src/HOL/Library/Multiset.thy
 author haftmann Wed Mar 27 10:55:05 2013 +0100 (2013-03-27) changeset 51548 757fa47af981 parent 51161 6ed12ae3b3e1 child 51599 1559e9266280 permissions -rw-r--r--
centralized various multiset operations in theory multiset;
more conversions between multisets and lists respectively
1 (*  Title:      HOL/Library/Multiset.thy
2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
3 *)
5 header {* (Finite) multisets *}
7 theory Multiset
8 imports Main DAList (* FIXME too specific dependency for a generic theory *)
9 begin
11 subsection {* The type of multisets *}
13 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
15 typedef 'a multiset = "multiset :: ('a => nat) set"
16   morphisms count Abs_multiset
17   unfolding multiset_def
18 proof
19   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
20 qed
22 setup_lifting type_definition_multiset
24 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
25   "a :# M == 0 < count M a"
27 notation (xsymbols)
28   Melem (infix "\<in>#" 50)
30 lemma multiset_eq_iff:
31   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
32   by (simp only: count_inject [symmetric] fun_eq_iff)
34 lemma multiset_eqI:
35   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
36   using multiset_eq_iff by auto
38 text {*
39  \medskip Preservation of the representing set @{term multiset}.
40 *}
42 lemma const0_in_multiset:
43   "(\<lambda>a. 0) \<in> multiset"
44   by (simp add: multiset_def)
46 lemma only1_in_multiset:
47   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
48   by (simp add: multiset_def)
50 lemma union_preserves_multiset:
51   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
52   by (simp add: multiset_def)
54 lemma diff_preserves_multiset:
55   assumes "M \<in> multiset"
56   shows "(\<lambda>a. M a - N a) \<in> multiset"
57 proof -
58   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
59     by auto
60   with assms show ?thesis
61     by (auto simp add: multiset_def intro: finite_subset)
62 qed
64 lemma filter_preserves_multiset:
65   assumes "M \<in> multiset"
66   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
67 proof -
68   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
69     by auto
70   with assms show ?thesis
71     by (auto simp add: multiset_def intro: finite_subset)
72 qed
74 lemmas in_multiset = const0_in_multiset only1_in_multiset
75   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
78 subsection {* Representing multisets *}
80 text {* Multiset enumeration *}
82 instantiation multiset :: (type) cancel_comm_monoid_add
83 begin
85 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
86 by (rule const0_in_multiset)
88 abbreviation Mempty :: "'a multiset" ("{#}") where
89   "Mempty \<equiv> 0"
91 lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
92 by (rule union_preserves_multiset)
94 instance
95 by default (transfer, simp add: fun_eq_iff)+
97 end
99 lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
100 by (rule only1_in_multiset)
102 syntax
103   "_multiset" :: "args => '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 {* Basic operations *}
117 subsubsection {* Union *}
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 {* Difference *}
125 instantiation multiset :: (type) comm_monoid_diff
126 begin
128 lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
129 by (rule diff_preserves_multiset)
131 instance
132 by default (transfer, simp add: fun_eq_iff)+
134 end
136 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
137   by (simp add: minus_multiset.rep_eq)
139 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
140 by(simp add: multiset_eq_iff)
142 lemma diff_cancel[simp]: "A - A = {#}"
143 by (rule multiset_eqI) simp
145 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
146 by(simp add: multiset_eq_iff)
148 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
149 by(simp add: multiset_eq_iff)
151 lemma insert_DiffM:
152   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
153   by (clarsimp simp: multiset_eq_iff)
155 lemma insert_DiffM2 [simp]:
156   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
157   by (clarsimp simp: multiset_eq_iff)
159 lemma diff_right_commute:
160   "(M::'a multiset) - N - Q = M - Q - N"
161   by (auto simp add: multiset_eq_iff)
164   "(M::'a multiset) - (N + Q) = M - N - Q"
165 by (simp add: multiset_eq_iff)
167 lemma diff_union_swap:
168   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
169   by (auto simp add: multiset_eq_iff)
171 lemma diff_union_single_conv:
172   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
173   by (simp add: multiset_eq_iff)
176 subsubsection {* Equality of multisets *}
178 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
179   by (simp add: multiset_eq_iff)
181 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
182   by (auto simp add: multiset_eq_iff)
184 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
185   by (auto simp add: multiset_eq_iff)
187 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
188   by (auto simp add: multiset_eq_iff)
190 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
191   by (auto simp add: multiset_eq_iff)
193 lemma diff_single_trivial:
194   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
195   by (auto simp add: multiset_eq_iff)
