src/HOL/Library/Multiset.thy
 author wenzelm Mon Dec 28 21:47:32 2015 +0100 (2015-12-28) changeset 61955 e96292f32c3c parent 61890 f6ded81f5690 child 62082 614ef6d7a6b6 permissions -rw-r--r--
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
1 (*  Title:      HOL/Library/Multiset.thy
2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
3     Author:     Andrei Popescu, TU Muenchen
4     Author:     Jasmin Blanchette, Inria, LORIA, MPII
5     Author:     Dmitriy Traytel, TU Muenchen
6     Author:     Mathias Fleury, MPII
7 *)
9 section \<open>(Finite) multisets\<close>
11 theory Multiset
12 imports Main
13 begin
15 subsection \<open>The type of multisets\<close>
17 definition "multiset = {f :: 'a \<Rightarrow> nat. finite {x. f x > 0}}"
19 typedef 'a multiset = "multiset :: ('a \<Rightarrow> nat) set"
20   morphisms count Abs_multiset
21   unfolding multiset_def
22 proof
23   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
24 qed
26 setup_lifting type_definition_multiset
28 abbreviation Melem :: "'a \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<in>#" 50)
29   where "a \<in># M \<equiv> 0 < count M a"
31 notation (ASCII)
32   Melem  ("(_/ :# _)" [50, 51] 50)  (* FIXME !? *)
34 lemma multiset_eq_iff: "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
35   by (simp only: count_inject [symmetric] fun_eq_iff)
37 lemma multiset_eqI: "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
38   using multiset_eq_iff by auto
40 text \<open>Preservation of the representing set @{term multiset}.\<close>
42 lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
43   by (simp add: multiset_def)
45 lemma only1_in_multiset: "(\<lambda>b. if b = a then n else 0) \<in> multiset"
46   by (simp add: multiset_def)
48 lemma union_preserves_multiset: "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
49   by (simp add: multiset_def)
51 lemma diff_preserves_multiset:
52   assumes "M \<in> multiset"
53   shows "(\<lambda>a. M a - N a) \<in> multiset"
54 proof -
55   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
56     by auto
57   with assms show ?thesis
58     by (auto simp add: multiset_def intro: finite_subset)
59 qed
61 lemma filter_preserves_multiset:
62   assumes "M \<in> multiset"
63   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
64 proof -
65   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
66     by auto
67   with assms show ?thesis
68     by (auto simp add: multiset_def intro: finite_subset)
69 qed
71 lemmas in_multiset = const0_in_multiset only1_in_multiset
72   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
75 subsection \<open>Representing multisets\<close>
77 text \<open>Multiset enumeration\<close>
79 instantiation multiset :: (type) cancel_comm_monoid_add
80 begin
82 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
83 by (rule const0_in_multiset)
85 abbreviation Mempty :: "'a multiset" ("{#}") where
86   "Mempty \<equiv> 0"
88 lift_definition plus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
89 by (rule union_preserves_multiset)
91 lift_definition minus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
92 by (rule diff_preserves_multiset)
94 instance
95   by (standard; transfer; simp add: fun_eq_iff)
97 end
99 lift_definition single :: "'a \<Rightarrow> 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
100 by (rule only1_in_multiset)
102 syntax
103   "_multiset" :: "args \<Rightarrow> 'a multiset"    ("{#(_)#}")
104 translations
105   "{#x, xs#}" == "{#x#} + {#xs#}"
106   "{#x#}" == "CONST single x"
108 lemma count_empty [simp]: "count {#} a = 0"
109   by (simp add: zero_multiset.rep_eq)
111 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
112   by (simp add: single.rep_eq)
115 subsection \<open>Basic operations\<close>
117 subsubsection \<open>Union\<close>
119 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
120   by (simp add: plus_multiset.rep_eq)
123 subsubsection \<open>Difference\<close>
125 instantiation multiset :: (type) comm_monoid_diff
126 begin
128 instance
129   by (standard; transfer; simp add: fun_eq_iff)
131 end
133 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
134   by (simp add: minus_multiset.rep_eq)
136 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
137   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
139 lemma diff_cancel[simp]: "A - A = {#}"
140   by (fact Groups.diff_cancel)
142 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
143   by (fact add_diff_cancel_right')
145 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
146   by (fact add_diff_cancel_left')
148 lemma diff_right_commute:
149   fixes M N Q :: "'a multiset"
150   shows "M - N - Q = M - Q - N"
151   by (fact diff_right_commute)
154   fixes M N Q :: "'a multiset"
155   shows "M - (N + Q) = M - N - Q"
156   by (rule sym) (fact diff_diff_add)
158 lemma insert_DiffM: "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
159   by (clarsimp simp: multiset_eq_iff)
161 lemma insert_DiffM2 [simp]: "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
162   by (clarsimp simp: multiset_eq_iff)
164 lemma diff_union_swap: "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
165   by (auto simp add: multiset_eq_iff)
167 lemma diff_union_single_conv: "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
168   by (simp add: multiset_eq_iff)
171 subsubsection \<open>Equality of multisets\<close>
173 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
174   by (simp add: multiset_eq_iff)
176 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
177   by (auto simp add: multiset_eq_iff)
179 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
180   by (auto simp add: multiset_eq_iff)
182 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
183   by (auto simp add: multiset_eq_iff)
185 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
186   by (auto simp add: multiset_eq_iff)
188 lemma diff_single_trivial: "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
189   by (auto simp add: multiset_eq_iff)
191 lemma diff_single_eq_union: "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
192   by auto
194 lemma union_single_eq_diff: "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
195   by (auto dest: sym)
197 lemma union_single_eq_member: "M + {#x#} = N \<Longrightarrow> x \<in># N"
198   by auto
200 lemma union_is_single: "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}"
201   (is "?lhs = ?rhs")
202 proof
203   show ?lhs if ?rhs using that by auto
204   show ?rhs if ?lhs
205     using that by (simp add: multiset_eq_iff split: if_splits) (metis add_is_1)
206 qed
208 lemma single_is_union: "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
209   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
212   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"
213   (is "?lhs \<longleftrightarrow> ?rhs")
214 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
215 proof
216   show ?lhs if ?rhs
217     using that
219       (drule sym, simp add: add.assoc [symmetric])
220   show ?rhs if ?lhs
221   proof (cases "a = b")
222     case True with \<open>?lhs\<close> show ?thesis by simp
223   next
224     case False
225     from \<open>?lhs\<close> have "a \<in># N + {#b#}" by (rule union_single_eq_member)
226     with False have "a \<in># N" by auto
227     moreover from \<open>?lhs\<close> have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
228     moreover note False
229     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
230   qed
231 qed
233 lemma insert_noteq_member:
234   assumes BC: "B + {#b#} = C + {#c#}"
235    and bnotc: "b \<noteq> c"
236   shows "c \<in># B"
237 proof -
238   have "c \<in># C + {#c#}" by simp
239   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
240   then have "c \<in># B + {#b#}" using BC by simp
241   then show "c \<in># B" using nc by simp
242 qed
245   "(M + {#a#} = N + {#b#}) =
246     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
249 lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
250   by (rule exI [where x = "M - {#x#}"]) simp
253   assumes "c \<in># B"
254     and "b \<noteq> c"
255   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
256 proof -
257   from \<open>c \<in># B\<close> obtain A where B: "B = A + {#c#}"
258     by (blast dest: multi_member_split)
259   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
260   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
261     by (simp add: ac_simps)
262   then show ?thesis using B by simp
263 qed
266 subsubsection \<open>Pointwise ordering induced by count\<close>
268 definition subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<subseteq>#" 50)
269   where "A \<subseteq># B = (\<forall>a. count A a \<le> count B a)"
271 definition subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subset>#" 50)
272   where "A \<subset># B = (A \<subseteq># B \<and> A \<noteq> B)"
274 notation (input)
275   subseteq_mset  (infix "\<le>#" 50)
277 notation (ASCII)
278   subseteq_mset  (infix "<=#" 50) and
279   subset_mset  (infix "<#" 50)
281 interpretation subset_mset: ordered_ab_semigroup_add_imp_le "op +" "op -" "op \<subseteq>#" "op \<subset>#"
282   by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym)
284 lemma mset_less_eqI: "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le># B"
285   by (simp add: subseteq_mset_def)
287 lemma mset_le_exists_conv: "(A::'a multiset) \<le># B \<longleftrightarrow> (\<exists>C. B = A + C)"
288   unfolding subseteq_mset_def
289   apply (rule iffI)
290    apply (rule exI [where x = "B - A"])
291    apply (auto intro: multiset_eq_iff [THEN iffD2])
292   done
294 interpretation subset_mset: ordered_cancel_comm_monoid_diff  "op +" "op -" 0 "op \<le>#" "op <#"
295   by standard (simp, fact mset_le_exists_conv)
297 lemma mset_le_mono_add_right_cancel [simp]: "(A::'a multiset) + C \<le># B + C \<longleftrightarrow> A \<le># B"
