show that a function
1328   @{text "f"} with @{text "f xs = ys"} behaves like sort.
1329 *}
1331 lemma properties_for_sort_key:
1332   assumes "multiset_of ys = multiset_of xs"
1333   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
1334   and "sorted (map f ys)"
1335   shows "sort_key f xs = ys"
1336 using assms
1337 proof (induct xs arbitrary: ys)
1338   case Nil then show ?case by simp
1339 next
1340   case (Cons x xs)
1341   from Cons.prems(2) have
1342     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
1343     by (simp add: filter_remove1)
1344   with Cons.prems have "sort_key f xs = remove1 x ys"
1345     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
1346   moreover from Cons.prems have "x \<in> set ys"
1347     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
1348   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
1349 qed
1351 lemma properties_for_sort:
1352   assumes multiset: "multiset_of ys = multiset_of xs"
1353   and "sorted ys"
1354   shows "sort xs = ys"
1355 proof (rule properties_for_sort_key)
1356   from multiset show "multiset_of ys = multiset_of xs" .
1357   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
1358   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
1359     by (rule multiset_of_eq_length_filter)
1360   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
1361     by simp
1362   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
1363     by (simp add: replicate_length_filter)
1364 qed
1366 lemma sort_key_by_quicksort:
1367   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
1368     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
1369     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
1370 proof (rule properties_for_sort_key)
1371   show "multiset_of ?rhs = multiset_of ?lhs"
1372     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
1373 next
1374   show "sorted (map f ?rhs)"
1375     by (auto simp add: sorted_append intro: sorted_map_same)
1376 next
1377   fix l
1378   assume "l \<in> set ?rhs"
1379   let ?pivot = "f (xs ! (length xs div 2))"
1380   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
1381   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
1382     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
1383   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
1384   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
1385   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
1386     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
1387   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
1388   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
1389   proof (cases "f l" ?pivot rule: linorder_cases)
1390     case less
1391     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
1392     with less show ?thesis
1393       by (simp add: filter_sort [symmetric] ** ***)
1394   next
1395     case equal then show ?thesis
1396       by (simp add: * less_le)
1397   next
1398     case greater
1399     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
1400     with greater show ?thesis
1401       by (simp add: filter_sort [symmetric] ** ***)
1402   qed
1403 qed
1405 lemma sort_by_quicksort:
1406   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
1407     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
1408     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
1409   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
1411 text {* A stable parametrized quicksort *}
1413 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
1414   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
1416 lemma part_code [code]:
1417   "part f pivot [] = ([], [], [])"
1418   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
1419      if x' < pivot then (x # lts, eqs, gts)
1420      else if x' > pivot then (lts, eqs, x # gts)
1421      else (lts, x # eqs, gts))"
1422   by (auto simp add: part_def Let_def split_def)
1424 lemma sort_key_by_quicksort_code [code]:
1425   "sort_key f xs = (case xs of [] \<Rightarrow> []
1426     | [x] \<Rightarrow> xs
1427     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
1428     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1429        in sort_key f lts @ eqs @ sort_key f gts))"
1430 proof (cases xs)
1431   case Nil then show ?thesis by simp
1432 next
1433   case (Cons _ ys) note hyps = Cons show ?thesis
1434   proof (cases ys)
1435     case Nil with hyps show ?thesis by simp
1436   next
1437     case (Cons _ zs) note hyps = hyps Cons show ?thesis
1438     proof (cases zs)
1439       case Nil with hyps show ?thesis by auto
1440     next
1441       case Cons
1442       from sort_key_by_quicksort [of f xs]
1443       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1444         in sort_key f lts @ eqs @ sort_key f gts)"
1445       by (simp only: split_def Let_def part_def fst_conv snd_conv)
1446       with hyps Cons show ?thesis by (simp only: list.cases)
1447     qed
1448   qed
1449 qed
1451 end
1453 hide_const (open) part
1455 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
1456   by (induct xs) (auto intro: order_trans)
1458 lemma multiset_of_update:
1459   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
1460 proof (induct ls arbitrary: i)
1461   case Nil then show ?case by simp
1462 next
1463   case (Cons x xs)
1464   show ?case
1465   proof (cases i)
1466     case 0 then show ?thesis by simp
1467   next
1468     case (Suc i')
1469     with Cons show ?thesis
1470       apply simp
1471       apply (subst add_assoc)
1472       apply (subst add_commute [of "{#v#}" "{#x#}"])
1473       apply (subst add_assoc [symmetric])
1474       apply simp
1475       apply (rule mset_le_multiset_union_diff_commute)
1476       apply (simp add: mset_le_single nth_mem_multiset_of)
1477       done
1478   qed
1479 qed
1481 lemma multiset_of_swap:
1482   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
1483     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
1484   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
1487 subsection {* The multiset order *}
1489 subsubsection {* Well-foundedness *}
1491 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1492   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1493       (\<forall>b. b :# K --> (b, a) \<in> r)}"
1495 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1496   "mult r = (mult1 r)\<^sup>+"
1498 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1499 by (simp add: mult1_def)
1501 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
1502     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1503     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
1504   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
1505 proof (unfold mult1_def)
1506   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
1507   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1508   let ?case1 = "?case1 {(N, M). ?R N M}"
1510   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
1511   then have "\<exists>a' M0' K.
1512       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
1513   then show "?case1 \<or> ?case2"
1514   proof (elim exE conjE)
1515     fix a' M0' K
1516     assume N: "N = M0' + K" and r: "?r K a'"
1517     assume "M0 + {#a#} = M0' + {#a'#}"
1518     then have "M0 = M0' \<and> a = a' \<or>
1519         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
1520       by (simp only: add_eq_conv_ex)
1521     then show ?thesis
1522     proof (elim disjE conjE exE)
1523       assume "M0 = M0'" "a = a'"
1524       with N r have "?r K a \<and> N = M0 + K" by simp
1525       then have ?case2 .. then show ?thesis ..
1526     next
1527       fix K'
1528       assume "M0' = K' + {#a#}"
1529       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
1531       assume "M0 = K' + {#a'#}"
1532       with r have "?R (K' + K) M0" by blast
1533       with n have ?case1 by simp then show ?thesis ..