197 lemma diff_single_eq_union:
198   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
199   by auto
201 lemma union_single_eq_diff:
202   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
203   by (auto dest: sym)
205 lemma union_single_eq_member:
206   "M + {#x#} = N \<Longrightarrow> x \<in># N"
207   by auto
209 lemma union_is_single:
210   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
211 proof
212   assume ?rhs then show ?lhs by auto
213 next
214   assume ?lhs then show ?rhs
215     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
216 qed
218 lemma single_is_union:
219   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
220   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
223   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
224 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
225 proof
226   assume ?rhs then show ?lhs
228     (drule sym, simp add: add_assoc [symmetric])
229 next
230   assume ?lhs
231   show ?rhs
232   proof (cases "a = b")
233     case True with `?lhs` show ?thesis by simp
234   next
235     case False
236     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
237     with False have "a \<in># N" by auto
238     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
239     moreover note False
240     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
241   qed
242 qed
244 lemma insert_noteq_member:
245   assumes BC: "B + {#b#} = C + {#c#}"
246    and bnotc: "b \<noteq> c"
247   shows "c \<in># B"
248 proof -
249   have "c \<in># C + {#c#}" by simp
250   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
251   then have "c \<in># B + {#b#}" using BC by simp
252   then show "c \<in># B" using nc by simp
253 qed
256   "(M + {#a#} = N + {#b#}) =
257     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
261 subsubsection {* Pointwise ordering induced by count *}
263 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
264 begin
266 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)"
267 by simp
268 lemmas mset_le_def = less_eq_multiset_def
270 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
271   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
273 instance
274   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
276 end
278 lemma mset_less_eqI:
279   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
280   by (simp add: mset_le_def)
282 lemma mset_le_exists_conv:
283   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
284 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
285 apply (auto intro: multiset_eq_iff [THEN iffD2])
286 done
288 lemma mset_le_mono_add_right_cancel [simp]:
289   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
290   by (fact add_le_cancel_right)
292 lemma mset_le_mono_add_left_cancel [simp]:
293   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
294   by (fact add_le_cancel_left)
297   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
298   by (fact add_mono)
300 lemma mset_le_add_left [simp]:
301   "(A::'a multiset) \<le> A + B"
302   unfolding mset_le_def by auto
304 lemma mset_le_add_right [simp]:
305   "B \<le> (A::'a multiset) + B"
306   unfolding mset_le_def by auto
308 lemma mset_le_single:
309   "a :# B \<Longrightarrow> {#a#} \<le> B"
310   by (simp add: mset_le_def)
312 lemma multiset_diff_union_assoc:
313   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
314   by (simp add: multiset_eq_iff mset_le_def)
316 lemma mset_le_multiset_union_diff_commute:
317   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
318 by (simp add: multiset_eq_iff mset_le_def)
320 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
321 by(simp add: mset_le_def)
323 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
324 apply (clarsimp simp: mset_le_def mset_less_def)
325 apply (erule_tac x=x in allE)
326 apply auto
327 done
329 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
330 apply (clarsimp simp: mset_le_def mset_less_def)
331 apply (erule_tac x = x in allE)
332 apply auto
333 done
335 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
336 apply (rule conjI)
337  apply (simp add: mset_lessD)
338 apply (clarsimp simp: mset_le_def mset_less_def)
339 apply safe
340  apply (erule_tac x = a in allE)
341  apply (auto split: split_if_asm)
342 done
344 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
345 apply (rule conjI)
346  apply (simp add: mset_leD)
347 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
348 done
350 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
351   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
353 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
354   by (auto simp: mset_le_def mset_less_def)
356 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
357   by simp
360   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
361   by (fact add_less_imp_less_right)
363 lemma mset_less_empty_nonempty:
364   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
365   by (auto simp: mset_le_def mset_less_def)
367 lemma mset_less_diff_self:
368   "c \<in># B \<Longrightarrow> B - {#c#} < B"
369   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
372 subsubsection {* Intersection *}
374 instantiation multiset :: (type) semilattice_inf
375 begin
377 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