298   by (fact subset_mset.add_le_cancel_right)
300 lemma mset_le_mono_add_left_cancel [simp]: "C + (A::'a multiset) \<le># C + B \<longleftrightarrow> A \<le># B"
301   by (fact subset_mset.add_le_cancel_left)
303 lemma mset_le_mono_add: "(A::'a multiset) \<le># B \<Longrightarrow> C \<le># D \<Longrightarrow> A + C \<le># B + D"
304   by (fact subset_mset.add_mono)
306 lemma mset_le_add_left [simp]: "(A::'a multiset) \<le># A + B"
307   unfolding subseteq_mset_def by auto
309 lemma mset_le_add_right [simp]: "B \<le># (A::'a multiset) + B"
310   unfolding subseteq_mset_def by auto
312 lemma mset_le_single: "a \<in># B \<Longrightarrow> {#a#} \<le># B"
313   by (simp add: subseteq_mset_def)
315 lemma multiset_diff_union_assoc:
316   fixes A B C D :: "'a multiset"
317   shows "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
320 lemma mset_le_multiset_union_diff_commute:
321   fixes A B C D :: "'a multiset"
322   shows "B \<le># A \<Longrightarrow> A - B + C = A + C - B"
325 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le># M"
326 by(simp add: subseteq_mset_def)
328 lemma mset_lessD: "A <# B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
329 apply (clarsimp simp: subset_mset_def subseteq_mset_def)
330 apply (erule allE [where x = x])
331 apply auto
332 done
334 lemma mset_leD: "A \<le># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
335 apply (clarsimp simp: subset_mset_def subseteq_mset_def)
336 apply (erule allE [where x = x])
337 apply auto
338 done
340 lemma mset_less_insertD: "(A + {#x#} <# B) \<Longrightarrow> (x \<in># B \<and> A <# B)"
341 apply (rule conjI)
342  apply (simp add: mset_lessD)
343 apply (clarsimp simp: subset_mset_def subseteq_mset_def)
344 apply safe
345  apply (erule_tac x = a in allE)
346  apply (auto split: split_if_asm)
347 done
349 lemma mset_le_insertD: "(A + {#x#} \<le># B) \<Longrightarrow> (x \<in># B \<and> A \<le># B)"
350 apply (rule conjI)
351  apply (simp add: mset_leD)
352 apply (force simp: subset_mset_def subseteq_mset_def split: split_if_asm)
353 done
355 lemma mset_less_of_empty[simp]: "A <# {#} \<longleftrightarrow> False"
356   by (auto simp add: subseteq_mset_def subset_mset_def multiset_eq_iff)
358 lemma empty_le[simp]: "{#} \<le># A"
359   unfolding mset_le_exists_conv by auto
361 lemma le_empty[simp]: "(M \<le># {#}) = (M = {#})"
362   unfolding mset_le_exists_conv by auto
364 lemma multi_psub_of_add_self[simp]: "A <# A + {#x#}"
365   by (auto simp: subset_mset_def subseteq_mset_def)
367 lemma multi_psub_self[simp]: "(A::'a multiset) <# A = False"
368   by simp
370 lemma mset_less_add_bothsides: "N + {#x#} <# M + {#x#} \<Longrightarrow> N <# M"
371   by (fact subset_mset.add_less_imp_less_right)
373 lemma mset_less_empty_nonempty: "{#} <# S \<longleftrightarrow> S \<noteq> {#}"
374   by (auto simp: subset_mset_def subseteq_mset_def)
376 lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} <# B"
377   by (auto simp: subset_mset_def subseteq_mset_def multiset_eq_iff)
380 subsubsection \<open>Intersection\<close>
382 definition inf_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
383   multiset_inter_def: "inf_subset_mset A B = A - (A - B)"
385 interpretation subset_mset: semilattice_inf inf_subset_mset "op \<le>#" "op <#"
386 proof -
387   have [simp]: "m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" for m n q :: nat
388     by arith
389   show "class.semilattice_inf op #\<inter> op \<le># op <#"
390     by standard (auto simp add: multiset_inter_def subseteq_mset_def)
391 qed
394 lemma multiset_inter_count [simp]:
395   fixes A B :: "'a multiset"
396   shows "count (A #\<inter> B) x = min (count A x) (count B x)"
397   by (simp add: multiset_inter_def)
399 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
400   by (rule multiset_eqI) auto
402 lemma multiset_union_diff_commute:
403   assumes "B #\<inter> C = {#}"
404   shows "A + B - C = A - C + B"
405 proof (rule multiset_eqI)
406   fix x
407   from assms have "min (count B x) (count C x) = 0"
408     by (auto simp add: multiset_eq_iff)
409   then have "count B x = 0 \<or> count C x = 0"
410     by auto
411   then show "count (A + B - C) x = count (A - C + B) x"
412     by auto
413 qed
415 lemma empty_inter [simp]: "{#} #\<inter> M = {#}"
416   by (simp add: multiset_eq_iff)
418 lemma inter_empty [simp]: "M #\<inter> {#} = {#}"
419   by (simp add: multiset_eq_iff)
421 lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
422   by (simp add: multiset_eq_iff)
424 lemma inter_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
425   by (simp add: multiset_eq_iff)
427 lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
428   by (simp add: multiset_eq_iff)
430 lemma inter_add_right2: "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
431   by (simp add: multiset_eq_iff)
434 subsubsection \<open>Bounded union\<close>
436 definition sup_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"(infixl "#\<union>" 70)
437   where "sup_subset_mset A B = A + (B - A)"
439 interpretation subset_mset: semilattice_sup sup_subset_mset "op \<le>#" "op <#"
440 proof -
441   have [simp]: "m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" for m n q :: nat
442     by arith
443   show "class.semilattice_sup op #\<union> op \<le># op <#"
444     by standard (auto simp add: sup_subset_mset_def subseteq_mset_def)
445 qed
447 lemma sup_subset_mset_count [simp]: "count (A #\<union> B) x = max (count A x) (count B x)"
448   by (simp add: sup_subset_mset_def)
450 lemma empty_sup [simp]: "{#} #\<union> M = M"
451   by (simp add: multiset_eq_iff)
453 lemma sup_empty [simp]: "M #\<union> {#} = M"
454   by (simp add: multiset_eq_iff)
456 lemma sup_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
457   by (simp add: multiset_eq_iff)
459 lemma sup_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
460   by (simp add: multiset_eq_iff)
462 lemma sup_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
463   by (simp add: multiset_eq_iff)
465 lemma sup_add_right2: "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
466   by (simp add: multiset_eq_iff)
468 subsubsection \<open>Subset is an order\<close>
469 interpretation subset_mset: order "op \<le>#" "op <#" by unfold_locales auto
471 subsubsection \<open>Filter (with comprehension syntax)\<close>
473 text \<open>Multiset comprehension\<close>
475 lift_definition filter_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"
476 is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
477 by (rule filter_preserves_multiset)
479 lemma count_filter_mset [simp]: "count (filter_mset P M) a = (if P a then count M a else 0)"
480   by (simp add: filter_mset.rep_eq)
482 lemma filter_empty_mset [simp]: "filter_mset P {#} = {#}"
483   by (rule multiset_eqI) simp
485 lemma filter_single_mset [simp]: "filter_mset P {#x#} = (if P x then {#x#} else {#})"
486   by (rule multiset_eqI) simp
488 lemma filter_union_mset [simp]: "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
489   by (rule multiset_eqI) simp
491 lemma filter_diff_mset [simp]: "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
492   by (rule multiset_eqI) simp
494 lemma filter_inter_mset [simp]: "filter_mset P (M #\<inter> N) = filter_mset P M #\<inter> filter_mset P N"
495   by (rule multiset_eqI) simp
497 lemma multiset_filter_subset[simp]: "filter_mset f M \<le># M"
498   by (simp add: mset_less_eqI)
500 lemma multiset_filter_mono:
501   assumes "A \<le># B"
502   shows "filter_mset f A \<le># filter_mset f B"
503 proof -
504   from assms[unfolded mset_le_exists_conv]
505   obtain C where B: "B = A + C" by auto
506   show ?thesis unfolding B by auto
507 qed
509 syntax (ASCII)
510   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
511 syntax
512   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
513 translations
514   "{#x \<in># M. P#}" == "CONST filter_mset (\<lambda>x. P) M"
517 subsubsection \<open>Set of elements\<close>
519 definition set_mset :: "'a multiset \<Rightarrow> 'a set"
520   where "set_mset M = {x. x \<in># M}"
522 lemma set_mset_empty [simp]: "set_mset {#} = {}"
523 by (simp add: set_mset_def)
525 lemma set_mset_single [simp]: "set_mset {#b#} = {b}"
526 by (simp add: set_mset_def)
528 lemma set_mset_union [simp]: "set_mset (M + N) = set_mset M \<union> set_mset N"
529 by (auto simp add: set_mset_def)
531 lemma set_mset_eq_empty_iff [simp]: "(set_mset M = {}) = (M = {#})"
532 by (auto simp add: set_mset_def multiset_eq_iff)
534 lemma mem_set_mset_iff [simp]: "(x \<in> set_mset M) = (x \<in># M)"
535 by (auto simp add: set_mset_def)
537 lemma set_mset_filter [simp]: "set_mset {# x\<in>#M. P x #} = set_mset M \<inter> {x. P x}"
538 by (auto simp add: set_mset_def)
540 lemma finite_set_mset [iff]: "finite (set_mset M)"
541   using count [of M] by (simp add: multiset_def set_mset_def)
543 lemma finite_Collect_mem [iff]: "finite {x. x \<in># M}"
544   unfolding set_mset_def[symmetric] by simp
546 lemma set_mset_mono: "A \<le># B \<Longrightarrow> set_mset A \<subseteq> set_mset B"
547   by (metis mset_leD subsetI mem_set_mset_iff)
549 lemma ball_set_mset_iff: "(\<forall>x \<in> set_mset M. P x) \<longleftrightarrow> (\<forall>x. x \<in># M \<longrightarrow> P x)"
550   by auto
553 subsubsection \<open>Size\<close>
555 definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
557 lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
558   by (auto simp: wcount_def add_mult_distrib)
560 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
561   "size_multiset f M = setsum (wcount f M) (set_mset M)"
563 lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
565 instantiation multiset :: (type) size
566 begin
568 definition size_multiset where
569   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
570 instance ..