1534     qed
1535   qed
1536 qed
1538 lemma all_accessible: "wf r ==> \<forall>M. M \<in> Wellfounded.acc (mult1 r)"
1539 proof
1540   let ?R = "mult1 r"
1541   let ?W = "Wellfounded.acc ?R"
1542   {
1543     fix M M0 a
1544     assume M0: "M0 \<in> ?W"
1545       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1546       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
1547     have "M0 + {#a#} \<in> ?W"
1548     proof (rule accI [of "M0 + {#a#}"])
1549       fix N
1550       assume "(N, M0 + {#a#}) \<in> ?R"
1551       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
1552           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
1553         by (rule less_add)
1554       then show "N \<in> ?W"
1555       proof (elim exE disjE conjE)
1556         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
1557         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
1558         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
1559         then show "N \<in> ?W" by (simp only: N)
1560       next
1561         fix K
1562         assume N: "N = M0 + K"
1563         assume "\<forall>b. b :# K --> (b, a) \<in> r"
1564         then have "M0 + K \<in> ?W"
1565         proof (induct K)
1566           case empty
1567           from M0 show "M0 + {#} \<in> ?W" by simp
1568         next
1569           case (add K x)
1570           from add.prems have "(x, a) \<in> r" by simp
1571           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
1572           moreover from add have "M0 + K \<in> ?W" by simp
1573           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
1574           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
1575         qed
1576         then show "N \<in> ?W" by (simp only: N)
1577       qed
1578     qed
1579   } note tedious_reasoning = this
1581   assume wf: "wf r"
1582   fix M
1583   show "M \<in> ?W"
1584   proof (induct M)
1585     show "{#} \<in> ?W"
1586     proof (rule accI)
1587       fix b assume "(b, {#}) \<in> ?R"
1588       with not_less_empty show "b \<in> ?W" by contradiction
1589     qed
1591     fix M a assume "M \<in> ?W"
1592     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1593     proof induct
1594       fix a
1595       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1596       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1597       proof
1598         fix M assume "M \<in> ?W"
1599         then show "M + {#a#} \<in> ?W"
1600           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
1601       qed
1602     qed
1603     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
1604   qed
1605 qed
1607 theorem wf_mult1: "wf r ==> wf (mult1 r)"
1608 by (rule acc_wfI) (rule all_accessible)
1610 theorem wf_mult: "wf r ==> wf (mult r)"
1611 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
1614 subsubsection {* Closure-free presentation *}
1616 text {* One direction. *}
1618 lemma mult_implies_one_step:
1619   "trans r ==> (M, N) \<in> mult r ==>
1620     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
1621     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
1622 apply (unfold mult_def mult1_def set_of_def)
1623 apply (erule converse_trancl_induct, clarify)
1624  apply (rule_tac x = M0 in exI, simp, clarify)
1625 apply (case_tac "a :# K")
1626  apply (rule_tac x = I in exI)
1627  apply (simp (no_asm))
1628  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1629  apply (simp (no_asm_simp) add: add_assoc [symmetric])
1630  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
1631  apply (simp add: diff_union_single_conv)
1632  apply (simp (no_asm_use) add: trans_def)
1633  apply blast
1634 apply (subgoal_tac "a :# I")
1635  apply (rule_tac x = "I - {#a#}" in exI)
1636  apply (rule_tac x = "J + {#a#}" in exI)
1637  apply (rule_tac x = "K + Ka" in exI)
1638  apply (rule conjI)
1639   apply (simp add: multiset_eq_iff split: nat_diff_split)
1640  apply (rule conjI)
1641   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
1642   apply (simp add: multiset_eq_iff split: nat_diff_split)
1643  apply (simp (no_asm_use) add: trans_def)
1644  apply blast
1645 apply (subgoal_tac "a :# (M0 + {#a#})")
1646  apply simp
1647 apply (simp (no_asm))
1648 done
1650 lemma one_step_implies_mult_aux:
1651   "trans r ==>
1652     \<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))
1653       --> (I + K, I + J) \<in> mult r"
1654 apply (induct_tac n, auto)
1655 apply (frule size_eq_Suc_imp_eq_union, clarify)
1656 apply (rename_tac "J'", simp)
1657 apply (erule notE, auto)
1658 apply (case_tac "J' = {#}")
1659  apply (simp add: mult_def)
1660  apply (rule r_into_trancl)
1661  apply (simp add: mult1_def set_of_def, blast)
1662 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
1663 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
1664 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
1665 apply (erule ssubst)
1666 apply (simp add: Ball_def, auto)
1667 apply (subgoal_tac
1668   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
1669     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
1670  prefer 2
1671  apply force
1672 apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
1673 apply (erule trancl_trans)
1674 apply (rule r_into_trancl)
1675 apply (simp add: mult1_def set_of_def)
1676 apply (rule_tac x = a in exI)
1677 apply (rule_tac x = "I + J'" in exI)
1679 done
1681 lemma one_step_implies_mult:
1682   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
1683     ==> (I + K, I + J) \<in> mult r"
1684 using one_step_implies_mult_aux by blast
1687 subsubsection {* Partial-order properties *}
1689 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
1690   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
1692 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
1693   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
1695 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
1696 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
1698 interpretation multiset_order: order le_multiset less_multiset
1699 proof -
1700   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
1701   proof
1702     fix M :: "'a multiset"
1703     assume "M \<subset># M"
1704     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
1705     have "trans {(x'::'a, x). x' < x}"
1706       by (rule transI) simp
1707     moreover note MM
1708     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
1709       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
1710       by (rule mult_implies_one_step)
1711     then obtain I J K where "M = I + J" and "M = I + K"
1712       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
1713     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
1714     have "finite (set_of K)" by simp
1715     moreover note aux2
1716     ultimately have "set_of K = {}"
1717       by (induct rule: finite_induct) (auto intro: order_less_trans)
1718     with aux1 show False by simp
1719   qed
1720   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
1721     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
1722   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
1723     by default (auto simp add: le_multiset_def irrefl dest: trans)
1724 qed
1726 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
1727   by simp
1730 subsubsection {* Monotonicity of multiset union *}
1732 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
1733 apply (unfold mult1_def)
1734 apply auto
1735 apply (rule_tac x = a in exI)
1736 apply (rule_tac x = "C + M0" in exI)
1738 done
1740 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
1741 apply (unfold less_multiset_def mult_def)
1742 apply (erule trancl_induct)
1743  apply (blast intro: mult1_union)
1744 apply (blast intro: mult1_union trancl_trans)
1745 done
1747 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
1748 apply (subst add_commute [of B C])
1749 apply (subst add_commute [of D C])
1750 apply (erule union_less_mono2)
1751 done
1753 lemma union_less_mono:
1754   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
1755   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
1757 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
1758 proof
1759 qed (auto simp add: le_multiset_def intro: union_less_mono2)
1762 subsection {* Termination proofs with multiset orders *}
1764 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
1765   and multi_member_this: "x \<in># {# x #} + XS"
1766   and multi_member_last: "x \<in># {# x #}"
1767   by auto
1769 definition "ms_strict = mult pair_less"
1770 definition "ms_weak = ms_strict \<union> Id"
1772 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
1773 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
1774 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
1776 lemma smsI:
1777   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
1778   unfolding ms_strict_def
1779 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
1781 lemma wmsI:
1782   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
1783   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
1784 unfolding ms_weak_def ms_strict_def
1785 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
1787 inductive pw_leq
1788 where
1789   pw_leq_empty: "pw_leq {#} {#}"
1790 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
1792 lemma pw_leq_lstep:
1793   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
1794 by (drule pw_leq_step) (rule pw_leq_empty, simp)
1796 lemma pw_leq_split:
1797   assumes "pw_leq X Y"
1798   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 = {#}))"
1799   using assms
1800 proof (induct)
1801   case pw_leq_empty thus ?case by auto
1802 next
1803   case (pw_leq_step x y X Y)
1804   then obtain A B Z where
1805     [simp]: "X = A + Z" "Y = B + Z"
1806       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
1807     by auto
1808   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
1809     unfolding pair_leq_def by auto
1810   thus ?case
1811   proof
1812     assume [simp]: "x = y"
1813     have
1814       "{#x#} + X = A + ({#y#}+Z)
1815       \<and> {#y#} + Y = B + ({#y#}+Z)
1816       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1817       by (auto simp: add_ac)
1818     thus ?case by (intro exI)
1819   next
1820     assume A: "(x, y) \<in> pair_less"
1821     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
1822     have "{#x#} + X = ?A' + Z"
1823       "{#y#} + Y = ?B' + Z"
1825     moreover have
1826       "(set_of ?A', set_of ?B') \<in> max_strict"
1827       using 1 A unfolding max_strict_def
1828       by (auto elim!: max_ext.cases)
1829     ultimately show ?thesis by blast
1830   qed
1831 qed
1833 lemma
1834   assumes pwleq: "pw_leq Z Z'"
1835   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
1836   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
1837   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
1838 proof -
1839   from pw_leq_split[OF pwleq]
1840   obtain A' B' Z''
1841     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
1842     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
1843     by blast
1844   {
1845     assume max: "(set_of A, set_of B) \<in> max_strict"
1846     from mx_or_empty
1847     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
1848     proof
1849       assume max': "(set_of A', set_of B') \<in> max_strict"
1850       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
1851         by (auto simp: max_strict_def intro: max_ext_additive)
1852       thus ?thesis by (rule smsI)
1853     next
1854       assume [simp]: "A' = {#} \<and> B' = {#}"
1855       show ?thesis by (rule smsI) (auto intro: max)
1856     qed
1857     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
1858     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
1859   }
1860   from mx_or_empty
1861   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
1862   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