378   multiset_inter_def: "inf_multiset A B = A - (A - B)"
380 instance
381 proof -
382   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
383   show "OFCLASS('a multiset, semilattice_inf_class)"
384     by default (auto simp add: multiset_inter_def mset_le_def aux)
385 qed
387 end
389 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
390   "multiset_inter \<equiv> inf"
392 lemma multiset_inter_count [simp]:
393   "count (A #\<inter> B) x = min (count A x) (count B x)"
394   by (simp add: multiset_inter_def)
396 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
397   by (rule multiset_eqI) auto
399 lemma multiset_union_diff_commute:
400   assumes "B #\<inter> C = {#}"
401   shows "A + B - C = A - C + B"
402 proof (rule multiset_eqI)
403   fix x
404   from assms have "min (count B x) (count C x) = 0"
405     by (auto simp add: multiset_eq_iff)
406   then have "count B x = 0 \<or> count C x = 0"
407     by auto
408   then show "count (A + B - C) x = count (A - C + B) x"
409     by auto
410 qed
413 subsubsection {* Filter (with comprehension syntax) *}
415 text {* Multiset comprehension *}
417 lift_definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
418 by (rule filter_preserves_multiset)
420 hide_const (open) filter
422 lemma count_filter [simp]:
423   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
424   by (simp add: filter.rep_eq)
426 lemma filter_empty [simp]:
427   "Multiset.filter P {#} = {#}"
428   by (rule multiset_eqI) simp
430 lemma filter_single [simp]:
431   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
432   by (rule multiset_eqI) simp
434 lemma filter_union [simp]:
435   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
436   by (rule multiset_eqI) simp
438 lemma filter_diff [simp]:
439   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
440   by (rule multiset_eqI) simp
442 lemma filter_inter [simp]:
443   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
444   by (rule multiset_eqI) simp
446 syntax
447   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
448 syntax (xsymbol)
449   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
450 translations
451   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
454 subsubsection {* Set of elements *}
456 definition set_of :: "'a multiset => 'a set" where
457   "set_of M = {x. x :# M}"
459 lemma set_of_empty [simp]: "set_of {#} = {}"
460 by (simp add: set_of_def)
462 lemma set_of_single [simp]: "set_of {#b#} = {b}"
463 by (simp add: set_of_def)
465 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
466 by (auto simp add: set_of_def)
468 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
469 by (auto simp add: set_of_def multiset_eq_iff)
471 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
472 by (auto simp add: set_of_def)
474 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
475 by (auto simp add: set_of_def)
477 lemma finite_set_of [iff]: "finite (set_of M)"
478   using count [of M] by (simp add: multiset_def set_of_def)
480 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
481   unfolding set_of_def[symmetric] by simp
483 subsubsection {* Size *}
485 instantiation multiset :: (type) size
486 begin
488 definition size_def:
489   "size M = setsum (count M) (set_of M)"
491 instance ..
493 end
495 lemma size_empty [simp]: "size {#} = 0"
496 by (simp add: size_def)
498 lemma size_single [simp]: "size {#b#} = 1"
499 by (simp add: size_def)
501 lemma setsum_count_Int:
502   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
503 apply (induct rule: finite_induct)
504  apply simp
505 apply (simp add: Int_insert_left set_of_def)
506 done
508 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
509 apply (unfold size_def)
510 apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
511  prefer 2
512  apply (rule ext, simp)
513 apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
514 apply (subst Int_commute)
515 apply (simp (no_asm_simp) add: setsum_count_Int)
516 done
518 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
519 by (auto simp add: size_def multiset_eq_iff)
521 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
522 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
524 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
525 apply (unfold size_def)
526 apply (drule setsum_SucD)
527 apply auto
528 done
530 lemma size_eq_Suc_imp_eq_union:
531   assumes "size M = Suc n"
532   shows "\<exists>a N. M = N + {#a#}"
533 proof -
534   from assms obtain a where "a \<in># M"
535     by (erule size_eq_Suc_imp_elem [THEN exE])
536   then have "M = M - {#a#} + {#a#}" by simp
537   then show ?thesis by blast
538 qed
541 subsection {* Induction and case splits *}
543 theorem multiset_induct [case_names empty add, induct type: multiset]:
544   assumes empty: "P {#}"
545   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
546   shows "P M"
547 proof (induct n \<equiv> "size M" arbitrary: M)
548   case 0 thus "P M" by (simp add: empty)
549 next
550   case (Suc k)
551   obtain N x where "M = N + {#x#}"
552     using `Suc k = size M` [symmetric]
553     using size_eq_Suc_imp_eq_union by fast
554   with Suc add show "P M" by simp
555 qed
557 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
558 by (induct M) auto
560 lemma multiset_cases [cases type, case_names empty add]:
561 assumes em:  "M = {#} \<Longrightarrow> P"
562 assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
563 shows "P"
564 using assms by (induct M) simp_all
566 lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
567 by (rule_tac x="M - {#x#}" in exI, simp)
569 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
570 by (cases "B = {#}") (auto dest: multi_member_split)
572 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
573 apply (subst multiset_eq_iff)
574 apply auto
575 done
577 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
578 proof (induct A arbitrary: B)
579   case (empty M)
580   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
581   then obtain M' x where "M = M' + {#x#}"
582     by (blast dest: multi_nonempty_split)
583   then show ?case by simp
584 next
585   case (add S x T)
586   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
587   have SxsubT: "S + {#x#} < T" by fact
588   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
589   then obtain T' where T: "T = T' + {#x#}"
590     by (blast dest: multi_member_split)
591   then have "S < T'" using SxsubT
592     by (blast intro: mset_less_add_bothsides)
593   then have "size S < size T'" using IH by simp
594   then show ?case using T by simp
595 qed
598 subsubsection {* Strong induction and subset induction for multisets *}
600 text {* Well-foundedness of proper subset operator: *}
602 text {* proper multiset subset *}
604 definition
605   mset_less_rel :: "('a multiset * 'a multiset) set" where
606   "mset_less_rel = {(A,B). A < B}"
609   assumes "c \<in># B" and "b \<noteq> c"
610   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
611 proof -
612   from `c \<in># B` obtain A where B: "B = A + {#c#}"
613     by (blast dest: multi_member_split)
614   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
615   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
617   then show ?thesis using B by simp
618 qed
620 lemma wf_mset_less_rel: "wf mset_less_rel"
621 apply (unfold mset_less_rel_def)
622 apply (rule wf_measure [THEN wf_subset, where f1=size])
623 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
624 done
626 text {* The induction rules: *}
628 lemma full_multiset_induct [case_names less]:
629 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
630 shows "P B"
631 apply (rule wf_mset_less_rel [THEN wf_induct])
632 apply (rule ih, auto simp: mset_less_rel_def)
633 done
635 lemma multi_subset_induct [consumes 2, case_names empty add]:
636 assumes "F \<le> A"
637   and empty: "P {#}"
638   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
639 shows "P F"
640 proof -
641   from `F \<le> A`
642   show ?thesis
643   proof (induct F)
644     show "P {#}" by fact
645   next
646     fix x F
647     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
648     show "P (F + {#x#})"
649     proof (rule insert)
650       from i show "x \<in># A" by (auto dest: mset_le_insertD)
651       from i have "F \<le> A" by (auto dest: mset_le_insertD)
652       with P show "P F" .
653     qed
654   qed
655 qed
658 subsection {* The fold combinator *}
660 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
661 where
662   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
664 lemma fold_mset_empty [simp]:
665   "fold f s {#} = s"
666   by (simp add: fold_def)
668 context comp_fun_commute
669 begin
671 lemma fold_mset_insert:
672   "fold f s (M + {#x#}) = f x (fold f s M)"
673 proof -
674   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
675     by (fact comp_fun_commute_funpow)
676   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
677     by (fact comp_fun_commute_funpow)
678   show ?thesis
679   proof (cases "x \<in> set_of M")
680     case False
681     then have *: "count (M + {#x#}) x = 1" by simp
682     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
683       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
684       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
685     with False * show ?thesis
686       by (simp add: fold_def del: count_union)
687   next
688     case True
689     def N \<equiv> "set_of M - {x}"
690     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
691     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
692       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
693       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
694     with * show ?thesis by (simp add: fold_def del: count_union) simp
695   qed
696 qed
698 corollary fold_mset_single [simp]:
699   "fold f s {#x#} = f x s"
700 proof -
701   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
702   then show ?thesis by simp
703 qed
705 lemma fold_mset_fun_left_comm:
706   "f x (fold f s M) = fold f (f x s) M"
707   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
709 lemma fold_mset_union [simp]:
710   "fold f s (M + N) = fold f (fold f s M) N"
711 proof (induct M)
712   case empty then show ?case by simp
713 next
714   case (add M x)
715   have "M + {#x#} + N = (M + N) + {#x#}"
717   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
718 qed
720 lemma fold_mset_fusion:
721   assumes "comp_fun_commute g"
722   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold g w A) = fold f (h w) A" (is "PROP ?P")
723 proof -
724   interpret comp_fun_commute g by (fact assms)