572 end
574 lemmas size_multiset_overloaded_eq =
575   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
577 lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
578 by (simp add: size_multiset_def)
580 lemma size_empty [simp]: "size {#} = 0"
583 lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
584 by (simp add: size_multiset_eq)
586 lemma size_single [simp]: "size {#b#} = 1"
589 lemma setsum_wcount_Int:
590   "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_mset N) = setsum (wcount f N) A"
591 apply (induct rule: finite_induct)
592  apply simp
593 apply (simp add: Int_insert_left set_mset_def wcount_def)
594 done
596 lemma size_multiset_union [simp]:
597   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
598 apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
599 apply (subst Int_commute)
600 apply (simp add: setsum_wcount_Int)
601 done
603 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
606 lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
607 by (auto simp add: size_multiset_eq multiset_eq_iff)
609 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
612 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
613 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
615 lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a \<in># M"
616 apply (unfold size_multiset_overloaded_eq)
617 apply (drule setsum_SucD)
618 apply auto
619 done
621 lemma size_eq_Suc_imp_eq_union:
622   assumes "size M = Suc n"
623   shows "\<exists>a N. M = N + {#a#}"
624 proof -
625   from assms obtain a where "a \<in># M"
626     by (erule size_eq_Suc_imp_elem [THEN exE])
627   then have "M = M - {#a#} + {#a#}" by simp
628   then show ?thesis by blast
629 qed
631 lemma size_mset_mono:
632   fixes A B :: "'a multiset"
633   assumes "A \<le># B"
634   shows "size A \<le> size B"
635 proof -
636   from assms[unfolded mset_le_exists_conv]
637   obtain C where B: "B = A + C" by auto
638   show ?thesis unfolding B by (induct C) auto
639 qed
641 lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) \<le> size M"
642 by (rule size_mset_mono[OF multiset_filter_subset])
644 lemma size_Diff_submset:
645   "M \<le># M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)"
646 by (metis add_diff_cancel_left' size_union mset_le_exists_conv)
648 subsection \<open>Induction and case splits\<close>
650 theorem multiset_induct [case_names empty add, induct type: multiset]:
651   assumes empty: "P {#}"
652   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
653   shows "P M"
654 proof (induct n \<equiv> "size M" arbitrary: M)
655   case 0 thus "P M" by (simp add: empty)
656 next
657   case (Suc k)
658   obtain N x where "M = N + {#x#}"
659     using \<open>Suc k = size M\<close> [symmetric]
660     using size_eq_Suc_imp_eq_union by fast
661   with Suc add show "P M" by simp
662 qed
664 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
665 by (induct M) auto
667 lemma multiset_cases [cases type]:
668   obtains (empty) "M = {#}"
669     | (add) N x where "M = N + {#x#}"
670   using assms by (induct M) simp_all
672 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
673 by (cases "B = {#}") (auto dest: multi_member_split)
675 lemma multiset_partition: "M = {# x\<in>#M. P x #} + {# x\<in>#M. \<not> P x #}"
676 apply (subst multiset_eq_iff)
677 apply auto
678 done
680 lemma mset_less_size: "(A::'a multiset) <# B \<Longrightarrow> size A < size B"
681 proof (induct A arbitrary: B)
682   case (empty M)
683   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
684   then obtain M' x where "M = M' + {#x#}"
685     by (blast dest: multi_nonempty_split)
686   then show ?case by simp
687 next
688   case (add S x T)
689   have IH: "\<And>B. S <# B \<Longrightarrow> size S < size B" by fact
690   have SxsubT: "S + {#x#} <# T" by fact
691   then have "x \<in># T" and "S <# T" by (auto dest: mset_less_insertD)
692   then obtain T' where T: "T = T' + {#x#}"
693     by (blast dest: multi_member_split)
694   then have "S <# T'" using SxsubT
695     by (blast intro: mset_less_add_bothsides)
696   then have "size S < size T'" using IH by simp
697   then show ?case using T by simp
698 qed
701 lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}"
702 by (cases M) auto
704 subsubsection \<open>Strong induction and subset induction for multisets\<close>
706 text \<open>Well-foundedness of strict subset relation\<close>
708 lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M <# N}"
709 apply (rule wf_measure [THEN wf_subset, where f1=size])
710 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
711 done
713 lemma full_multiset_induct [case_names less]:
714 assumes ih: "\<And>B. \<forall>(A::'a multiset). A <# B \<longrightarrow> P A \<Longrightarrow> P B"
715 shows "P B"
716 apply (rule wf_less_mset_rel [THEN wf_induct])
717 apply (rule ih, auto)
718 done
720 lemma multi_subset_induct [consumes 2, case_names empty add]:
721   assumes "F \<le># A"
722     and empty: "P {#}"
723     and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
724   shows "P F"
725 proof -
726   from \<open>F \<le># A\<close>
727   show ?thesis
728   proof (induct F)
729     show "P {#}" by fact
730   next
731     fix x F
732     assume P: "F \<le># A \<Longrightarrow> P F" and i: "F + {#x#} \<le># A"
733     show "P (F + {#x#})"
734     proof (rule insert)
735       from i show "x \<in># A" by (auto dest: mset_le_insertD)
736       from i have "F \<le># A" by (auto dest: mset_le_insertD)
737       with P show "P F" .
738     qed
739   qed
740 qed
743 subsection \<open>The fold combinator\<close>
745 definition fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
746 where
747   "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"
749 lemma fold_mset_empty [simp]: "fold_mset f s {#} = s"
750   by (simp add: fold_mset_def)
752 context comp_fun_commute
753 begin
755 lemma fold_mset_insert: "fold_mset f s (M + {#x#}) = f x (fold_mset f s M)"
756 proof -
757   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
758     by (fact comp_fun_commute_funpow)
759   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
760     by (fact comp_fun_commute_funpow)
761   show ?thesis
762   proof (cases "x \<in> set_mset M")
763     case False
764     then have *: "count (M + {#x#}) x = 1" by simp
765     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_mset M) =
766       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_mset M)"
767       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
768     with False * show ?thesis
769       by (simp add: fold_mset_def del: count_union)
770   next
771     case True
772     def N \<equiv> "set_mset M - {x}"
773     from N_def True have *: "set_mset M = insert x N" "x \<notin> N" "finite N" by auto
774     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
775       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
776       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
777     with * show ?thesis by (simp add: fold_mset_def del: count_union) simp
778   qed
779 qed
781 corollary fold_mset_single [simp]: "fold_mset f s {#x#} = f x s"
782 proof -
783   have "fold_mset f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
784   then show ?thesis by simp
785 qed
787 lemma fold_mset_fun_left_comm: "f x (fold_mset f s M) = fold_mset f (f x s) M"
788   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
790 lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
791 proof (induct M)
792   case empty then show ?case by simp
793 next
794   case (add M x)
795   have "M + {#x#} + N = (M + N) + {#x#}"
796     by (simp add: ac_simps)
797   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
798 qed
800 lemma fold_mset_fusion:
801   assumes "comp_fun_commute g"
802     and *: "\<And>x y. h (g x y) = f x (h y)"
803   shows "h (fold_mset g w A) = fold_mset f (h w) A"
804 proof -
805   interpret comp_fun_commute g by (fact assms)
806   from * show ?thesis by (induct A) auto
807 qed
809 end
811 text \<open>
812   A note on code generation: When defining some function containing a
813   subterm @{term "fold_mset F"}, code generation is not automatic. When
814   interpreting locale \<open>left_commutative\<close> with \<open>F\<close>, the
815   would be code thms for @{const fold_mset} become thms like
816   @{term "fold_mset F z {#} = z"} where \<open>F\<close> is not a pattern but
817   contains defined symbols, i.e.\ is not a code thm. Hence a separate
818   constant with its own code thms needs to be introduced for \<open>F\<close>. See the image operator below.
819 \<close>
822 subsection \<open>Image\<close>
824 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
825   "image_mset f = fold_mset (plus \<circ> single \<circ> f) {#}"
827 lemma comp_fun_commute_mset_image: "comp_fun_commute (plus \<circ> single \<circ> f)"
828 proof
829 qed (simp add: ac_simps fun_eq_iff)
831 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
832   by (simp add: image_mset_def)
834 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
835 proof -
836   interpret comp_fun_commute "plus \<circ> single \<circ> f"
837     by (fact comp_fun_commute_mset_image)
838   show ?thesis by (simp add: image_mset_def)
839 qed
841 lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
842 proof -
843   interpret comp_fun_commute "plus \<circ> single \<circ> f"
844     by (fact comp_fun_commute_mset_image)
845   show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
846 qed
848 corollary image_mset_insert: "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
849   by simp
851 lemma set_image_mset [simp]: "set_mset (image_mset f M) = image f (set_mset M)"
852   by (induct M) simp_all
854 lemma size_image_mset [simp]: "size (image_mset f M) = size M"
855   by (induct M) simp_all
857 lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
858   by (cases M) auto
860 syntax (ASCII)
861   "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ :# _#})")
862 syntax
863   "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ \<in># _#})")
864 translations
865   "{#e. x \<in># M#}" \<rightleftharpoons> "CONST image_mset (\<lambda>x. e) M"
867 syntax (ASCII)
868   "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ :# _./ _#})")
869 syntax
870   "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ \<in># _./ _#})")
871 translations
872   "{#e | x\<in>#M. P#}" \<rightharpoonup> "{#e. x \<in># {# x\<in>#M. P#}#}"
874 text \<open>
875   This allows to write not just filters like @{term "{#x\<in>#M. x<c#}"}
876   but also images like @{term "{#x+x. x\<in>#M #}"} and @{term [source]
877   "{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as
878   @{term "{#x+x|x\<in>#M. x<c#}"}.
879 \<close>
881 lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M"
882 by (metis mem_set_mset_iff set_image_mset)
884 functor image_mset: image_mset
885 proof -
886   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
887   proof
888     fix A
889     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
890       by (induct A) simp_all
891   qed
892   show "image_mset id = id"
893   proof
894     fix A
895     show "image_mset id A = id A"
896       by (induct A) simp_all
897   qed
898 qed
900 declare
901   image_mset.id [simp]
902   image_mset.identity [simp]
904 lemma image_mset_id[simp]: "image_mset id x = x"
905   unfolding id_def by auto
907 lemma image_mset_cong: "(\<And>x. x \<in># M \<Longrightarrow> f x = g x) \<Longrightarrow> {#f x. x \<in># M#} = {#g x. x \<in># M#}"
908   by (induct M) auto
910 lemma image_mset_cong_pair:
911   "(\<forall>x y. (x, y) \<in># M \<longrightarrow> f x y = g x y) \<Longrightarrow> {#f x y. (x, y) \<in># M#} = {#g x y. (x, y) \<in># M#}"
912   by (metis image_mset_cong split_cong)
915 subsection \<open>Further conversions\<close>
917 primrec mset :: "'a list \<Rightarrow> 'a multiset" where
918   "mset [] = {#}" |
919   "mset (a # x) = mset x + {# a #}"
921 lemma in_multiset_in_set:
922   "x \<in># mset xs \<longleftrightarrow> x \<in> set xs"
923   by (induct xs) simp_all
925 lemma count_mset:
926   "count (mset xs) x = length (filter (\<lambda>y. x = y) xs)"
927   by (induct xs) simp_all
929 lemma mset_zero_iff[simp]: "(mset x = {#}) = (x = [])"
930   by (induct x) auto
932 lemma mset_zero_iff_right[simp]: "({#} = mset x) = (x = [])"
933 by (induct x) auto