1863 qed
1865 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
1866 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
1867 and nonempty_single: "{# x #} \<noteq> {#}"
1868 by auto
1870 setup {*
1871 let
1872   fun msetT T = Type (@{type_name multiset}, [T]);
1874   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
1875     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
1876     | mk_mset T (x :: xs) =
1877           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
1878                 mk_mset T [x] \$ mk_mset T xs
1880   fun mset_member_tac m i =
1881       (if m <= 0 then
1882            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
1883        else
1884            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
1886   val mset_nonempty_tac =
1887       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
1889   val regroup_munion_conv =
1890       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
1891         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))
1893   fun unfold_pwleq_tac i =
1894     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
1895       ORELSE (rtac @{thm pw_leq_lstep} i)
1896       ORELSE (rtac @{thm pw_leq_empty} i)
1898   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
1899                       @{thm Un_insert_left}, @{thm Un_empty_left}]
1900 in
1901   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
1902   {
1903     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
1904     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
1905     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
1906     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
1907     reduction_pair= @{thm ms_reduction_pair}
1908   })
1909 end
1910 *}
1913 subsection {* Legacy theorem bindings *}
1915 lemmas multi_count_eq = multiset_eq_iff [symmetric]
1917 lemma union_commute: "M + N = N + (M::'a multiset)"
1918   by (fact add_commute)
1920 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
1921   by (fact add_assoc)
1923 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
1924   by (fact add_left_commute)
1926 lemmas union_ac = union_assoc union_commute union_lcomm
1928 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
1929   by (fact add_right_cancel)
1931 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
1932   by (fact add_left_cancel)
1934 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
1935   by (fact add_imp_eq)
1937 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
1938   by (fact order_less_trans)
1940 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
1941   by (fact inf.commute)
1943 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
1944   by (fact inf.assoc [symmetric])
1946 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
1947   by (fact inf.left_commute)
1949 lemmas multiset_inter_ac =
1950   multiset_inter_commute
1951   multiset_inter_assoc
1952   multiset_inter_left_commute
1954 lemma mult_less_not_refl:
1955   "\<not> M \<subset># (M::'a::order multiset)"
1956   by (fact multiset_order.less_irrefl)
1958 lemma mult_less_trans:
1959   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
1960   by (fact multiset_order.less_trans)
1962 lemma mult_less_not_sym:
1963   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
1964   by (fact multiset_order.less_not_sym)
1966 lemma mult_less_asym:
1967   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
1968   by (fact multiset_order.less_asym)
1970 ML {*
1971 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
1972                       (Const _ \$ t') =
1973     let
1974       val (maybe_opt, ps) =
1975         Nitpick_Model.dest_plain_fun t' ||> op ~~
1976         ||> map (apsnd (snd o HOLogic.dest_number))
1977       fun elems_for t =
1978         case AList.lookup (op =) ps t of
1979           SOME n => replicate n t
1980         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
1981     in
1982       case maps elems_for (all_values elem_T) @
1983            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
1984             else []) of
1985         [] => Const (@{const_name zero_class.zero}, T)
1986       | ts => foldl1 (fn (t1, t2) =>
1987                          Const (@{const_name plus_class.plus}, T --> T --> T)
1988                          \$ t1 \$ t2)
1989                      (map (curry (op \$) (Const (@{const_name single},
1990                                                 elem_T --> T))) ts)
1991     end
1992   | multiset_postproc _ _ _ _ t = t
1993 *}
1995 declaration {*
1996 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
1997     multiset_postproc
1998 *}
2000 hide_const (open) fold
2003 subsection {* Naive implementation using lists *}
2005 code_datatype multiset_of
2007 lemma [code]:
2008   "{#} = multiset_of []"
2009   by simp
2011 lemma [code]:
2012   "{#x#} = multiset_of [x]"
2013   by simp
2015 lemma union_code [code]:
2016   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
2017   by simp
2019 lemma [code]:
2020   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
2021   by (simp add: multiset_of_map)
2023 lemma [code]:
2024   "Multiset.filter f (multiset_of xs) = multiset_of (filter f xs)"
2025   by (simp add: multiset_of_filter)
2027 lemma [code]:
2028   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
2029   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
2031 lemma [code]:
2032   "multiset_of xs #\<inter> multiset_of ys =
2033     multiset_of (snd (fold (\<lambda>x (ys, zs).
2034       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
2035 proof -
2036   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
2037     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
2038       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
2039     by (induct xs arbitrary: ys)
2041   then show ?thesis by simp
2042 qed
2044 lemma [code]:
2045   "multiset_of xs #\<union> multiset_of ys =
2046     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
2047 proof -
2048   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
2049       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
2050     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
2051   then show ?thesis by simp
2052 qed
2054 lemma [code_unfold]:
2055   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
2056   by (simp add: in_multiset_of)
2058 lemma [code]:
2059   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
2060 proof -
2061   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
2062     by (induct xs) simp_all
2063   then show ?thesis by simp
2064 qed
2066 lemma [code]:
2067   "set_of (multiset_of xs) = set xs"
2068   by simp
2070 lemma [code]:
2071   "sorted_list_of_multiset (multiset_of xs) = sort xs"
2072   by (induct xs) simp_all
2074 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
2075   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
2076   apply (cases "finite A")
2077   apply simp_all
2078   apply (induct A rule: finite_induct)
2079   apply (simp_all add: union_commute)
2080   done
2082 lemma [code]:
2083   "mcard (multiset_of xs) = length xs"
2084   by (simp add: mcard_multiset_of)
2086 lemma [code]:
2087   "A \<le> B \<longleftrightarrow> A #\<inter> B = A"
2088   by (auto simp add: inf.order_iff)
2090 instantiation multiset :: (equal) equal
2091 begin
2093 definition
2094   [code]: "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"
2096 instance
2097   by default (simp add: equal_multiset_def eq_iff)
2099 end
2101 lemma [code]:
2102   "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
2103   by auto
2105 lemma [code]:
2106   "msetsum (multiset_of xs) = listsum xs"
2107   by (induct xs) (simp_all add: add.commute)
2109 lemma [code]:
2110   "msetprod (multiset_of xs) = fold times xs 1"
2111 proof -
2112   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
2113     by (induct xs) (simp_all add: mult.assoc)
2114   then show ?thesis by simp
2115 qed
2117 lemma [code]:
2118   "size = mcard"
2119   by (fact size_eq_mcard)
2121 text {*
2122   Exercise for the casual reader: add implementations for @{const le_multiset}
2123   and @{const less_multiset} (multiset order).
2124 *}
2126 text {* Quickcheck generators *}
2128 definition (in term_syntax)
2129   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
2130     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
2131   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
2133 notation fcomp (infixl "\<circ>>" 60)
2134 notation scomp (infixl "\<circ>\<rightarrow>" 60)
2136 instantiation multiset :: (random) random
2137 begin
2139 definition
2140   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
2142 instance ..
2144 end
2146 no_notation fcomp (infixl "\<circ>>" 60)
2147 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
2149 instantiation multiset :: (full_exhaustive) full_exhaustive
2150 begin
2152 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
2153 where
2154   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
2156 instance ..
2158 end
2160 hide_const (open) msetify
2163 subsection {* BNF setup *}
2165 lemma setsum_gt_0_iff:
2166 fixes f :: "'a \<Rightarrow> nat" assumes "finite A"
2167 shows "setsum f A > 0 \<longleftrightarrow> (\<exists> a \<in> A. f a > 0)"
2168 (is "?L \<longleftrightarrow> ?R")
2169 proof-
2170   have "?L \<longleftrightarrow> \<not> setsum f A = 0" by fast
2171   also have "... \<longleftrightarrow> (\<exists> a \<in> A. f a \<noteq> 0)" using assms by simp
2172   also have "... \<longleftrightarrow> ?R" by simp
2173   finally show ?thesis .
2174 qed
2176 lift_definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" is
2177   "\<lambda>h f b. setsum f {a. h a = b \<and> f a > 0} :: nat"
2178 unfolding multiset_def proof safe
2179   fix h :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat"
2180   assume fin: "finite {a. 0 < f a}"  (is "finite ?A")
2181   show "finite {b. 0 < setsum f {a. h a = b \<and> 0 < f a}}"
2182   (is "finite {b. 0 < setsum f (?As b)}")
2183   proof- let ?B = "{b. 0 < setsum f (?As b)}"
2184     have "\<And> b. finite (?As b)" using fin by simp
2185     hence B: "?B = {b. ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
2186     hence "?B \<subseteq> h ` ?A" by auto
2187     thus ?thesis using finite_surj[OF fin] by auto
2188   qed
2189 qed
2191 lemma mmap_id0: "mmap id = id"
2192 proof (intro ext multiset_eqI)
2193   fix f a show "count (mmap id f) a = count (id f) a"
2194   proof (cases "count f a = 0")
2195     case False
2196     hence 1: "{aa. aa = a \<and> aa \<in># f} = {a}" by auto
2197     thus ?thesis by transfer auto
2198   qed (transfer, simp)
2199 qed
2201 lemma inj_on_setsum_inv:
2202 assumes 1: "(0::nat) < setsum (count f) {a. h a = b' \<and> a \<in># f}" (is "0 < setsum (count f) ?A'")
2203 and     2: "{a. h a = b \<and> a \<in># f} = {a. h a = b' \<and> a \<in># f}" (is "?A = ?A'")
2204 shows "b = b'"
2205 using assms by (auto simp add: setsum_gt_0_iff)
2207 lemma mmap_comp:
2208 fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
2209 shows "mmap (h2 o h1) = mmap h2 o mmap h1"
2210 proof (intro ext multiset_eqI)
2211   fix f :: "'a multiset" fix c :: 'c
2212   let ?A = "{a. h2 (h1 a) = c \<and> a \<in># f}"
2213   let ?As = "\<lambda> b. {a. h1 a = b \<and> a \<in># f}"
2214   let ?B = "{b. h2 b = c \<and> 0 < setsum (count f) (?As b)}"
2215   have 0: "{?As b | b.  b \<in> ?B} = ?As ` ?B" by auto
2216   have "\<And> b. finite (?As b)" by transfer (simp add: multiset_def)
2217   hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
2218   hence A: "?A = \<Union> {?As b | b.  b \<in> ?B}" by auto
2219   have "setsum (count f) ?A = setsum (setsum (count f)) {?As b | b.  b \<in> ?B}"
2220     unfolding A by transfer (intro setsum_Union_disjoint, auto simp: multiset_def)
2221   also have "... = setsum (setsum (count f)) (?As ` ?B)" unfolding 0 ..