725   show "PROP ?P" by (induct A) auto
726 qed
728 end
730 text {*
731   A note on code generation: When defining some function containing a
732   subterm @{term "fold F"}, code generation is not automatic. When
733   interpreting locale @{text left_commutative} with @{text F}, the
734   would be code thms for @{const fold} become thms like
735   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
736   contains defined symbols, i.e.\ is not a code thm. Hence a separate
737   constant with its own code thms needs to be introduced for @{text
738   F}. See the image operator below.
739 *}
742 subsection {* Image *}
744 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
745   "image_mset f = fold (plus o single o f) {#}"
747 lemma comp_fun_commute_mset_image:
748   "comp_fun_commute (plus o single o f)"
749 proof
752 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
753   by (simp add: image_mset_def)
755 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
756 proof -
757   interpret comp_fun_commute "plus o single o f"
758     by (fact comp_fun_commute_mset_image)
759   show ?thesis by (simp add: image_mset_def)
760 qed
762 lemma image_mset_union [simp]:
763   "image_mset f (M + N) = image_mset f M + image_mset f N"
764 proof -
765   interpret comp_fun_commute "plus o single o f"
766     by (fact comp_fun_commute_mset_image)
767   show ?thesis by (induct N) (simp_all add: image_mset_def add_ac)
768 qed
770 corollary image_mset_insert:
771   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
772   by simp
774 lemma set_of_image_mset [simp]:
775   "set_of (image_mset f M) = image f (set_of M)"
776   by (induct M) simp_all
778 lemma size_image_mset [simp]:
779   "size (image_mset f M) = size M"
780   by (induct M) simp_all
782 lemma image_mset_is_empty_iff [simp]:
783   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
784   by (cases M) auto
786 syntax
787   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
788       ("({#_/. _ :# _#})")
789 translations
790   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
792 syntax
793   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
794       ("({#_/ | _ :# _./ _#})")
795 translations
796   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
798 text {*
799   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
800   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
801   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
802   @{term "{#x+x|x:#M. x<c#}"}.
803 *}
805 enriched_type image_mset: image_mset
806 proof -
807   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
808   proof
809     fix A
810     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
811       by (induct A) simp_all
812   qed
813   show "image_mset id = id"
814   proof
815     fix A
816     show "image_mset id A = id A"
817       by (induct A) simp_all
818   qed
819 qed
821 declare image_mset.identity [simp]
824 subsection {* Further conversions *}
826 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
827   "multiset_of [] = {#}" |
828   "multiset_of (a # x) = multiset_of x + {# a #}"
830 lemma in_multiset_in_set:
831   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
832   by (induct xs) simp_all
834 lemma count_multiset_of:
835   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
836   by (induct xs) simp_all
838 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
839 by (induct x) auto
841 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
842 by (induct x) auto
844 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
845 by (induct x) auto
847 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
848 by (induct xs) auto
850 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
851   by (induct xs) simp_all
853 lemma multiset_of_append [simp]:
854   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
855   by (induct xs arbitrary: ys) (auto simp: add_ac)
857 lemma multiset_of_filter:
858   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
859   by (induct xs) simp_all
861 lemma multiset_of_rev [simp]:
862   "multiset_of (rev xs) = multiset_of xs"
863   by (induct xs) simp_all
865 lemma surj_multiset_of: "surj multiset_of"
866 apply (unfold surj_def)
867 apply (rule allI)
868 apply (rule_tac M = y in multiset_induct)
869  apply auto
870 apply (rule_tac x = "x # xa" in exI)
871 apply auto
872 done
874 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
875 by (induct x) auto
877 lemma distinct_count_atmost_1:
878   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
879 apply (induct x, simp, rule iffI, simp_all)
880 apply (rule conjI)
881 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
882 apply (erule_tac x = a in allE, simp, clarify)
883 apply (erule_tac x = aa in allE, simp)
884 done
886 lemma multiset_of_eq_setD:
887   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
888 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
890 lemma set_eq_iff_multiset_of_eq_distinct:
891   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
892     (set x = set y) = (multiset_of x = multiset_of y)"
893 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
895 lemma set_eq_iff_multiset_of_remdups_eq:
896    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
897 apply (rule iffI)
898 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
899 apply (drule distinct_remdups [THEN distinct_remdups