935 lemma set_mset_mset[simp]: "set_mset (mset x) = set x"
936 by (induct x) auto
938 lemma mem_set_multiset_eq: "x \<in> set xs = (x \<in># mset xs)"
939 by (induct xs) auto
941 lemma size_mset [simp]: "size (mset xs) = length xs"
942   by (induct xs) simp_all
944 lemma mset_append [simp]: "mset (xs @ ys) = mset xs + mset ys"
945   by (induct xs arbitrary: ys) (auto simp: ac_simps)
947 lemma mset_filter: "mset (filter P xs) = {#x \<in># mset xs. P x #}"
948   by (induct xs) simp_all
950 lemma mset_rev [simp]:
951   "mset (rev xs) = mset xs"
952   by (induct xs) simp_all
954 lemma surj_mset: "surj mset"
955 apply (unfold surj_def)
956 apply (rule allI)
957 apply (rule_tac M = y in multiset_induct)
958  apply auto
959 apply (rule_tac x = "x # xa" in exI)
960 apply auto
961 done
963 lemma set_count_greater_0: "set x = {a. count (mset x) a > 0}"
964 by (induct x) auto
966 lemma distinct_count_atmost_1:
967   "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
968   apply (induct x, simp, rule iffI, simp_all)
969   subgoal for a b
970     apply (rule conjI)
971     apply (simp_all add: set_mset_mset [symmetric] del: set_mset_mset)
972     apply (erule_tac x = a in allE, simp)
973     apply clarify
974     apply (erule_tac x = aa in allE, simp)
975     done
976   done
978 lemma mset_eq_setD: "mset xs = mset ys \<Longrightarrow> set xs = set ys"
979 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
981 lemma set_eq_iff_mset_eq_distinct:
982   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
983     (set x = set y) = (mset x = mset y)"
984 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
986 lemma set_eq_iff_mset_remdups_eq:
987    "(set x = set y) = (mset (remdups x) = mset (remdups y))"
988 apply (rule iffI)
989 apply (simp add: set_eq_iff_mset_eq_distinct[THEN iffD1])
990 apply (drule distinct_remdups [THEN distinct_remdups
991       [THEN set_eq_iff_mset_eq_distinct [THEN iffD2]]])
992 apply simp
993 done
995 lemma mset_compl_union [simp]: "mset [x\<leftarrow>xs. P x] + mset [x\<leftarrow>xs. \<not>P x] = mset xs"
996   by (induct xs) (auto simp: ac_simps)
998 lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls"
999 proof (induct ls arbitrary: i)
1000   case Nil
1001   then show ?case by simp
1002 next
1003   case Cons
1004   then show ?case by (cases i) auto
1005 qed
1007 lemma mset_remove1[simp]: "mset (remove1 a xs) = mset xs - {#a#}"
1008   by (induct xs) (auto simp add: multiset_eq_iff)
1010 lemma mset_eq_length:
1011   assumes "mset xs = mset ys"
1012   shows "length xs = length ys"
1013   using assms by (metis size_mset)
1015 lemma mset_eq_length_filter:
1016   assumes "mset xs = mset ys"
1017   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
1018   using assms by (metis count_mset)
1020 lemma fold_multiset_equiv:
1021   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"
1022     and equiv: "mset xs = mset ys"
1023   shows "List.fold f xs = List.fold f ys"
1024   using f equiv [symmetric]
1025 proof (induct xs arbitrary: ys)
1026   case Nil
1027   then show ?case by simp
1028 next
1029   case (Cons x xs)
1030   then have *: "set ys = set (x # xs)"
1031     by (blast dest: mset_eq_setD)
1032   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
1033     by (rule Cons.prems(1)) (simp_all add: *)
1034   moreover from * have "x \<in> set ys"
1035     by simp
1036   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x"
1037     by (fact fold_remove1_split)
1038   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)"
1039     by (auto intro: Cons.hyps)
1040   ultimately show ?case by simp
1041 qed
1043 lemma mset_insort [simp]: "mset (insort x xs) = mset xs + {#x#}"
1044   by (induct xs) (simp_all add: ac_simps)
1046 lemma mset_map: "mset (map f xs) = image_mset f (mset xs)"
1047   by (induct xs) simp_all
1049 global_interpretation mset_set: folding "\<lambda>x M. {#x#} + M" "{#}"
1050   defines mset_set = "folding.F (\<lambda>x M. {#x#} + M) {#}"
1051   by standard (simp add: fun_eq_iff ac_simps)
1053 lemma count_mset_set [simp]:
1054   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (mset_set A) x = 1" (is "PROP ?P")
1055   "\<not> finite A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?Q")
1056   "x \<notin> A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?R")
1057 proof -
1058   have *: "count (mset_set A) x = 0" if "x \<notin> A" for A
1059   proof (cases "finite A")
1060     case False then show ?thesis by simp
1061   next
1062     case True from True \<open>x \<notin> A\<close> show ?thesis by (induct A) auto
1063   qed
1064   then show "PROP ?P" "PROP ?Q" "PROP ?R"
1065   by (auto elim!: Set.set_insert)
1066 qed \<comment> \<open>TODO: maybe define @{const mset_set} also in terms of @{const Abs_multiset}\<close>
1068 lemma elem_mset_set[simp, intro]: "finite A \<Longrightarrow> x \<in># mset_set A \<longleftrightarrow> x \<in> A"
1069   by (induct A rule: finite_induct) simp_all
1071 context linorder
1072 begin
1074 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
1075 where
1076   "sorted_list_of_multiset M = fold_mset insort [] M"
1078 lemma sorted_list_of_multiset_empty [simp]:
1079   "sorted_list_of_multiset {#} = []"
1080   by (simp add: sorted_list_of_multiset_def)
1082 lemma sorted_list_of_multiset_singleton [simp]:
1083   "sorted_list_of_multiset {#x#} = [x]"
1084 proof -
1085   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1086   show ?thesis by (simp add: sorted_list_of_multiset_def)
1087 qed
1089 lemma sorted_list_of_multiset_insert [simp]:
1090   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
1091 proof -
1092   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1093   show ?thesis by (simp add: sorted_list_of_multiset_def)
1094 qed
1096 end
1098 lemma mset_sorted_list_of_multiset [simp]:
1099   "mset (sorted_list_of_multiset M) = M"
1100 by (induct M) simp_all
1102 lemma sorted_list_of_multiset_mset [simp]:
1103   "sorted_list_of_multiset (mset xs) = sort xs"
1104 by (induct xs) simp_all
1106 lemma finite_set_mset_mset_set[simp]:
1107   "finite A \<Longrightarrow> set_mset (mset_set A) = A"
1108 by (induct A rule: finite_induct) simp_all
1110 lemma infinite_set_mset_mset_set:
1111   "\<not> finite A \<Longrightarrow> set_mset (mset_set A) = {}"
1112 by simp
1114 lemma set_sorted_list_of_multiset [simp]:
1115   "set (sorted_list_of_multiset M) = set_mset M"
1116 by (induct M) (simp_all add: set_insort)
1118 lemma sorted_list_of_mset_set [simp]:
1119   "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
1120 by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
1123 subsection \<open>Replicate operation\<close>
1125 definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
1126   "replicate_mset n x = ((op + {#x#}) ^^ n) {#}"
1128 lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}"
1129   unfolding replicate_mset_def by simp
1131 lemma replicate_mset_Suc[simp]: "replicate_mset (Suc n) x = replicate_mset n x + {#x#}"
1132   unfolding replicate_mset_def by (induct n) (auto intro: add.commute)
1134 lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y"
1135   unfolding replicate_mset_def by (induct n) simp_all
1137 lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)"
1138   unfolding replicate_mset_def by (induct n) simp_all
1140 lemma set_mset_replicate_mset_subset[simp]: "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
1141   by (auto split: if_splits)
1143 lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n"
1144   by (induct n, simp_all)
1146 lemma count_le_replicate_mset_le: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<le># M"
1147   by (auto simp add: assms mset_less_eqI) (metis count_replicate_mset subseteq_mset_def)
1149 lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
1150   by (induct D) simp_all
1152 lemma replicate_count_mset_eq_filter_eq:
1153   "replicate (count (mset xs) k) k = filter (HOL.eq k) xs"
1154   by (induct xs) auto
1157 subsection \<open>Big operators\<close>
1159 no_notation times (infixl "*" 70)
1160 no_notation Groups.one ("1")
1162 locale comm_monoid_mset = comm_monoid
1163 begin
1165 definition F :: "'a multiset \<Rightarrow> 'a"
1166   where eq_fold: "F M = fold_mset f 1 M"
1168 lemma empty [simp]: "F {#} = 1"
1169   by (simp add: eq_fold)
1171 lemma singleton [simp]: "F {#x#} = x"
1172 proof -
1173   interpret comp_fun_commute
1174     by standard (simp add: fun_eq_iff left_commute)
1175   show ?thesis by (simp add: eq_fold)
1176 qed
1178 lemma union [simp]: "F (M + N) = F M * F N"
1179 proof -
1180   interpret comp_fun_commute f
1181     by standard (simp add: fun_eq_iff left_commute)
1182   show ?thesis
1183     by (induct N) (simp_all add: left_commute eq_fold)
1184 qed
1186 end
1188 lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + :: 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
1189   by standard (simp add: add_ac comp_def)
1191 declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp]
1193 lemma in_mset_fold_plus_iff[iff]: "x \<in># fold_mset (op +) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)"
1194   by (induct NN) auto
1196 notation times (infixl "*" 70)
1197 notation Groups.one ("1")
1200 begin
1202 sublocale msetsum: comm_monoid_mset plus 0
1203   defines msetsum = msetsum.F ..
1205 lemma (in semiring_1) msetsum_replicate_mset [simp]:
1206   "msetsum (replicate_mset n a) = of_nat n * a"
1207   by (induct n) (simp_all add: algebra_simps)
1209 lemma setsum_unfold_msetsum:
1210   "setsum f A = msetsum (image_mset f (mset_set A))"
1211   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1213 end
1215 lemma msetsum_diff:
1216   fixes M N :: "('a :: ordered_cancel_comm_monoid_diff) multiset"
1217   shows "N \<le># M \<Longrightarrow> msetsum (M - N) = msetsum M - msetsum N"
1220 lemma size_eq_msetsum: "size M = msetsum (image_mset (\<lambda>_. 1) M)"
1221 proof (induct M)
1222   case empty then show ?case by simp
1223 next
1224   case (add M x) then show ?case
1225     by (cases "x \<in> set_mset M")
1226       (simp_all del: mem_set_mset_iff add: size_multiset_overloaded_eq setsum.distrib setsum.delta' insert_absorb, simp)
1227 qed
1230 abbreviation Union_mset :: "'a multiset multiset \<Rightarrow> 'a multiset"  ("\<Union>#_"  900)
1231   where "\<Union># MM \<equiv> msetsum MM"
1233 lemma set_mset_Union_mset[simp]: "set_mset (\<Union># MM) = (\<Union>M \<in> set_mset MM. set_mset M)"
1234   by (induct MM) auto
1236 lemma in_Union_mset_iff[iff]: "x \<in># \<Union># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
1237   by (induct MM) auto
1239 syntax (ASCII)
1240   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3SUM _:#_. _)" [0, 51, 10] 10)
1241 syntax
1242   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1243 translations
1244   "\<Sum>i \<in># A. b" \<rightleftharpoons> "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"
1246 context comm_monoid_mult
1247 begin
1249 sublocale msetprod: comm_monoid_mset times 1
1250   defines msetprod = msetprod.F ..
1252 lemma msetprod_empty:
1253   "msetprod {#} = 1"
1254   by (fact msetprod.empty)
1256 lemma msetprod_singleton:
1257   "msetprod {#x#} = x"
1258   by (fact msetprod.singleton)
1260 lemma msetprod_Un:
1261   "msetprod (A + B) = msetprod A * msetprod B"
1262   by (fact msetprod.union)
1264 lemma msetprod_replicate_mset [simp]:
1265   "msetprod (replicate_mset n a) = a ^ n"
1266   by (induct n) (simp_all add: ac_simps)
1268 lemma setprod_unfold_msetprod:
1269   "setprod f A = msetprod (image_mset f (mset_set A))"
1270   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1272 lemma msetprod_multiplicity:
1273   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_mset M)"
1274   by (simp add: fold_mset_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
1276 end
1278 syntax (ASCII)
1279   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3PROD _:#_. _)" [0, 51, 10] 10)
1280 syntax
1281   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1282 translations
1283   "\<Prod>i \<in># A. b" \<rightleftharpoons> "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"
1285 lemma (in comm_semiring_1) dvd_msetprod:
1286   assumes "x \<in># A"
1287   shows "x dvd msetprod A"
1288 proof -
1289   from assms have "A = (A - {#x#}) + {#x#}" by simp
1290   then obtain B where "A = B + {#x#}" ..