2222   also have "... = setsum (setsum (count f) o ?As) ?B"
2223     by(intro setsum_reindex) (auto simp add: setsum_gt_0_iff inj_on_def)
2224   also have "... = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" unfolding comp_def ..
2225   finally have "setsum (count f) ?A = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" .
2226   thus "count (mmap (h2 \<circ> h1) f) c = count ((mmap h2 \<circ> mmap h1) f) c"
2227     by transfer (unfold comp_apply, blast)
2228 qed
2230 lemma mmap_cong:
2231 assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
2232 shows "mmap f M = mmap g M"
2233 using assms by transfer (auto intro!: setsum_cong)
2235 context
2236 begin
2237 interpretation lifting_syntax .
2239 lemma set_of_transfer[transfer_rule]: "(pcr_multiset op = ===> op =) (\<lambda>f. {a. 0 < f a}) set_of"
2240   unfolding set_of_def pcr_multiset_def cr_multiset_def fun_rel_def by auto
2242 end
2244 lemma set_of_mmap: "set_of o mmap h = image h o set_of"
2245 proof (rule ext, unfold comp_apply)
2246   fix M show "set_of (mmap h M) = h ` set_of M"
2247     by transfer (auto simp add: multiset_def setsum_gt_0_iff)
2248 qed
2250 lemma multiset_of_surj:
2251   "multiset_of ` {as. set as \<subseteq> A} = {M. set_of M \<subseteq> A}"
2252 proof safe
2253   fix M assume M: "set_of M \<subseteq> A"
2254   obtain as where eq: "M = multiset_of as" using surj_multiset_of unfolding surj_def by auto
2255   hence "set as \<subseteq> A" using M by auto
2256   thus "M \<in> multiset_of ` {as. set as \<subseteq> A}" using eq by auto
2257 next
2258   show "\<And>x xa xb. \<lbrakk>set xa \<subseteq> A; xb \<in> set_of (multiset_of xa)\<rbrakk> \<Longrightarrow> xb \<in> A"
2259   by (erule set_mp) (unfold set_of_multiset_of)
2260 qed
2262 lemma card_of_set_of:
2263 "(card_of {M. set_of M \<subseteq> A}, card_of {as. set as \<subseteq> A}) \<in> ordLeq"
2264 apply(rule card_of_ordLeqI2[of _ multiset_of]) using multiset_of_surj by auto
2266 lemma nat_sum_induct:
2267 assumes "\<And>n1 n2. (\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow> phi m1 m2) \<Longrightarrow> phi n1 n2"
2268 shows "phi (n1::nat) (n2::nat)"
2269 proof-
2270   let ?chi = "\<lambda> n1n2 :: nat * nat. phi (fst n1n2) (snd n1n2)"
2271   have "?chi (n1,n2)"
2272   apply(induct rule: measure_induct[of "\<lambda> n1n2. fst n1n2 + snd n1n2" ?chi])
2273   using assms by (metis fstI sndI)
2274   thus ?thesis by simp
2275 qed
2277 lemma matrix_count:
2278 fixes ct1 ct2 :: "nat \<Rightarrow> nat"
2279 assumes "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
2280 shows
2281 "\<exists> ct. (\<forall> i1 \<le> n1. setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ct1 i1) \<and>
2282        (\<forall> i2 \<le> n2. setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ct2 i2)"
2283 (is "?phi ct1 ct2 n1 n2")
2284 proof-
2285   have "\<forall> ct1 ct2 :: nat \<Rightarrow> nat.
2286         setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"
2287   proof(induct rule: nat_sum_induct[of
2288 "\<lambda> n1 n2. \<forall> ct1 ct2 :: nat \<Rightarrow> nat.
2289      setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"],
2290       clarify)
2291   fix n1 n2 :: nat and ct1 ct2 :: "nat \<Rightarrow> nat"
2292   assume IH: "\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow>
2293                 \<forall> dt1 dt2 :: nat \<Rightarrow> nat.
2294                 setsum dt1 {..<Suc m1} = setsum dt2 {..<Suc m2} \<longrightarrow> ?phi dt1 dt2 m1 m2"
2295   and ss: "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
2296   show "?phi ct1 ct2 n1 n2"
2297   proof(cases n1)
2298     case 0 note n1 = 0
2299     show ?thesis
2300     proof(cases n2)
2301       case 0 note n2 = 0
2302       let ?ct = "\<lambda> i1 i2. ct2 0"
2303       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by simp
2304     next
2305       case (Suc m2) note n2 = Suc
2306       let ?ct = "\<lambda> i1 i2. ct2 i2"
2307       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
2308     qed
2309   next
2310     case (Suc m1) note n1 = Suc
2311     show ?thesis
2312     proof(cases n2)
2313       case 0 note n2 = 0
2314       let ?ct = "\<lambda> i1 i2. ct1 i1"
2315       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
2316     next
2317       case (Suc m2) note n2 = Suc
2318       show ?thesis
2319       proof(cases "ct1 n1 \<le> ct2 n2")
2320         case True
2321         def dt2 \<equiv> "\<lambda> i2. if i2 = n2 then ct2 i2 - ct1 n1 else ct2 i2"
2322         have "setsum ct1 {..<Suc m1} = setsum dt2 {..<Suc n2}"
2323         unfolding dt2_def using ss n1 True by auto
2324         hence "?phi ct1 dt2 m1 n2" using IH[of m1 n2] n1 by simp
2325         then obtain dt where
2326         1: "\<And> i1. i1 \<le> m1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc n2} = ct1 i1" and
2327         2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc m1} = dt2 i2" by auto
2328         let ?ct = "\<lambda> i1 i2. if i1 = n1 then (if i2 = n2 then ct1 n1 else 0)
2329                                        else dt i1 i2"
2330         show ?thesis apply(rule exI[of _ ?ct])
2331         using n1 n2 1 2 True unfolding dt2_def by simp
2332       next
2333         case False
2334         hence False: "ct2 n2 < ct1 n1" by simp
2335         def dt1 \<equiv> "\<lambda> i1. if i1 = n1 then ct1 i1 - ct2 n2 else ct1 i1"
2336         have "setsum dt1 {..<Suc n1} = setsum ct2 {..<Suc m2}"
2337         unfolding dt1_def using ss n2 False by auto
2338         hence "?phi dt1 ct2 n1 m2" using IH[of n1 m2] n2 by simp
2339         then obtain dt where
2340         1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc m2} = dt1 i1" and
2341         2: "\<And> i2. i2 \<le> m2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc n1} = ct2 i2" by force
2342         let ?ct = "\<lambda> i1 i2. if i2 = n2 then (if i1 = n1 then ct2 n2 else 0)
2343                                        else dt i1 i2"
2344         show ?thesis apply(rule exI[of _ ?ct])
2345         using n1 n2 1 2 False unfolding dt1_def by simp
2346       qed
2347     qed
2348   qed
2349   qed
2350   thus ?thesis using assms by auto
2351 qed
2353 definition
2354 "inj2 u B1 B2 \<equiv>
2355  \<forall> b1 b1' b2 b2'. {b1,b1'} \<subseteq> B1 \<and> {b2,b2'} \<subseteq> B2 \<and> u b1 b2 = u b1' b2'
2356                   \<longrightarrow> b1 = b1' \<and> b2 = b2'"
2358 lemma matrix_setsum_finite:
2359 assumes B1: "B1 \<noteq> {}" "finite B1" and B2: "B2 \<noteq> {}" "finite B2" and u: "inj2 u B1 B2"
2360 and ss: "setsum N1 B1 = setsum N2 B2"
2361 shows "\<exists> M :: 'a \<Rightarrow> nat.