900       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
901 apply simp
902 done
904 lemma multiset_of_compl_union [simp]:
905   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
906   by (induct xs) (auto simp: add_ac)
908 lemma count_multiset_of_length_filter:
909   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
910   by (induct xs) auto
912 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
913 apply (induct ls arbitrary: i)
914  apply simp
915 apply (case_tac i)
916  apply auto
917 done
919 lemma multiset_of_remove1[simp]:
920   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
921 by (induct xs) (auto simp add: multiset_eq_iff)
923 lemma multiset_of_eq_length:
924   assumes "multiset_of xs = multiset_of ys"
925   shows "length xs = length ys"
926   using assms by (metis size_multiset_of)
928 lemma multiset_of_eq_length_filter:
929   assumes "multiset_of xs = multiset_of ys"
930   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
931   using assms by (metis count_multiset_of)
933 lemma fold_multiset_equiv:
934   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"
935     and equiv: "multiset_of xs = multiset_of ys"
936   shows "List.fold f xs = List.fold f ys"
937 using f equiv [symmetric]
938 proof (induct xs arbitrary: ys)
939   case Nil then show ?case by simp
940 next
941   case (Cons x xs)
942   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
943   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
944     by (rule Cons.prems(1)) (simp_all add: *)
945   moreover from * have "x \<in> set ys" by simp
946   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
947   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
948   ultimately show ?case by simp
949 qed
951 lemma multiset_of_insort [simp]:
952   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
953   by (induct xs) (simp_all add: ac_simps)
955 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
956 where
957   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
959 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
960 where
961   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
962 proof -
963   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
964   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
965   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
966 qed
968 context linorder
969 begin
971 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
972 where
973   "sorted_list_of_multiset M = fold insort [] M"
975 lemma sorted_list_of_multiset_empty [simp]:
976   "sorted_list_of_multiset {#} = []"
977   by (simp add: sorted_list_of_multiset_def)
979 lemma sorted_list_of_multiset_singleton [simp]:
980   "sorted_list_of_multiset {#x#} = [x]"
981 proof -
982   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
983   show ?thesis by (simp add: sorted_list_of_multiset_def)
984 qed
986 lemma sorted_list_of_multiset_insert [simp]:
987   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
988 proof -
989   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
990   show ?thesis by (simp add: sorted_list_of_multiset_def)
991 qed
993 end
995 lemma multiset_of_sorted_list_of_multiset [simp]:
996   "multiset_of (sorted_list_of_multiset M) = M"
997   by (induct M) simp_all
999 lemma sorted_list_of_multiset_multiset_of [simp]:
1000   "sorted_list_of_multiset (multiset_of xs) = sort xs"
1001   by (induct xs) simp_all
1003 lemma finite_set_of_multiset_of_set:
1004   assumes "finite A"
1005   shows "set_of (multiset_of_set A) = A"
1006   using assms by (induct A) simp_all
1008 lemma infinite_set_of_multiset_of_set:
1009   assumes "\<not> finite A"
1010   shows "set_of (multiset_of_set A) = {}"
1011   using assms by simp
1013 lemma set_sorted_list_of_multiset [simp]:
1014   "set (sorted_list_of_multiset M) = set_of M"
1015   by (induct M) (simp_all add: set_insort)
1017 lemma sorted_list_of_multiset_of_set [simp]:
1018   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
1019   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
1022 subsection {* Big operators *}
1024 no_notation times (infixl "*" 70)
1025 no_notation Groups.one ("1")
1027 locale comm_monoid_mset = comm_monoid
1028 begin
1030 definition F :: "'a multiset \<Rightarrow> 'a"
1031 where
1032   eq_fold: "F M = Multiset.fold f 1 M"
1034 lemma empty [simp]:
1035   "F {#} = 1"
1036   by (simp add: eq_fold)
1038 lemma singleton [simp]:
1039   "F {#x#} = x"
1040 proof -
1041   interpret comp_fun_commute
1042     by default (simp add: fun_eq_iff left_commute)
1043   show ?thesis by (simp add: eq_fold)
1044 qed
1046 lemma union [simp]:
1047   "F (M + N) = F M * F N"
1048 proof -
1049   interpret comp_fun_commute f
1050     by default (simp add: fun_eq_iff left_commute)
1051   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
1052 qed
1054 end
1056 notation times (infixl "*" 70)
1057 notation Groups.one ("1")
1059 definition (in comm_monoid_add) msetsum :: "'a multiset \<Rightarrow> 'a"
1060 where
1061   "msetsum = comm_monoid_mset.F plus 0"
1063 definition (in comm_monoid_mult) msetprod :: "'a multiset \<Rightarrow> 'a"
1064 where
1065   "msetprod = comm_monoid_mset.F times 1"
1067 sublocale comm_monoid_add < msetsum!: comm_monoid_mset plus 0
1068 where
1069   "comm_monoid_mset.F plus 0 = msetsum"
1070 proof -
1071   show "comm_monoid_mset plus 0" ..