1291   then show ?thesis by simp
1292 qed
1295 subsection \<open>Alternative representations\<close>
1297 subsubsection \<open>Lists\<close>
1299 context linorder
1300 begin
1302 lemma mset_insort [simp]:
1303   "mset (insort_key k x xs) = {#x#} + mset xs"
1304   by (induct xs) (simp_all add: ac_simps)
1306 lemma mset_sort [simp]:
1307   "mset (sort_key k xs) = mset xs"
1308   by (induct xs) (simp_all add: ac_simps)
1310 text \<open>
1311   This lemma shows which properties suffice to show that a function
1312   \<open>f\<close> with \<open>f xs = ys\<close> behaves like sort.
1313 \<close>
1315 lemma properties_for_sort_key:
1316   assumes "mset ys = mset xs"
1317     and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
1318     and "sorted (map f ys)"
1319   shows "sort_key f xs = ys"
1320   using assms
1321 proof (induct xs arbitrary: ys)
1322   case Nil then show ?case by simp
1323 next
1324   case (Cons x xs)
1325   from Cons.prems(2) have
1326     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
1327     by (simp add: filter_remove1)
1328   with Cons.prems have "sort_key f xs = remove1 x ys"
1329     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
1330   moreover from Cons.prems have "x \<in> set ys"
1331     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
1332   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
1333 qed
1335 lemma properties_for_sort:
1336   assumes multiset: "mset ys = mset xs"
1337     and "sorted ys"
1338   shows "sort xs = ys"
1339 proof (rule properties_for_sort_key)
1340   from multiset show "mset ys = mset xs" .
1341   from \<open>sorted ys\<close> show "sorted (map (\<lambda>x. x) ys)" by simp
1342   from multiset have "length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)" for k
1343     by (rule mset_eq_length_filter)
1344   then have "replicate (length (filter (\<lambda>y. k = y) ys)) k =
1345     replicate (length (filter (\<lambda>x. k = x) xs)) k" for k
1346     by simp
1347   then show "k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs" for k
1348     by (simp add: replicate_length_filter)
1349 qed
1351 lemma sort_key_inj_key_eq:
1352   assumes mset_equal: "mset xs = mset ys"
1353     and "inj_on f (set xs)"
1354     and "sorted (map f ys)"
1355   shows "sort_key f xs = ys"
1356 proof (rule properties_for_sort_key)
1357   from mset_equal
1358   show "mset ys = mset xs" by simp
1359   from \<open>sorted (map f ys)\<close>
1360   show "sorted (map f ys)" .
1361   show "[x\<leftarrow>ys . f k = f x] = [x\<leftarrow>xs . f k = f x]" if "k \<in> set ys" for k
1362   proof -
1363     from mset_equal
1364     have set_equal: "set xs = set ys" by (rule mset_eq_setD)
1365     with that have "insert k (set ys) = set ys" by auto
1366     with \<open>inj_on f (set xs)\<close> have inj: "inj_on f (insert k (set ys))"
1367       by (simp add: set_equal)
1368     from inj have "[x\<leftarrow>ys . f k = f x] = filter (HOL.eq k) ys"
1369       by (auto intro!: inj_on_filter_key_eq)
1370     also have "\<dots> = replicate (count (mset ys) k) k"
1371       by (simp add: replicate_count_mset_eq_filter_eq)
1372     also have "\<dots> = replicate (count (mset xs) k) k"
1373       using mset_equal by simp
1374     also have "\<dots> = filter (HOL.eq k) xs"
1375       by (simp add: replicate_count_mset_eq_filter_eq)
1376     also have "\<dots> = [x\<leftarrow>xs . f k = f x]"
1377       using inj by (auto intro!: inj_on_filter_key_eq [symmetric] simp add: set_equal)
1378     finally show ?thesis .
1379   qed
1380 qed
1382 lemma sort_key_eq_sort_key:
1383   assumes "mset xs = mset ys"
1384     and "inj_on f (set xs)"
1385   shows "sort_key f xs = sort_key f ys"
1386   by (rule sort_key_inj_key_eq) (simp_all add: assms)
1388 lemma sort_key_by_quicksort:
1389   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
1390     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
1391     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
1392 proof (rule properties_for_sort_key)
1393   show "mset ?rhs = mset ?lhs"
1394     by (rule multiset_eqI) (auto simp add: mset_filter)
1395   show "sorted (map f ?rhs)"
1396     by (auto simp add: sorted_append intro: sorted_map_same)
1397 next
1398   fix l
1399   assume "l \<in> set ?rhs"
1400   let ?pivot = "f (xs ! (length xs div 2))"
1401   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
1402   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
1403     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
1404   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
1405   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
1406   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
1407     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
1408   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
1409   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
1410   proof (cases "f l" ?pivot rule: linorder_cases)
1411     case less
1412     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
1413     with less show ?thesis
1414       by (simp add: filter_sort [symmetric] ** ***)
1415   next
1416     case equal then show ?thesis
1417       by (simp add: * less_le)
1418   next
1419     case greater
1420     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
1421     with greater show ?thesis
1422       by (simp add: filter_sort [symmetric] ** ***)
1423   qed
1424 qed
1426 lemma sort_by_quicksort:
1427   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
1428     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
1429     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
1430   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
1432 text \<open>A stable parametrized quicksort\<close>
1434 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
1435   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
1437 lemma part_code [code]:
1438   "part f pivot [] = ([], [], [])"
1439   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
1440      if x' < pivot then (x # lts, eqs, gts)
1441      else if x' > pivot then (lts, eqs, x # gts)
1442      else (lts, x # eqs, gts))"
1443   by (auto simp add: part_def Let_def split_def)
1445 lemma sort_key_by_quicksort_code [code]:
1446   "sort_key f xs =
1447     (case xs of
1448       [] \<Rightarrow> []
1449     | [x] \<Rightarrow> xs
1450     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
1451     | _ \<Rightarrow>
1452         let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1453         in sort_key f lts @ eqs @ sort_key f gts)"
1454 proof (cases xs)
1455   case Nil then show ?thesis by simp
1456 next
1457   case (Cons _ ys) note hyps = Cons show ?thesis
1458   proof (cases ys)
1459     case Nil with hyps show ?thesis by simp
1460   next
1461     case (Cons _ zs) note hyps = hyps Cons show ?thesis
1462     proof (cases zs)
1463       case Nil with hyps show ?thesis by auto
1464     next
1465       case Cons
1466       from sort_key_by_quicksort [of f xs]
1467       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1468         in sort_key f lts @ eqs @ sort_key f gts)"
1469       by (simp only: split_def Let_def part_def fst_conv snd_conv)
1470       with hyps Cons show ?thesis by (simp only: list.cases)
1471     qed
1472   qed
1473 qed
1475 end
1477 hide_const (open) part
1479 lemma mset_remdups_le: "mset (remdups xs) \<le># mset xs"
1480   by (induct xs) (auto intro: subset_mset.order_trans)
1482 lemma mset_update:
1483   "i < length ls \<Longrightarrow> mset (ls[i := v]) = mset ls - {#ls ! i#} + {#v#}"
1484 proof (induct ls arbitrary: i)
1485   case Nil then show ?case by simp
1486 next
1487   case (Cons x xs)
1488   show ?case
1489   proof (cases i)
1490     case 0 then show ?thesis by simp
1491   next
1492     case (Suc i')
1493     with Cons show ?thesis
1494       apply simp
1495       apply (subst add.assoc)
1496       apply (subst add.commute [of "{#v#}" "{#x#}"])
1497       apply (subst add.assoc [symmetric])
1498       apply simp
1499       apply (rule mset_le_multiset_union_diff_commute)
1500       apply (simp add: mset_le_single nth_mem_mset)
1501       done
1502   qed
1503 qed
1505 lemma mset_swap:
1506   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
1507     mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
1508   by (cases "i = j") (simp_all add: mset_update nth_mem_mset)
1511 subsection \<open>The multiset order\<close>
1513 subsubsection \<open>Well-foundedness\<close>
1515 definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
1516   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1517       (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
1519 definition mult :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
1520   "mult r = (mult1 r)\<^sup>+"
1522 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1523 by (simp add: mult1_def)
1526   assumes mult1: "(N, M0 + {#a#}) \<in> mult1 r"
1527   shows
1528     "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1529      (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
1530 proof -
1531   let ?r = "\<lambda>K a. \<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r"
1532   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1533   obtain a' M0' K where M0: "M0 + {#a#} = M0' + {#a'#}"
1534     and N: "N = M0' + K"
1535     and r: "?r K a'"
1536     using mult1 unfolding mult1_def by auto
1537   show ?thesis (is "?case1 \<or> ?case2")
1538   proof -
1539     from M0 consider "M0 = M0'" "a = a'"
1540       | K' where "M0 = K' + {#a'#}" "M0' = K' + {#a#}"
1541       by atomize_elim (simp only: add_eq_conv_ex)
1542     then show ?thesis
1543     proof cases
1544       case 1
1545       with N r have "?r K a \<and> N = M0 + K" by simp
1546       then have ?case2 ..
1547       then show ?thesis ..
1548     next
1549       case 2
1550       from N 2(2) have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
1551       with r 2(1) have "?R (K' + K) M0" by blast
1552       with n have ?case1 by (simp add: mult1_def)
1553       then show ?thesis ..
1554     qed
1555   qed
1556 qed
1558 lemma all_accessible:
1559   assumes "wf r"
1560   shows "\<forall>M. M \<in> Wellfounded.acc (mult1 r)"
1561 proof
1562   let ?R = "mult1 r"
1563   let ?W = "Wellfounded.acc ?R"
1564   {
1565     fix M M0 a
1566     assume M0: "M0 \<in> ?W"
1567       and wf_hyp: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1568       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W"
1569     have "M0 + {#a#} \<in> ?W"
1570     proof (rule accI [of "M0 + {#a#}"])
1571       fix N
1572       assume "(N, M0 + {#a#}) \<in> ?R"
1573       then consider M where "(M, M0) \<in> ?R" "N = M + {#a#}"
1574         | K where "\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r" "N = M0 + K"
1575         by atomize_elim (rule less_add)
1576       then show "N \<in> ?W"
1577       proof cases
1578         case 1
1579         from acc_hyp have "(M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W" ..
1580         from this and \<open>(M, M0) \<in> ?R\<close> have "M + {#a#} \<in> ?W" ..