2362             (\<forall> b1 \<in> B1. setsum (\<lambda> b2. M (u b1 b2)) B2 = N1 b1) \<and>
2363             (\<forall> b2 \<in> B2. setsum (\<lambda> b1. M (u b1 b2)) B1 = N2 b2)"
2364 proof-
2365   obtain n1 where "card B1 = Suc n1" using B1 by (metis card_insert finite.simps)
2366   then obtain e1 where e1: "bij_betw e1 {..<Suc n1} B1"
2367   using ex_bij_betw_finite_nat[OF B1(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
2368   hence e1_inj: "inj_on e1 {..<Suc n1}" and e1_surj: "e1 ` {..<Suc n1} = B1"
2369   unfolding bij_betw_def by auto
2370   def f1 \<equiv> "inv_into {..<Suc n1} e1"
2371   have f1: "bij_betw f1 B1 {..<Suc n1}"
2372   and f1e1[simp]: "\<And> i1. i1 < Suc n1 \<Longrightarrow> f1 (e1 i1) = i1"
2373   and e1f1[simp]: "\<And> b1. b1 \<in> B1 \<Longrightarrow> e1 (f1 b1) = b1" unfolding f1_def
2374   apply (metis bij_betw_inv_into e1, metis bij_betw_inv_into_left e1 lessThan_iff)
2375   by (metis e1_surj f_inv_into_f)
2376   (*  *)
2377   obtain n2 where "card B2 = Suc n2" using B2 by (metis card_insert finite.simps)
2378   then obtain e2 where e2: "bij_betw e2 {..<Suc n2} B2"
2379   using ex_bij_betw_finite_nat[OF B2(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
2380   hence e2_inj: "inj_on e2 {..<Suc n2}" and e2_surj: "e2 ` {..<Suc n2} = B2"
2381   unfolding bij_betw_def by auto
2382   def f2 \<equiv> "inv_into {..<Suc n2} e2"
2383   have f2: "bij_betw f2 B2 {..<Suc n2}"
2384   and f2e2[simp]: "\<And> i2. i2 < Suc n2 \<Longrightarrow> f2 (e2 i2) = i2"
2385   and e2f2[simp]: "\<And> b2. b2 \<in> B2 \<Longrightarrow> e2 (f2 b2) = b2" unfolding f2_def
2386   apply (metis bij_betw_inv_into e2, metis bij_betw_inv_into_left e2 lessThan_iff)
2387   by (metis e2_surj f_inv_into_f)
2388   (*  *)
2389   let ?ct1 = "N1 o e1"  let ?ct2 = "N2 o e2"
2390   have ss: "setsum ?ct1 {..<Suc n1} = setsum ?ct2 {..<Suc n2}"
2391   unfolding setsum_reindex[OF e1_inj, symmetric] setsum_reindex[OF e2_inj, symmetric]
2392   e1_surj e2_surj using ss .
2393   obtain ct where
2394   ct1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ?ct1 i1" and
2395   ct2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ?ct2 i2"
2396   using matrix_count[OF ss] by blast
2397   (*  *)
2398   def A \<equiv> "{u b1 b2 | b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"
2399   have "\<forall> a \<in> A. \<exists> b1b2 \<in> B1 <*> B2. u (fst b1b2) (snd b1b2) = a"
2400   unfolding A_def Ball_def mem_Collect_eq by auto
2401   then obtain h1h2 where h12:
2402   "\<And>a. a \<in> A \<Longrightarrow> u (fst (h1h2 a)) (snd (h1h2 a)) = a \<and> h1h2 a \<in> B1 <*> B2" by metis
2403   def h1 \<equiv> "fst o h1h2"  def h2 \<equiv> "snd o h1h2"
2404   have h12[simp]: "\<And>a. a \<in> A \<Longrightarrow> u (h1 a) (h2 a) = a"
2405                   "\<And> a. a \<in> A \<Longrightarrow> h1 a \<in> B1"  "\<And> a. a \<in> A \<Longrightarrow> h2 a \<in> B2"
2406   using h12 unfolding h1_def h2_def by force+
2407   {fix b1 b2 assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2"
2408    hence inA: "u b1 b2 \<in> A" unfolding A_def by auto
2409    hence "u b1 b2 = u (h1 (u b1 b2)) (h2 (u b1 b2))" by auto
2410    moreover have "h1 (u b1 b2) \<in> B1" "h2 (u b1 b2) \<in> B2" using inA by auto
2411    ultimately have "h1 (u b1 b2) = b1 \<and> h2 (u b1 b2) = b2"
2412    using u b1 b2 unfolding inj2_def by fastforce
2413   }
2414   hence h1[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h1 (u b1 b2) = b1" and
2415         h2[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h2 (u b1 b2) = b2" by auto
2416   def M \<equiv> "\<lambda> a. ct (f1 (h1 a)) (f2 (h2 a))"
2417   show ?thesis
2418   apply(rule exI[of _ M]) proof safe
2419     fix b1 assume b1: "b1 \<in> B1"
2420     hence f1b1: "f1 b1 \<le> n1" using f1 unfolding bij_betw_def
2421     by (metis image_eqI lessThan_iff less_Suc_eq_le)
2422     have "(\<Sum>b2\<in>B2. M (u b1 b2)) = (\<Sum>i2<Suc n2. ct (f1 b1) (f2 (e2 i2)))"
2423     unfolding e2_surj[symmetric] setsum_reindex[OF e2_inj]
2424     unfolding M_def comp_def apply(intro setsum_cong) apply force
2425     by (metis e2_surj b1 h1 h2 imageI)
2426     also have "... = N1 b1" using b1 ct1[OF f1b1] by simp
2427     finally show "(\<Sum>b2\<in>B2. M (u b1 b2)) = N1 b1" .
2428   next
2429     fix b2 assume b2: "b2 \<in> B2"
2430     hence f2b2: "f2 b2 \<le> n2" using f2 unfolding bij_betw_def
2431     by (metis image_eqI lessThan_iff less_Suc_eq_le)
2432     have "(\<Sum>b1\<in>B1. M (u b1 b2)) = (\<Sum>i1<Suc n1. ct (f1 (e1 i1)) (f2 b2))"
2433     unfolding e1_surj[symmetric] setsum_reindex[OF e1_inj]
2434     unfolding M_def comp_def apply(intro setsum_cong) apply force
2435     by (metis e1_surj b2 h1 h2 imageI)
2436     also have "... = N2 b2" using b2 ct2[OF f2b2] by simp
2437     finally show "(\<Sum>b1\<in>B1. M (u b1 b2)) = N2 b2" .