1072   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
1073 qed
1076 begin
1078 lemma setsum_unfold_msetsum:
1079   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
1080   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1082 abbreviation msetsum_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1083 where
1084   "msetsum_image f M \<equiv> msetsum (image_mset f M)"
1086 end
1088 syntax
1089   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1090       ("(3SUM _:#_. _)" [0, 51, 10] 10)
1092 syntax (xsymbols)
1093   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1094       ("(3\<Sum>_:#_. _)" [0, 51, 10] 10)
1096 syntax (HTML output)
1097   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1098       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1100 translations
1101   "SUM i :# A. b" == "CONST msetsum_image (\<lambda>i. b) A"
1103 sublocale comm_monoid_mult < msetprod!: comm_monoid_mset times 1
1104 where
1105   "comm_monoid_mset.F times 1 = msetprod"
1106 proof -
1107   show "comm_monoid_mset times 1" ..
1108   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
1109 qed
1111 context comm_monoid_mult
1112 begin
1114 lemma msetprod_empty:
1115   "msetprod {#} = 1"
1116   by (fact msetprod.empty)
1118 lemma msetprod_singleton:
1119   "msetprod {#x#} = x"
1120   by (fact msetprod.singleton)
1122 lemma msetprod_Un:
1123   "msetprod (A + B) = msetprod A * msetprod B"
1124   by (fact msetprod.union)
1126 lemma setprod_unfold_msetprod:
1127   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
1128   by (cases "finite A") (induct A rule: finite_induct, simp_all)
1130 lemma msetprod_multiplicity:
1131   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
1132   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
1134 abbreviation msetprod_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1135 where
1136   "msetprod_image f M \<equiv> msetprod (image_mset f M)"
1138 end
1140 syntax
1141   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1142       ("(3PROD _:#_. _)" [0, 51, 10] 10)
1144 syntax (xsymbols)
1145   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1146       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1148 syntax (HTML output)
1149   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1150       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1152 translations
1153   "PROD i :# A. b" == "CONST msetprod_image (\<lambda>i. b) A"
1155 lemma (in comm_semiring_1) dvd_msetprod:
1156   assumes "x \<in># A"
1157   shows "x dvd msetprod A"
1158 proof -
1159   from assms have "A = (A - {#x#}) + {#x#}" by simp
1160   then obtain B where "A = B + {#x#}" ..
1161   then show ?thesis by simp
1162 qed
1165 subsection {* Cardinality *}
1167 definition mcard :: "'a multiset \<Rightarrow> nat"
1168 where
1169   "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
1171 lemma mcard_empty [simp]:
1172   "mcard {#} = 0"
1173   by (simp add: mcard_def)
1175 lemma mcard_singleton [simp]:
1176   "mcard {#a#} = Suc 0"
1177   by (simp add: mcard_def)
1179 lemma mcard_plus [simp]:
1180   "mcard (M + N) = mcard M + mcard N"
1181   by (simp add: mcard_def)
1183 lemma mcard_empty_iff [simp]:
1184   "mcard M = 0 \<longleftrightarrow> M = {#}"
1185   by (induct M) simp_all
1187 lemma mcard_unfold_setsum:
1188   "mcard M = setsum (count M) (set_of M)"
1189 proof (induct M)
1190   case empty then show ?case by simp
1191 next
1192   case (add M x) then show ?case
1193     by (cases "x \<in> set_of M")
1194       (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
1195 qed
1198 subsection {* Alternative representations *}
1200 subsubsection {* Lists *}
1202 context linorder
1203 begin
1205 lemma multiset_of_insort [simp]:
1206   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
1207   by (induct xs) (simp_all add: ac_simps)
1209 lemma multiset_of_sort [simp]:
1210   "multiset_of (sort_key k xs) = multiset_of xs"
1211   by (induct xs) (simp_all add: ac_simps)
1213 text {*
1214   This lemma shows which properties suffice to show that a function
1215   @{text "f"} with @{text "f xs = ys"} behaves like sort.