1581         then show "N \<in> ?W" by (simp only: \<open>N = M + {#a#}\<close>)
1582       next
1583         case 2
1584         from this(1) have "M0 + K \<in> ?W"
1585         proof (induct K)
1586           case empty
1587           from M0 show "M0 + {#} \<in> ?W" by simp
1588         next
1589           case (add K x)
1590           from add.prems have "(x, a) \<in> r" by simp
1591           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
1592           moreover from add have "M0 + K \<in> ?W" by simp
1593           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
1594           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
1595         qed
1596         then show "N \<in> ?W" by (simp only: 2(2))
1597       qed
1598     qed
1599   } note tedious_reasoning = this
1601   show "M \<in> ?W" for M
1602   proof (induct M)
1603     show "{#} \<in> ?W"
1604     proof (rule accI)
1605       fix b assume "(b, {#}) \<in> ?R"
1606       with not_less_empty show "b \<in> ?W" by contradiction
1607     qed
1609     fix M a assume "M \<in> ?W"
1610     from \<open>wf r\<close> have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1611     proof induct
1612       fix a
1613       assume r: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1614       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1615       proof
1616         fix M assume "M \<in> ?W"
1617         then show "M + {#a#} \<in> ?W"
1618           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
1619       qed
1620     qed
1621     from this and \<open>M \<in> ?W\<close> show "M + {#a#} \<in> ?W" ..
1622   qed
1623 qed
1625 theorem wf_mult1: "wf r \<Longrightarrow> wf (mult1 r)"
1626 by (rule acc_wfI) (rule all_accessible)
1628 theorem wf_mult: "wf r \<Longrightarrow> wf (mult r)"
1629 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
1632 subsubsection \<open>Closure-free presentation\<close>
1634 text \<open>One direction.\<close>
1636 lemma mult_implies_one_step:
1637   "trans r \<Longrightarrow> (M, N) \<in> mult r \<Longrightarrow>
1638     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
1639     (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)"
1640 apply (unfold mult_def mult1_def set_mset_def)
1641 apply (erule converse_trancl_induct, clarify)
1642  apply (rule_tac x = M0 in exI, simp, clarify)
1643 apply (case_tac "a \<in># K")
1644  apply (rule_tac x = I in exI)
1645  apply (simp (no_asm))
1646  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1647  apply (simp (no_asm_simp) add: add.assoc [symmetric])
1648  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong)
1649  apply (simp add: diff_union_single_conv)
1650  apply (simp (no_asm_use) add: trans_def)
1651  apply blast
1652 apply (subgoal_tac "a \<in># I")
1653  apply (rule_tac x = "I - {#a#}" in exI)
1654  apply (rule_tac x = "J + {#a#}" in exI)
1655  apply (rule_tac x = "K + Ka" in exI)
1656  apply (rule conjI)
1657   apply (simp add: multiset_eq_iff split: nat_diff_split)
1658  apply (rule conjI)
1659   apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
1660   apply (simp add: multiset_eq_iff split: nat_diff_split)
1661  apply (simp (no_asm_use) add: trans_def)
1662  apply blast
1663 apply (subgoal_tac "a \<in># (M0 + {#a#})")
1664  apply simp
1665 apply (simp (no_asm))
1666 done
1668 lemma one_step_implies_mult_aux:
1669   "\<forall>I J K. size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)
1670     \<longrightarrow> (I + K, I + J) \<in> mult r"
1671 apply (induct n)
1672  apply auto
1673 apply (frule size_eq_Suc_imp_eq_union, clarify)
1674 apply (rename_tac "J'", simp)
1675 apply (erule notE, auto)
1676 apply (case_tac "J' = {#}")
1677  apply (simp add: mult_def)
1678  apply (rule r_into_trancl)
1679  apply (simp add: mult1_def set_mset_def, blast)
1680 txt \<open>Now we know @{term "J' \<noteq> {#}"}.\<close>
1681 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
1682 apply (erule_tac P = "\<forall>k \<in> set_mset K. P k" for P in rev_mp)
1683 apply (erule ssubst)
1684 apply (simp add: Ball_def, auto)
1685 apply (subgoal_tac
1686   "((I + {# x \<in># K. (x, a) \<in> r #}) + {# x \<in># K. (x, a) \<notin> r #},
1687     (I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r")
1688  prefer 2
1689  apply force
1690 apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
1691 apply (erule trancl_trans)
1692 apply (rule r_into_trancl)
1693 apply (simp add: mult1_def set_mset_def)
1694 apply (rule_tac x = a in exI)
1695 apply (rule_tac x = "I + J'" in exI)
1696 apply (simp add: ac_simps)
1697 done
1699 lemma one_step_implies_mult:
1700   "trans r \<Longrightarrow> J \<noteq> {#} \<Longrightarrow> \<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r
1701     \<Longrightarrow> (I + K, I + J) \<in> mult r"
1702 using one_step_implies_mult_aux by blast
1705 subsubsection \<open>Partial-order properties\<close>
1707 definition less_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "#\<subset>#" 50)
1708   where "M' #\<subset># M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
1710 definition le_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "#\<subseteq>#" 50)
1711   where "M' #\<subseteq># M \<longleftrightarrow> M' #\<subset># M \<or> M' = M"
1713 notation (ASCII)
1714   less_multiset (infix "#<#" 50) and
1715   le_multiset (infix "#<=#" 50)
1717 interpretation multiset_order: order le_multiset less_multiset
1718 proof -
1719   have irrefl: "\<not> M #\<subset># M" for M :: "'a multiset"
1720   proof
1721     assume "M #\<subset># M"
1722     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
1723     have "trans {(x'::'a, x). x' < x}"
1724       by (rule transI) simp
1725     moreover note MM
1726     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
1727       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})"
1728       by (rule mult_implies_one_step)
1729     then obtain I J K where "M = I + J" and "M = I + K"
1730       and "J \<noteq> {#}" and "(\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})" by blast
1731     then have *: "K \<noteq> {#}" and **: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. k < j" by auto
1732     have "finite (set_mset K)" by simp
1733     moreover note **
1734     ultimately have "set_mset K = {}"
1735       by (induct rule: finite_induct) (auto intro: order_less_trans)
1736     with * show False by simp
1737   qed
1738   have trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># N" for K M N :: "'a multiset"
1739     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
1740   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
1741     by standard (auto simp add: le_multiset_def irrefl dest: trans)
1742 qed
1744 lemma mult_less_irrefl [elim!]:
1745   fixes M :: "'a::order multiset"
1746   shows "M #\<subset># M \<Longrightarrow> R"
1747   by simp
1750 subsubsection \<open>Monotonicity of multiset union\<close>
1752 lemma mult1_union: "(B, D) \<in> mult1 r \<Longrightarrow> (C + B, C + D) \<in> mult1 r"
1753 apply (unfold mult1_def)
1754 apply auto
1755 apply (rule_tac x = a in exI)
1756 apply (rule_tac x = "C + M0" in exI)
1758 done
1760 lemma union_less_mono2: "B #\<subset># D \<Longrightarrow> C + B #\<subset># C + (D::'a::order multiset)"
1761 apply (unfold less_multiset_def mult_def)
1762 apply (erule trancl_induct)
1763  apply (blast intro: mult1_union)
1764 apply (blast intro: mult1_union trancl_trans)
1765 done
1767 lemma union_less_mono1: "B #\<subset># D \<Longrightarrow> B + C #\<subset># D + (C::'a::order multiset)"
1768 apply (subst add.commute [of B C])
1769 apply (subst add.commute [of D C])
1770 apply (erule union_less_mono2)
1771 done
1773 lemma union_less_mono:
1774   fixes A B C D :: "'a::order multiset"
1775   shows "A #\<subset># C \<Longrightarrow> B #\<subset># D \<Longrightarrow> A + B #\<subset># C + D"
1776   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
1778 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
1779   by standard (auto simp add: le_multiset_def intro: union_less_mono2)
1782 subsubsection \<open>Termination proofs with multiset orders\<close>
1784 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
1785   and multi_member_this: "x \<in># {# x #} + XS"
1786   and multi_member_last: "x \<in># {# x #}"
1787   by auto
1789 definition "ms_strict = mult pair_less"
1790 definition "ms_weak = ms_strict \<union> Id"
1792 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
1793 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
1794 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
1796 lemma smsI:
1797   "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
1798   unfolding ms_strict_def
1799 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
1801 lemma wmsI:
1802   "(set_mset A, set_mset B) \<in> max_strict \<or> A = {#} \<and> B = {#}
1803   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
1804 unfolding ms_weak_def ms_strict_def
1805 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
1807 inductive pw_leq
1808 where
1809   pw_leq_empty: "pw_leq {#} {#}"
1810 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
1812 lemma pw_leq_lstep:
1813   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
1814 by (drule pw_leq_step) (rule pw_leq_empty, simp)
1816 lemma pw_leq_split:
1817   assumes "pw_leq X Y"
1818   shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1819   using assms
1820 proof induct
1821   case pw_leq_empty thus ?case by auto
1822 next
1823   case (pw_leq_step x y X Y)
1824   then obtain A B Z where
1825     [simp]: "X = A + Z" "Y = B + Z"
1826       and 1[simp]: "(set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
1827     by auto
1828   from pw_leq_step consider "x = y" | "(x, y) \<in> pair_less"
1829     unfolding pair_leq_def by auto
1830   thus ?case
1831   proof cases
1832     case [simp]: 1
1833     have "{#x#} + X = A + ({#y#}+Z) \<and> {#y#} + Y = B + ({#y#}+Z) \<and>
1834       ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1835       by (auto simp: ac_simps)
1836     thus ?thesis by blast
1837   next
1838     case 2
1839     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
1840     have "{#x#} + X = ?A' + Z"
1841       "{#y#} + Y = ?B' + Z"
1842       by (auto simp add: ac_simps)
1843     moreover have
1844       "(set_mset ?A', set_mset ?B') \<in> max_strict"
1845       using 1 2 unfolding max_strict_def
1846       by (auto elim!: max_ext.cases)
1847     ultimately show ?thesis by blast
1848   qed
1849 qed
1851 lemma
1852   assumes pwleq: "pw_leq Z Z'"
1853   shows ms_strictI: "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
1854     and ms_weakI1:  "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
1855     and ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
1856 proof -
1857   from pw_leq_split[OF pwleq]
1858   obtain A' B' Z''
1859     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
1860     and mx_or_empty: "(set_mset A', set_mset B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
1861     by blast
1862   {
1863     assume max: "(set_mset A, set_mset B) \<in> max_strict"
1864     from mx_or_empty
1865     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
1866     proof
1867       assume max': "(set_mset A', set_mset B') \<in> max_strict"
1868       with max have "(set_mset (A + A'), set_mset (B + B')) \<in> max_strict"
1869         by (auto simp: max_strict_def intro: max_ext_additive)
1870       thus ?thesis by (rule smsI)