2438   qed
2439 qed
2441 lemma supp_vimage_mmap: "set_of M \<subseteq> f -` (set_of (mmap f M))"
2442   by transfer (auto simp: multiset_def setsum_gt_0_iff)
2444 lemma mmap_ge_0: "b \<in># mmap f M \<longleftrightarrow> (\<exists>a. a \<in># M \<and> f a = b)"
2445   by transfer (auto simp: multiset_def setsum_gt_0_iff)
2447 lemma finite_twosets:
2448 assumes "finite B1" and "finite B2"
2449 shows "finite {u b1 b2 |b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"  (is "finite ?A")
2450 proof-
2451   have A: "?A = (\<lambda> b1b2. u (fst b1b2) (snd b1b2)) ` (B1 <*> B2)" by force
2452   show ?thesis unfolding A using finite_cartesian_product[OF assms] by auto
2453 qed
2455 (* Weak pullbacks: *)
2456 definition wpull where
2457 "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow>
2458  (\<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))"
2460 (* Weak pseudo-pullbacks *)
2461 definition wppull where
2462 "wppull A B1 B2 f1 f2 e1 e2 p1 p2 \<longleftrightarrow>
2463  (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow>
2464            (\<exists> a \<in> A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2))"
2467 (* The pullback of sets *)
2468 definition thePull where
2469 "thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
2471 lemma wpull_thePull:
2472 "wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
2473 unfolding wpull_def thePull_def by auto
2475 lemma wppull_thePull:
2476 assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
2477 shows
2478 "\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
2479    j a' \<in> A \<and>
2480    e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
2481 (is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
2482 proof(rule bchoice[of ?A' ?phi], default)
2483   fix a' assume a': "a' \<in> ?A'"
2484   hence "fst a' \<in> B1" unfolding thePull_def by auto
2485   moreover
2486   from a' have "snd a' \<in> B2" unfolding thePull_def by auto
2487   moreover have "f1 (fst a') = f2 (snd a')"
2488   using a' unfolding csquare_def thePull_def by auto
2489   ultimately show "\<exists> ja'. ?phi a' ja'"
2490   using assms unfolding wppull_def by blast
2491 qed
2493 lemma wpull_wppull:
2494 assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
2495 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')"
2496 shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
2497 unfolding wppull_def proof safe
2498   fix b1 b2
2499   assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
2500   then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
2501   using wp unfolding wpull_def by blast
2502   show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
2503   apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
2504 qed
2506 lemma wppull_fstOp_sndOp:
2507 shows "wppull (Collect (split (P OO Q))) (Collect (split P)) (Collect (split Q))
2508   snd fst fst snd (BNF_Def.fstOp P Q) (BNF_Def.sndOp P Q)"
2509 using pick_middlep unfolding wppull_def fstOp_def sndOp_def relcompp.simps by auto
2511 lemma wpull_mmap:
2512 fixes A :: "'a set" and B1 :: "'b1 set" and B2 :: "'b2 set"
2513 assumes wp: "wpull A B1 B2 f1 f2 p1 p2"
2514 shows
2515 "wpull {M. set_of M \<subseteq> A}
2516        {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
2517        (mmap f1) (mmap f2) (mmap p1) (mmap p2)"
2518 unfolding wpull_def proof (safe, unfold Bex_def mem_Collect_eq)
2519   fix N1 :: "'b1 multiset" and N2 :: "'b2 multiset"
2520   assume mmap': "mmap f1 N1 = mmap f2 N2"
2521   and N1[simp]: "set_of N1 \<subseteq> B1"
2522   and N2[simp]: "set_of N2 \<subseteq> B2"
2523   def P \<equiv> "mmap f1 N1"
2524   have P1: "P = mmap f1 N1" and P2: "P = mmap f2 N2" unfolding P_def using mmap' by auto
2525   note P = P1 P2
2526   have fin_N1[simp]: "finite (set_of N1)"
2527    and fin_N2[simp]: "finite (set_of N2)"
2528    and fin_P[simp]: "finite (set_of P)" by auto
2530   def set1 \<equiv> "\<lambda> c. {b1 \<in> set_of N1. f1 b1 = c}"
2531   have set1[simp]: "\<And> c b1. b1 \<in> set1 c \<Longrightarrow> f1 b1 = c" unfolding set1_def by auto
2532   have fin_set1: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set1 c)"
2533     using N1(1) unfolding set1_def multiset_def by auto
2534   have set1_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<noteq> {}"
2535    unfolding set1_def set_of_def P mmap_ge_0 by auto
2536   have supp_N1_set1: "set_of N1 = (\<Union> c \<in> set_of P. set1 c)"
2537     using supp_vimage_mmap[of N1 f1] unfolding set1_def P1 by auto
2538   hence set1_inclN1: "\<And>c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> set_of N1" by auto
2539   hence set1_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> B1" using N1 by blast
2540   have set1_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set1 c \<inter> set1 c' = {}"
2541     unfolding set1_def by auto
2542   have setsum_set1: "\<And> c. setsum (count N1) (set1 c) = count P c"
2543     unfolding P1 set1_def by transfer (auto intro: setsum_cong)
2545   def set2 \<equiv> "\<lambda> c. {b2 \<in> set_of N2. f2 b2 = c}"
2546   have set2[simp]: "\<And> c b2. b2 \<in> set2 c \<Longrightarrow> f2 b2 = c" unfolding set2_def by auto
2547   have fin_set2: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set2 c)"
2548   using N2(1) unfolding set2_def multiset_def by auto
2549   have set2_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<noteq> {}"
2550     unfolding set2_def P2 mmap_ge_0 set_of_def by auto
2551   have supp_N2_set2: "set_of N2 = (\<Union> c \<in> set_of P. set2 c)"
2552     using supp_vimage_mmap[of N2 f2] unfolding set2_def P2 by auto
2553   hence set2_inclN2: "\<And>c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> set_of N2" by auto
2554   hence set2_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> B2" using N2 by blast
2555   have set2_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set2 c \<inter> set2 c' = {}"
2556     unfolding set2_def by auto
2557   have setsum_set2: "\<And> c. setsum (count N2) (set2 c) = count P c"
2558     unfolding P2 set2_def by transfer (auto intro: setsum_cong)
2560   have ss: "\<And> c. c \<in> set_of P \<Longrightarrow> setsum (count N1) (set1 c) = setsum (count N2) (set2 c)"
2561     unfolding setsum_set1 setsum_set2 ..
2562   have "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
2563           \<exists> a \<in> A. p1 a = fst b1b2 \<and> p2 a = snd b1b2"
2564     using wp set1_incl set2_incl unfolding wpull_def Ball_def mem_Collect_eq
2565     by simp (metis set1 set2 set_rev_mp)
2566   then obtain uu where uu:
2567   "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
2568      uu c b1b2 \<in> A \<and> p1 (uu c b1b2) = fst b1b2 \<and> p2 (uu c b1b2) = snd b1b2" by metis
2569   def u \<equiv> "\<lambda> c b1 b2. uu c (b1,b2)"
2570   have u[simp]:
2571   "\<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"
2572   "\<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"
2573   "\<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"
2574     using uu unfolding u_def by auto
2575   {fix c assume c: "c \<in> set_of P"
2576    have "inj2 (u c) (set1 c) (set2 c)" unfolding inj2_def proof clarify
2577      fix b1 b1' b2 b2'
2578      assume "{b1, b1'} \<subseteq> set1 c" "{b2, b2'} \<subseteq> set2 c" and 0: "u c b1 b2 = u c b1' b2'"
2579      hence "p1 (u c b1 b2) = b1 \<and> p2 (u c b1 b2) = b2 \<and>
2580             p1 (u c b1' b2') = b1' \<and> p2 (u c b1' b2') = b2'"
2581      using u(2)[OF c] u(3)[OF c] by simp metis
2582      thus "b1 = b1' \<and> b2 = b2'" using 0 by auto
2583    qed
2584   } note inj = this
2585   def sset \<equiv> "\<lambda> c. {u c b1 b2 | b1 b2. b1 \<in> set1 c \<and> b2 \<in> set2 c}"
2586   have fin_sset[simp]: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (sset c)" unfolding sset_def
2587     using fin_set1 fin_set2 finite_twosets by blast
2588   have sset_A: "\<And> c. c \<in> set_of P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
2589   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
2590    then obtain b1 b2 where b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
2591    and a: "a = u c b1 b2" unfolding sset_def by auto
2592    have "p1 a \<in> set1 c" and p2a: "p2 a \<in> set2 c"
2593    using ac a b1 b2 c u(2) u(3) by simp+
2594    hence "u c (p1 a) (p2 a) = a" unfolding a using b1 b2 inj[OF c]
2595    unfolding inj2_def by (metis c u(2) u(3))
2596   } note u_p12[simp] = this
2597   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
2598    hence "p1 a \<in> set1 c" unfolding sset_def by auto
2599   }note p1[simp] = this
2600   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
2601    hence "p2 a \<in> set2 c" unfolding sset_def by auto
2602   }note p2[simp] = this
2604   {fix c assume c: "c \<in> set_of P"
2605    hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = count N1 b1) \<and>
2606                (\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = count N2 b2)"
2607    unfolding sset_def
2608    using matrix_setsum_finite[OF set1_NE[OF c] fin_set1[OF c]
2609                                  set2_NE[OF c] fin_set2[OF c] inj[OF c] ss[OF c]] by auto
2610   }
2611   then obtain Ms where
2612   ss1: "\<And> c b1. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
2613                    setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = count N1 b1" and
2614   ss2: "\<And> c b2. \<lbrakk>c \<in> set_of P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
2615                    setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = count N2 b2"
2616   by metis
2617   def SET \<equiv> "\<Union> c \<in> set_of P. sset c"
2618   have fin_SET[simp]: "finite SET" unfolding SET_def apply(rule finite_UN_I) by auto
2619   have SET_A: "SET \<subseteq> A" unfolding SET_def using sset_A by blast
2620   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"
2621     unfolding SET_def sset_def by blast
2622   {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
2623    then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
2624     unfolding SET_def by auto
2625    hence "p1 a \<in> set1 c'" unfolding sset_def by auto
2626    hence eq: "c = c'" using p1a c c' set1_disj by auto
2627    hence "a \<in> sset c" using ac' by simp
2628   } note p1_rev = this
2629   {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
2630    then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
2631    unfolding SET_def by auto
2632    hence "p2 a \<in> set2 c'" unfolding sset_def by auto
2633    hence eq: "c = c'" using p2a c c' set2_disj by auto
2634    hence "a \<in> sset c" using ac' by simp
2635   } note p2_rev = this
2637   have "\<forall> a \<in> SET. \<exists> c \<in> set_of P. a \<in> sset c" unfolding SET_def by auto
2638   then obtain h where h: "\<forall> a \<in> SET. h a \<in> set_of P \<and> a \<in> sset (h a)" by metis
2639   have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
2640                       \<Longrightarrow> h (u c b1 b2) = c"
2641   by (metis h p2 set2 u(3) u_SET)
2642   have h_u1: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
2643                       \<Longrightarrow> h (u c b1 b2) = f1 b1"
2644   using h unfolding sset_def by auto
2645   have h_u2: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
2646                       \<Longrightarrow> h (u c b1 b2) = f2 b2"
2647   using h unfolding sset_def by auto
2648   def M \<equiv>
2649     "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)"
2650   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"
2651     unfolding multiset_def by auto
2652   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"
2653     unfolding M_def pcr_multiset_def cr_multiset_def by (auto simp: Abs_multiset_inverse)
2654   have sM: "set_of M \<subseteq> SET" "set_of M \<subseteq> p1 -` (set_of N1)" "set_of M \<subseteq> p2 -` set_of N2"
2655     by (transfer, auto split: split_if_asm)+
2656   show "\<exists>M. set_of M \<subseteq> A \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
2657   proof(rule exI[of _ M], safe)
2658     fix a assume *: "a \<in> set_of M"
2659     from SET_A show "a \<in> A"
2660     proof (cases "a \<in> SET")
2661       case False thus ?thesis using * by transfer' auto
2662     qed blast
2663   next
2664     show "mmap p1 M = N1"
2665     proof(intro multiset_eqI)
2666       fix b1
2667       let ?K = "{a. p1 a = b1 \<and> a \<in># M}"
2668       have "setsum (count M) ?K = count N1 b1"
2669       proof(cases "b1 \<in> set_of N1")
2670         case False
2671         hence "?K = {}" using sM(2) by auto
2672         thus ?thesis using False by auto
2673       next
2674         case True
2675         def c \<equiv> "f1 b1"
2676         have c: "c \<in> set_of P" and b1: "b1 \<in> set1 c"
2677           unfolding set1_def c_def P1 using True by (auto simp: comp_eq_dest[OF set_of_mmap])
2678         with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p1 a = b1 \<and> a \<in> SET}"
2679           by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
2680         also have "... = setsum (count M) ((\<lambda> b2. u c b1 b2) ` (set2 c))"
2681           apply(rule setsum_cong) using c b1 proof safe
2682           fix a assume p1a: "p1 a \<in> set1 c" and "c \<in> set_of P" and "a \<in> SET"
2683           hence ac: "a \<in> sset c" using p1_rev by auto
2684           hence "a = u c (p1 a) (p2 a)" using c by auto
2685           moreover have "p2 a \<in> set2 c" using ac c by auto
2686           ultimately show "a \<in> u c (p1 a) ` set2 c" by auto
2687         qed auto
2688         also have "... = setsum (\<lambda> b2. count M (u c b1 b2)) (set2 c)"
2689           unfolding comp_def[symmetric] apply(rule setsum_reindex)
2690           using inj unfolding inj_on_def inj2_def using b1 c u(3) by blast
2691         also have "... = count N1 b1" unfolding ss1[OF c b1, symmetric]
2692           apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b1 set2)
2693           using True h_u[OF c b1] set2_def u(2,3)[OF c b1] u_SET[OF c b1] by fastforce
2694         finally show ?thesis .
2695       qed
2696       thus "count (mmap p1 M) b1 = count N1 b1" by transfer
2697     qed
2698   next
2699     show "mmap p2 M = N2"
2700     proof(intro multiset_eqI)
2701       fix b2
2702       let ?K = "{a. p2 a = b2 \<and> a \<in># M}"
2703       have "setsum (count M) ?K = count N2 b2"
2704       proof(cases "b2 \<in> set_of N2")
2705         case False
2706         hence "?K = {}" using sM(3) by auto
2707         thus ?thesis using False by auto
2708       next
2709         case True
2710         def c \<equiv> "f2 b2"
2711         have c: "c \<in> set_of P" and b2: "b2 \<in> set2 c"
2712           unfolding set2_def c_def P2 using True by (auto simp: comp_eq_dest[OF set_of_mmap])
2713         with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p2 a = b2 \<and> a \<in> SET}"
2714           by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
2715         also have "... = setsum (count M) ((\<lambda> b1. u c b1 b2) ` (set1 c))"
2716           apply(rule setsum_cong) using c b2 proof safe
2717           fix a assume p2a: "p2 a \<in> set2 c" and "c \<in> set_of P" and "a \<in> SET"
2718           hence ac: "a \<in> sset c" using p2_rev by auto
2719           hence "a = u c (p1 a) (p2 a)" using c by auto
2720           moreover have "p1 a \<in> set1 c" using ac c by auto
2721           ultimately show "a \<in> (\<lambda>x. u c x (p2 a)) ` set1 c" by auto
2722         qed auto
2723         also have "... = setsum (count M o (\<lambda> b1. u c b1 b2)) (set1 c)"
2724           apply(rule setsum_reindex)
2725           using inj unfolding inj_on_def inj2_def using b2 c u(2) by blast
2726         also have "... = setsum (\<lambda> b1. count M (u c b1 b2)) (set1 c)" by simp
2727         also have "... = count N2 b2" unfolding ss2[OF c b2, symmetric] comp_def
2728           apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b2 set1)
2729           using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2] set1_def by fastforce
2730         finally show ?thesis .
2731       qed
2732       thus "count (mmap p2 M) b2 = count N2 b2" by transfer
2733     qed
2734   qed
2735 qed
2737 lemma set_of_bd: "(card_of (set_of x), natLeq) \<in> ordLeq"
2738   by transfer
2739     (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
2741 lemma wppull_mmap:
2742   assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
2743   shows "wppull {M. set_of M \<subseteq> A} {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
2744     (mmap f1) (mmap f2) (mmap e1) (mmap e2) (mmap p1) (mmap p2)"
2745 proof -
2746   from assms obtain j where j: "\<forall>a'\<in>thePull B1 B2 f1 f2.
2747     j a' \<in> A \<and> e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
2748     by (blast dest: wppull_thePull)
2749   then show ?thesis
2750     by (intro wpull_wppull[OF wpull_mmap[OF wpull_thePull], of _ _ _ _ "mmap j"])
2751       (auto simp: comp_eq_dest_lhs[OF mmap_comp[symmetric]] comp_eq_dest[OF set_of_mmap]
2752         intro!: mmap_cong simp del: mem_set_of_iff simp: mem_set_of_iff[symmetric])
2753 qed
2755 bnf "'a multiset"
2756   map: mmap
2757   sets: set_of
2758   bd: natLeq
2759   wits: "{#}"
2760 by (auto simp add: mmap_id0 mmap_comp set_of_mmap natLeq_card_order natLeq_cinfinite set_of_bd
2761   Grp_def relcompp.simps intro: mmap_cong)
2762   (metis wppull_mmap[OF wppull_fstOp_sndOp, unfolded wppull_def
2763     o_eq_dest_lhs[OF mmap_comp[symmetric]] fstOp_def sndOp_def comp_def, simplified])
2765 inductive rel_multiset' where
2766   Zero[intro]: "rel_multiset' R {#} {#}"
2767 | Plus[intro]: "\<lbrakk>R a b; rel_multiset' R M N\<rbrakk> \<Longrightarrow> rel_multiset' R (M + {#a#}) (N + {#b#})"
2769 lemma map_multiset_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"
2770 by (metis image_is_empty multiset.set_map set_of_eq_empty_iff)
2772 lemma map_multiset_Zero[simp]: "mmap f {#} = {#}" by simp
2774 lemma rel_multiset_Zero: "rel_multiset R {#} {#}"
2775 unfolding rel_multiset_def Grp_def by auto
2777 declare multiset.count[simp]
2778 declare Abs_multiset_inverse[simp]
2779 declare multiset.count_inverse[simp]
2780 declare union_preserves_multiset[simp]
2782 lemma map_multiset_Plus[simp]: "mmap f (M1 + M2) = mmap f M1 + mmap f M2"
2783 proof (intro multiset_eqI, transfer fixing: f)
2784   fix x :: 'a and M1 M2 :: "'b \<Rightarrow> nat"
2785   assume "M1 \<in> multiset" "M2 \<in> multiset"
2786   hence "setsum M1 {a. f a = x \<and> 0 < M1 a} = setsum M1 {a. f a = x \<and> 0 < M1 a + M2 a}"
2787         "setsum M2 {a. f a = x \<and> 0 < M2 a} = setsum M2 {a. f a = x \<and> 0 < M1 a + M2 a}"
2788     by (auto simp: multiset_def intro!: setsum_mono_zero_cong_left)
2789   then show "(\<Sum>a | f a = x \<and> 0 < M1 a + M2 a. M1 a + M2 a) =
2790        setsum M1 {a. f a = x \<and> 0 < M1 a} +
2791        setsum M2 {a. f a = x \<and> 0 < M2 a}"
2792     by (auto simp: setsum.distrib[symmetric])
2793 qed
2795 lemma map_multiset_single[simp]: "mmap f {#a#} = {#f a#}"
2796   by transfer auto
2798 lemma rel_multiset_Plus:
2799 assumes ab: "R a b" and MN: "rel_multiset R M N"
2800 shows "rel_multiset R (M + {#a#}) (N + {#b#})"
2801 proof-
2802   {fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
2803    hence "\<exists>ya. mmap fst y + {#a#} = mmap fst ya \<and>
2804                mmap snd y + {#b#} = mmap snd ya \<and>
2805                set_of ya \<subseteq> {(x, y). R x y}"
2806    apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
2807   }
2808   thus ?thesis
2809   using assms
2810   unfolding rel_multiset_def Grp_def by force
2811 qed
2813 lemma rel_multiset'_imp_rel_multiset:
2814 "rel_multiset' R M N \<Longrightarrow> rel_multiset R M N"
2815 apply(induct rule: rel_multiset'.induct)
2816 using rel_multiset_Zero rel_multiset_Plus by auto
2818 lemma mcard_mmap[simp]: "mcard (mmap f M) = mcard M"
2819 proof -
2820   def A \<equiv> "\<lambda> b. {a. f a = b \<and> a \<in># M}"
2821   let ?B = "{b. 0 < setsum (count M) (A b)}"
2822   have "{b. \<exists>a. f a = b \<and> a \<in># M} \<subseteq> f ` {a. a \<in># M}" by auto
2823   moreover have "finite (f ` {a. a \<in># M})" apply(rule finite_imageI)
2824   using finite_Collect_mem .