1871     next
1872       assume [simp]: "A' = {#} \<and> B' = {#}"
1873       show ?thesis by (rule smsI) (auto intro: max)
1874     qed
1875     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add: ac_simps)
1876     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
1877   }
1878   from mx_or_empty
1879   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
1880   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
1881 qed
1883 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
1884 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
1885 and nonempty_single: "{# x #} \<noteq> {#}"
1886 by auto
1888 setup \<open>
1889   let
1890     fun msetT T = Type (@{type_name multiset}, [T]);
1892     fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
1893       | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
1894       | mk_mset T (x :: xs) =
1895             Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
1896                   mk_mset T [x] \$ mk_mset T xs
1898     fun mset_member_tac ctxt m i =
1899       if m <= 0 then
1900         resolve_tac ctxt @{thms multi_member_this} i ORELSE
1901         resolve_tac ctxt @{thms multi_member_last} i
1902       else
1903         resolve_tac ctxt @{thms multi_member_skip} i THEN mset_member_tac ctxt (m - 1) i
1905     fun mset_nonempty_tac ctxt =
1906       resolve_tac ctxt @{thms nonempty_plus} ORELSE'
1907       resolve_tac ctxt @{thms nonempty_single}
1909     fun regroup_munion_conv ctxt =
1910       Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus}
1911         (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
1913     fun unfold_pwleq_tac ctxt i =
1914       (resolve_tac ctxt @{thms pw_leq_step} i THEN (fn st => unfold_pwleq_tac ctxt (i + 1) st))
1915         ORELSE (resolve_tac ctxt @{thms pw_leq_lstep} i)
1916         ORELSE (resolve_tac ctxt @{thms pw_leq_empty} i)
1918     val set_mset_simps = [@{thm set_mset_empty}, @{thm set_mset_single}, @{thm set_mset_union},
1919                         @{thm Un_insert_left}, @{thm Un_empty_left}]
1920   in
1921     ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
1922     {
1923       msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
1924       mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
1925       mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_mset_simps,
1926       smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
1927       reduction_pair = @{thm ms_reduction_pair}
1928     })
1929   end
1930 \<close>
1933 subsection \<open>Legacy theorem bindings\<close>
1935 lemmas multi_count_eq = multiset_eq_iff [symmetric]
1937 lemma union_commute: "M + N = N + (M::'a multiset)"
1938   by (fact add.commute)
1940 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
1941   by (fact add.assoc)
1943 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
1944   by (fact add.left_commute)
1946 lemmas union_ac = union_assoc union_commute union_lcomm
1948 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
1949   by (fact add_right_cancel)
1951 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
1952   by (fact add_left_cancel)
1954 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
1955   by (fact add_left_imp_eq)
1957 lemma mset_less_trans: "(M::'a multiset) <# K \<Longrightarrow> K <# N \<Longrightarrow> M <# N"
1958   by (fact subset_mset.less_trans)
1960 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
1961   by (fact subset_mset.inf.commute)
1963 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
1964   by (fact subset_mset.inf.assoc [symmetric])
1966 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
1967   by (fact subset_mset.inf.left_commute)
1969 lemmas multiset_inter_ac =
1970   multiset_inter_commute
1971   multiset_inter_assoc
1972   multiset_inter_left_commute
1974 lemma mult_less_not_refl: "\<not> M #\<subset># (M::'a::order multiset)"
1975   by (fact multiset_order.less_irrefl)
1977 lemma mult_less_trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># (N::'a::order multiset)"
1978   by (fact multiset_order.less_trans)
1980 lemma mult_less_not_sym: "M #\<subset># N \<Longrightarrow> \<not> N #\<subset># (M::'a::order multiset)"
1981   by (fact multiset_order.less_not_sym)
1983 lemma mult_less_asym: "M #\<subset># N \<Longrightarrow> (\<not> P \<Longrightarrow> N #\<subset># (M::'a::order multiset)) \<Longrightarrow> P"
1984   by (fact multiset_order.less_asym)
1986 declaration \<open>
1987   let
1988     fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T])) (Const _ \$ t') =
1989           let
1990             val (maybe_opt, ps) =
1991               Nitpick_Model.dest_plain_fun t'
1992               ||> op ~~
1993               ||> map (apsnd (snd o HOLogic.dest_number))
1994             fun elems_for t =
1995               (case AList.lookup (op =) ps t of
1996                 SOME n => replicate n t
1997               | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t])
1998           in
1999             (case maps elems_for (all_values elem_T) @
2000                  (if maybe_opt then [Const (Nitpick_Model.unrep_mixfix (), elem_T)] else []) of
2001               [] => Const (@{const_name zero_class.zero}, T)
2002             | ts =>
2003                 foldl1 (fn (t1, t2) =>
2004                     Const (@{const_name plus_class.plus}, T --> T --> T) \$ t1 \$ t2)
2005                   (map (curry (op \$) (Const (@{const_name single}, elem_T --> T))) ts))
2006           end
2007       | multiset_postproc _ _ _ _ t = t
2008   in Nitpick_Model.register_term_postprocessor @{typ "'a multiset"} multiset_postproc end
2009 \<close>
2012 subsection \<open>Naive implementation using lists\<close>
2014 code_datatype mset
2016 lemma [code]: "{#} = mset []"
2017   by simp
2019 lemma [code]: "{#x#} = mset [x]"
2020   by simp
2022 lemma union_code [code]: "mset xs + mset ys = mset (xs @ ys)"
2023   by simp
2025 lemma [code]: "image_mset f (mset xs) = mset (map f xs)"
2026   by (simp add: mset_map)
2028 lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)"
2029   by (simp add: mset_filter)
2031 lemma [code]: "mset xs - mset ys = mset (fold remove1 ys xs)"
2032   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
2034 lemma [code]:
2035   "mset xs #\<inter> mset ys =
2036     mset (snd (fold (\<lambda>x (ys, zs).
2037       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
2038 proof -
2039   have "\<And>zs. mset (snd (fold (\<lambda>x (ys, zs).
2040     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
2041       (mset xs #\<inter> mset ys) + mset zs"
2042     by (induct xs arbitrary: ys)
2044   then show ?thesis by simp
2045 qed
2047 lemma [code]:
2048   "mset xs #\<union> mset ys =
2049     mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
2050 proof -
2051   have "\<And>zs. mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
2052       (mset xs #\<union> mset ys) + mset zs"
2053     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
2054   then show ?thesis by simp
2055 qed
2057 declare in_multiset_in_set [code_unfold]
2059 lemma [code]: "count (mset 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 (mset xs) x + n"
2062     by (induct xs) simp_all
2063   then show ?thesis by simp
2064 qed
2066 declare set_mset_mset [code]
2068 declare sorted_list_of_multiset_mset [code]
2070 lemma [code]: \<comment> \<open>not very efficient, but representation-ignorant!\<close>
2071   "mset_set A = mset (sorted_list_of_set A)"
2072   apply (cases "finite A")
2073   apply simp_all
2074   apply (induct A rule: finite_induct)
2076   done
2078 declare size_mset [code]
2080 fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
2081   "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
2082 | "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
2083      None \<Rightarrow> None
2084    | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
2086 lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> mset xs \<le># mset ys) \<and>
2087   (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> mset xs <# mset ys) \<and>
2088   (ms_lesseq_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)"
2089 proof (induct xs arbitrary: ys)
2090   case (Nil ys)
2091   show ?case by (auto simp: mset_less_empty_nonempty)
2092 next
2093   case (Cons x xs ys)
2094   show ?case
2095   proof (cases "List.extract (op = x) ys")
2096     case None
2097     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
2098     {
2099       assume "mset (x # xs) \<le># mset ys"
2100       from set_mset_mono[OF this] x have False by simp
2101     } note nle = this
2102     moreover
2103     {
2104       assume "mset (x # xs) <# mset ys"
2105       hence "mset (x # xs) \<le># mset ys" by auto
2106       from nle[OF this] have False .
2107     }
2108     ultimately show ?thesis using None by auto
2109   next
2110     case (Some res)
2111     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
2112     note Some = Some[unfolded res]
2113     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
2114     hence id: "mset ys = mset (ys1 @ ys2) + {#x#}"
2115       by (auto simp: ac_simps)
2116     show ?thesis unfolding ms_lesseq_impl.simps
2117       unfolding Some option.simps split
2118       unfolding id
2119       using Cons[of "ys1 @ ys2"]
2120       unfolding subset_mset_def subseteq_mset_def by auto
2121   qed
2122 qed
2124 lemma [code]: "mset xs \<le># mset ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
2125   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2127 lemma [code]: "mset xs <# mset ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
2128   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2130 instantiation multiset :: (equal) equal
2131 begin
2133 definition
2134   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
2135 lemma [code]: "HOL.equal (mset xs) (mset ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
2136   unfolding equal_multiset_def
2137   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
2139 instance
2140   by standard (simp add: equal_multiset_def)
2142 end
2144 lemma [code]: "msetsum (mset xs) = listsum xs"
2145   by (induct xs) (simp_all add: add.commute)
2147 lemma [code]: "msetprod (mset xs) = fold times xs 1"
2148 proof -
2149   have "\<And>x. fold times xs x = msetprod (mset xs) * x"
2150     by (induct xs) (simp_all add: mult.assoc)
2151   then show ?thesis by simp
2152 qed
2154 text \<open>
2155   Exercise for the casual reader: add implementations for @{const le_multiset}
2156   and @{const less_multiset} (multiset order).
2157 \<close>
2159 text \<open>Quickcheck generators\<close>
2161 definition (in term_syntax)
2162   msetify :: "'a::typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
2163     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
2164   [code_unfold]: "msetify xs = Code_Evaluation.valtermify mset {\<cdot>} xs"
2166 notation fcomp (infixl "\<circ>>" 60)
2167 notation scomp (infixl "\<circ>\<rightarrow>" 60)
2169 instantiation multiset :: (random) random
2170 begin
2172 definition
2173   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
2175 instance ..
2177 end
2179 no_notation fcomp (infixl "\<circ>>" 60)
2180 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
2182 instantiation multiset :: (full_exhaustive) full_exhaustive
2183 begin
2185 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
2186 where
2187   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
2189 instance ..