2825   ultimately have fin: "finite {b. \<exists>a. f a = b \<and> a \<in># M}" by(rule finite_subset)
2826   have i: "inj_on A ?B" unfolding inj_on_def A_def apply clarsimp
2827     by (metis (lifting, full_types) mem_Collect_eq neq0_conv setsum.neutral)
2828   have 0: "\<And> b. 0 < setsum (count M) (A b) \<longleftrightarrow> (\<exists> a \<in> A b. count M a > 0)"
2829   apply safe
2830     apply (metis less_not_refl setsum_gt_0_iff setsum_infinite)
2831     by (metis A_def finite_Collect_conjI finite_Collect_mem setsum_gt_0_iff)
2832   hence AB: "A ` ?B = {A b | b. \<exists> a \<in> A b. count M a > 0}" by auto
2834   have "setsum (\<lambda> x. setsum (count M) (A x)) ?B = setsum (setsum (count M) o A) ?B"
2835   unfolding comp_def ..
2836   also have "... = (\<Sum>x\<in> A ` ?B. setsum (count M) x)"
2837   unfolding setsum.reindex [OF i, symmetric] ..
2838   also have "... = setsum (count M) (\<Union>x\<in>A ` {b. 0 < setsum (count M) (A b)}. x)"
2839   (is "_ = setsum (count M) ?J")
2840   apply(rule setsum.UNION_disjoint[symmetric])
2841   using 0 fin unfolding A_def by auto
2842   also have "?J = {a. a \<in># M}" unfolding AB unfolding A_def by auto
2843   finally have "setsum (\<lambda> x. setsum (count M) (A x)) ?B =
2844                 setsum (count M) {a. a \<in># M}" .
2845   then show ?thesis unfolding mcard_unfold_setsum A_def by transfer
2846 qed
2848 lemma rel_multiset_mcard:
2849 assumes "rel_multiset R M N"
2850 shows "mcard M = mcard N"
2851 using assms unfolding rel_multiset_def Grp_def by auto
2854 assumes empty: "P {#} {#}"
2855 and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
2856 and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
2857 shows "P M N"
2858 apply(induct N rule: multiset_induct)
2859   apply(induct M rule: multiset_induct, rule empty, erule addL)
2860   apply(induct M rule: multiset_induct, erule addR, erule addR)
2861 done
2863 lemma multiset_induct2_mcard[consumes 1, case_names empty add]:
2864 assumes c: "mcard M = mcard N"
2865 and empty: "P {#} {#}"
2866 and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
2867 shows "P M N"
2868 using c proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
2869   case (less M)  show ?case
2870   proof(cases "M = {#}")
2871     case True hence "N = {#}" using less.prems by auto
2872     thus ?thesis using True empty by auto
2873   next
2874     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2875     have "N \<noteq> {#}" using False less.prems by auto
2876     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
2877     have "mcard M1 = mcard N1" using less.prems unfolding M N by auto
2878     thus ?thesis using M N less.hyps add by auto
2879   qed
2880 qed
2882 lemma msed_map_invL:
2883 assumes "mmap f (M + {#a#}) = N"
2884 shows "\<exists> N1. N = N1 + {#f a#} \<and> mmap f M = N1"
2885 proof-
2886   have "f a \<in># N"
2887   using assms multiset.set_map[of f "M + {#a#}"] by auto
2888   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
2889   have "mmap f M = N1" using assms unfolding N by simp
2890   thus ?thesis using N by blast
2891 qed
2893 lemma msed_map_invR:
2894 assumes "mmap f M = N + {#b#}"
2895 shows "\<exists> M1 a. M = M1 + {#a#} \<and> f a = b \<and> mmap f M1 = N"
2896 proof-
2897   obtain a where a: "a \<in># M" and fa: "f a = b"
2898   using multiset.set_map[of f M] unfolding assms
2899   by (metis image_iff mem_set_of_iff union_single_eq_member)
2900   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
2901   have "mmap f M1 = N" using assms unfolding M fa[symmetric] by simp
2902   thus ?thesis using M fa by blast
2903 qed
2905 lemma msed_rel_invL:
2906 assumes "rel_multiset R (M + {#a#}) N"
2907 shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_multiset R M N1"
2908 proof-
2909   obtain K where KM: "mmap fst K = M + {#a#}"
2910   and KN: "mmap snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2911   using assms
2912   unfolding rel_multiset_def Grp_def by auto
2913   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
2914   and K1M: "mmap fst K1 = M" using msed_map_invR[OF KM] by auto
2915   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "mmap snd K1 = N1"
2916   using msed_map_invL[OF KN[unfolded K]] by auto
2917   have Rab: "R a (snd ab)" using sK a unfolding K by auto
2918   have "rel_multiset R M N1" using sK K1M K1N1
2919   unfolding K rel_multiset_def Grp_def by auto
2920   thus ?thesis using N Rab by auto
2921 qed
2923 lemma msed_rel_invR:
2924 assumes "rel_multiset R M (N + {#b#})"
2925 shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_multiset R M1 N"
2926 proof-
2927   obtain K where KN: "mmap snd K = N + {#b#}"
2928   and KM: "mmap fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
2929   using assms
2930   unfolding rel_multiset_def Grp_def by auto
2931   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
2932   and K1N: "mmap snd K1 = N" using msed_map_invR[OF KN] by auto
2933   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "mmap fst K1 = M1"
2934   using msed_map_invL[OF KM[unfolded K]] by auto
2935   have Rab: "R (fst ab) b" using sK b unfolding K by auto
2936   have "rel_multiset R M1 N" using sK K1N K1M1
2937   unfolding K rel_multiset_def Grp_def by auto
2938   thus ?thesis using M Rab by auto
2939 qed
2941 lemma rel_multiset_imp_rel_multiset':
2942 assumes "rel_multiset R M N"
2943 shows "rel_multiset' R M N"
2944 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
2945   case (less M)
2946   have c: "mcard M = mcard N" using rel_multiset_mcard[OF less.prems] .
2947   show ?case
2948   proof(cases "M = {#}")
2949     case True hence "N = {#}" using c by simp
2950     thus ?thesis using True rel_multiset'.Zero by auto
2951   next
2952     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2953     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_multiset R M1 N1"
2954     using msed_rel_invL[OF less.prems[unfolded M]] by auto
2955     have "rel_multiset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
2956     thus ?thesis using rel_multiset'.Plus[of R a b, OF R] unfolding M N by simp
2957   qed
2958 qed
2960 lemma rel_multiset_rel_multiset':
2961 "rel_multiset R M N = rel_multiset' R M N"
2962 using  rel_multiset_imp_rel_multiset' rel_multiset'_imp_rel_multiset by auto
2964 (* The main end product for rel_multiset: inductive characterization *)
2965 theorems rel_multiset_induct[case_names empty add, induct pred: rel_multiset] =
2966          rel_multiset'.induct[unfolded rel_multiset_rel_multiset'[symmetric]]
2968 end