2191 end
2193 hide_const (open) msetify
2196 subsection \<open>BNF setup\<close>
2198 definition rel_mset where
2199   "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
2201 lemma mset_zip_take_Cons_drop_twice:
2202   assumes "length xs = length ys" "j \<le> length xs"
2203   shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
2204     mset (zip xs ys) + {#(x, y)#}"
2205   using assms
2206 proof (induct xs ys arbitrary: x y j rule: list_induct2)
2207   case Nil
2208   thus ?case
2209     by simp
2210 next
2211   case (Cons x xs y ys)
2212   thus ?case
2213   proof (cases "j = 0")
2214     case True
2215     thus ?thesis
2216       by simp
2217   next
2218     case False
2219     then obtain k where k: "j = Suc k"
2220       by (cases j) simp
2221     hence "k \<le> length xs"
2222       using Cons.prems by auto
2223     hence "mset (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
2224       mset (zip xs ys) + {#(x, y)#}"
2225       by (rule Cons.hyps(2))
2226     thus ?thesis
2227       unfolding k by (auto simp: add.commute union_lcomm)
2228   qed
2229 qed
2231 lemma ex_mset_zip_left:
2232   assumes "length xs = length ys" "mset xs' = mset xs"
2233   shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
2234 using assms
2235 proof (induct xs ys arbitrary: xs' rule: list_induct2)
2236   case Nil
2237   thus ?case
2238     by auto
2239 next
2240   case (Cons x xs y ys xs')
2241   obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
2242     by (metis Cons.prems in_set_conv_nth list.set_intros(1) mset_eq_setD)
2244   def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
2245   have "mset xs' = {#x#} + mset xsa"
2246     unfolding xsa_def using j_len nth_j
2247     by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
2248       mset.simps(2) union_code add.commute)
2249   hence ms_x: "mset xsa = mset xs"
2250     by (metis Cons.prems add.commute add_right_imp_eq mset.simps(2))
2251   then obtain ysa where
2252     len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)"
2253     using Cons.hyps(2) by blast
2255   def ys' \<equiv> "take j ysa @ y # drop j ysa"
2256   have xs': "xs' = take j xsa @ x # drop j xsa"
2257     using ms_x j_len nth_j Cons.prems xsa_def
2258     by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
2259       length_drop size_mset)
2260   have j_len': "j \<le> length xsa"
2261     using j_len xs' xsa_def
2262     by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
2263   have "length ys' = length xs'"
2264     unfolding ys'_def using Cons.prems len_a ms_x
2265     by (metis add_Suc_right append_take_drop_id length_Cons length_append mset_eq_length)
2266   moreover have "mset (zip xs' ys') = mset (zip (x # xs) (y # ys))"
2267     unfolding xs' ys'_def
2268     by (rule trans[OF mset_zip_take_Cons_drop_twice])
2269       (auto simp: len_a ms_a j_len' add.commute)
2270   ultimately show ?case
2271     by blast
2272 qed
2274 lemma list_all2_reorder_left_invariance:
2275   assumes rel: "list_all2 R xs ys" and ms_x: "mset xs' = mset xs"
2276   shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
2277 proof -
2278   have len: "length xs = length ys"
2279     using rel list_all2_conv_all_nth by auto
2280   obtain ys' where
2281     len': "length xs' = length ys'" and ms_xy: "mset (zip xs' ys') = mset (zip xs ys)"
2282     using len ms_x by (metis ex_mset_zip_left)
2283   have "list_all2 R xs' ys'"
2284     using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: mset_eq_setD)
2285   moreover have "mset ys' = mset ys"
2286     using len len' ms_xy map_snd_zip mset_map by metis
2287   ultimately show ?thesis
2288     by blast
2289 qed
2291 lemma ex_mset: "\<exists>xs. mset xs = X"
2292   by (induct X) (simp, metis mset.simps(2))
2294 bnf "'a multiset"
2295   map: image_mset
2296   sets: set_mset
2297   bd: natLeq
2298   wits: "{#}"
2299   rel: rel_mset
2300 proof -
2301   show "image_mset id = id"
2302     by (rule image_mset.id)
2303   show "image_mset (g \<circ> f) = image_mset g \<circ> image_mset f" for f g
2304     unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality)
2305   show "(\<And>z. z \<in> set_mset X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X" for f g X
2306     by (induct X) (simp_all (no_asm),
2307       metis One_nat_def Un_iff count_single mem_set_mset_iff set_mset_union zero_less_Suc)
2308   show "set_mset \<circ> image_mset f = op ` f \<circ> set_mset" for f
2309     by auto
2310   show "card_order natLeq"
2311     by (rule natLeq_card_order)
2312   show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
2313     by (rule natLeq_cinfinite)
2314   show "ordLeq3 (card_of (set_mset X)) natLeq" for X
2315     by transfer
2316       (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
2317   show "rel_mset R OO rel_mset S \<le> rel_mset (R OO S)" for R S
2318     unfolding rel_mset_def[abs_def] OO_def
2319     apply clarify
2320     subgoal for X Z Y xs ys' ys zs
2321       apply (drule list_all2_reorder_left_invariance [where xs = ys' and ys = zs and xs' = ys])
2322       apply (auto intro: list_all2_trans)
2323       done
2324     done
2325   show "rel_mset R =
2326     (BNF_Def.Grp {x. set_mset x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
2327     BNF_Def.Grp {x. set_mset x \<subseteq> {(x, y). R x y}} (image_mset snd)" for R
2328     unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
2329     apply (rule ext)+
2330     apply auto
2331      apply (rule_tac x = "mset (zip xs ys)" in exI; auto)
2332         apply (metis list_all2_lengthD map_fst_zip mset_map)
2333        apply (auto simp: list_all2_iff)
2334       apply (metis list_all2_lengthD map_snd_zip mset_map)
2335      apply (auto simp: list_all2_iff)
2336     apply (rename_tac XY)
2337     apply (cut_tac X = XY in ex_mset)
2338     apply (erule exE)
2339     apply (rename_tac xys)
2340     apply (rule_tac x = "map fst xys" in exI)
2341     apply (auto simp: mset_map)
2342     apply (rule_tac x = "map snd xys" in exI)
2343     apply (auto simp: mset_map list_all2I subset_eq zip_map_fst_snd)
2344     done
2345   show "z \<in> set_mset {#} \<Longrightarrow> False" for z
2346     by auto
2347 qed
2349 inductive rel_mset'
2350 where
2351   Zero[intro]: "rel_mset' R {#} {#}"
2352 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
2354 lemma rel_mset_Zero: "rel_mset R {#} {#}"
2355 unfolding rel_mset_def Grp_def by auto
2357 declare multiset.count[simp]
2358 declare Abs_multiset_inverse[simp]
2359 declare multiset.count_inverse[simp]
2360 declare union_preserves_multiset[simp]
2362 lemma rel_mset_Plus:
2363   assumes ab: "R a b"
2364     and MN: "rel_mset R M N"
2365   shows "rel_mset R (M + {#a#}) (N + {#b#})"
2366 proof -
2367   have "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
2368     image_mset snd y + {#b#} = image_mset snd ya \<and>
2369     set_mset ya \<subseteq> {(x, y). R x y}"
2370     if "R a b" and "set_mset y \<subseteq> {(x, y). R x y}" for y
2371     using that by (intro exI[of _ "y + {#(a,b)#}"]) auto
2372   thus ?thesis
2373   using assms
2374   unfolding multiset.rel_compp_Grp Grp_def by blast
2375 qed
2377 lemma rel_mset'_imp_rel_mset: "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
2378   by (induct rule: rel_mset'.induct) (auto simp: rel_mset_Zero rel_mset_Plus)
2380 lemma rel_mset_size: "rel_mset R M N \<Longrightarrow> size M = size N"
2381   unfolding multiset.rel_compp_Grp Grp_def by auto
2384   assumes empty: "P {#} {#}"
2385     and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
2386     and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
2387   shows "P M N"
2388 apply(induct N rule: multiset_induct)
2389   apply(induct M rule: multiset_induct, rule empty, erule addL)
2390   apply(induct M rule: multiset_induct, erule addR, erule addR)
2391 done
2393 lemma multiset_induct2_size[consumes 1, case_names empty add]:
2394   assumes c: "size M = size N"
2395     and empty: "P {#} {#}"
2396     and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
2397   shows "P M N"
2398   using c
2399 proof (induct M arbitrary: N rule: measure_induct_rule[of size])
2400   case (less M)
2401   show ?case
2402   proof(cases "M = {#}")
2403     case True hence "N = {#}" using less.prems by auto
2404     thus ?thesis using True empty by auto
2405   next
2406     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2407     have "N \<noteq> {#}" using False less.prems by auto
2408     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
2409     have "size M1 = size N1" using less.prems unfolding M N by auto
2410     thus ?thesis using M N less.hyps add by auto
2411   qed
2412 qed
2414 lemma msed_map_invL:
2415   assumes "image_mset f (M + {#a#}) = N"
2416   shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1"
2417 proof -
2418   have "f a \<in># N"
2419     using assms multiset.set_map[of f "M + {#a#}"] by auto
2420   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
2421   have "image_mset f M = N1" using assms unfolding N by simp
2422   thus ?thesis using N by blast
2423 qed
2425 lemma msed_map_invR:
2426   assumes "image_mset f M = N + {#b#}"
2427   shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N"
2428 proof -
2429   obtain a where a: "a \<in># M" and fa: "f a = b"
2430     using multiset.set_map[of f M] unfolding assms
2431     by (metis image_iff mem_set_mset_iff union_single_eq_member)
2432   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
2433   have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
2434   thus ?thesis using M fa by blast
2435 qed
2437 lemma msed_rel_invL:
2438   assumes "rel_mset R (M + {#a#}) N"
2439   shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1"
2440 proof -
2441   obtain K where KM: "image_mset fst K = M + {#a#}"
2442     and KN: "image_mset snd K = N" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
2443     using assms
2444     unfolding multiset.rel_compp_Grp Grp_def by auto
2445   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
2446     and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
2447   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1"
2448     using msed_map_invL[OF KN[unfolded K]] by auto
2449   have Rab: "R a (snd ab)" using sK a unfolding K by auto
2450   have "rel_mset R M N1" using sK K1M K1N1
2451     unfolding K multiset.rel_compp_Grp Grp_def by auto
2452   thus ?thesis using N Rab by auto
2453 qed
2455 lemma msed_rel_invR:
2456   assumes "rel_mset R M (N + {#b#})"
2457   shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N"
2458 proof -
2459   obtain K where KN: "image_mset snd K = N + {#b#}"
2460     and KM: "image_mset fst K = M" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
2461     using assms
2462     unfolding multiset.rel_compp_Grp Grp_def by auto
2463   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
2464     and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
2465   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1"
2466     using msed_map_invL[OF KM[unfolded K]] by auto
2467   have Rab: "R (fst ab) b" using sK b unfolding K by auto
2468   have "rel_mset R M1 N" using sK K1N K1M1
2469     unfolding K multiset.rel_compp_Grp Grp_def by auto
2470   thus ?thesis using M Rab by auto
2471 qed
2473 lemma rel_mset_imp_rel_mset':
2474   assumes "rel_mset R M N"
2475   shows "rel_mset' R M N"
2476 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of size])
2477   case (less M)
2478   have c: "size M = size N" using rel_mset_size[OF less.prems] .
2479   show ?case
2480   proof(cases "M = {#}")
2481     case True hence "N = {#}" using c by simp
2482     thus ?thesis using True rel_mset'.Zero by auto
2483   next
2484     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
2485     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1"
2486       using msed_rel_invL[OF less.prems[unfolded M]] by auto
2487     have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
2488     thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
2489   qed
2490 qed
2492 lemma rel_mset_rel_mset': "rel_mset R M N = rel_mset' R M N"
2493   using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
2495 text \<open>The main end product for @{const rel_mset}: inductive characterization:\<close>
2496 lemmas rel_mset_induct[case_names empty add, induct pred: rel_mset] =
2497   rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
2500 subsection \<open>Size setup\<close>
2502 lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
2503   apply (rule ext)
2504   subgoal for x by (induct x) auto
2505   done
2507 setup \<open>
2508   BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
2509     @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
2510       size_union}
2511     @{thms multiset_size_o_map}
2512 \<close>
2514 hide_const (open) wcount
2516 end