1216 *}
1218 lemma properties_for_sort_key:
1219   assumes "multiset_of ys = multiset_of xs"
1220   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
1221   and "sorted (map f ys)"
1222   shows "sort_key f xs = ys"
1223 using assms
1224 proof (induct xs arbitrary: ys)
1225   case Nil then show ?case by simp
1226 next
1227   case (Cons x xs)
1228   from Cons.prems(2) have
1229     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
1230     by (simp add: filter_remove1)
1231   with Cons.prems have "sort_key f xs = remove1 x ys"
1232     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
1233   moreover from Cons.prems have "x \<in> set ys"
1234     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
1235   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
1236 qed
1238 lemma properties_for_sort:
1239   assumes multiset: "multiset_of ys = multiset_of xs"
1240   and "sorted ys"
1241   shows "sort xs = ys"
1242 proof (rule properties_for_sort_key)
1243   from multiset show "multiset_of ys = multiset_of xs" .
1244   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
1245   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
1246     by (rule multiset_of_eq_length_filter)
1247   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
1248     by simp
1249   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
1250     by (simp add: replicate_length_filter)
1251 qed
1253 lemma sort_key_by_quicksort:
1254   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
1255     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
1256     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
1257 proof (rule properties_for_sort_key)
1258   show "multiset_of ?rhs = multiset_of ?lhs"
1259     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
1260 next
1261   show "sorted (map f ?rhs)"
1262     by (auto simp add: sorted_append intro: sorted_map_same)
1263 next
1264   fix l
1265   assume "l \<in> set ?rhs"
1266   let ?pivot = "f (xs ! (length xs div 2))"
1267   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
1268   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
1269     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
1270   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
1271   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
1272   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
1273     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
1274   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
1275   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
1276   proof (cases "f l" ?pivot rule: linorder_cases)
1277     case less
1278     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
1279     with less show ?thesis
1280       by (simp add: filter_sort [symmetric] ** ***)
1281   next
1282     case equal then show ?thesis
1283       by (simp add: * less_le)
1284   next
1285     case greater
1286     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
1287     with greater show ?thesis
1288       by (simp add: filter_sort [symmetric] ** ***)
1289   qed
1290 qed
1292 lemma sort_by_quicksort:
1293   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
1294     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
1295     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
1296   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
1298 text {* A stable parametrized quicksort *}
1300 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
1301   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
1303 lemma part_code [code]:
1304   "part f pivot [] = ([], [], [])"
1305   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
1306      if x' < pivot then (x # lts, eqs, gts)
1307      else if x' > pivot then (lts, eqs, x # gts)
1308      else (lts, x # eqs, gts))"
1309   by (auto simp add: part_def Let_def split_def)
1311 lemma sort_key_by_quicksort_code [code]:
1312   "sort_key f xs = (case xs of [] \<Rightarrow> []
1313     | [x] \<Rightarrow> xs
1314     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
1315     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1316        in sort_key f lts @ eqs @ sort_key f gts))"
1317 proof (cases xs)
1318   case Nil then show ?thesis by simp
1319 next
1320   case (Cons _ ys) note hyps = Cons show ?thesis
1321   proof (cases ys)
1322     case Nil with hyps show ?thesis by simp
1323   next
1324     case (Cons _ zs) note hyps = hyps Cons show ?thesis
1325     proof (cases zs)
1326       case Nil with hyps show ?thesis by auto
1327     next
1328       case Cons
1329       from sort_key_by_quicksort [of f xs]
1330       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
1331         in sort_key f lts @ eqs @ sort_key f gts)"
1332       by (simp only: split_def Let_def part_def fst_conv snd_conv)
1333       with hyps Cons show ?thesis by (simp only: list.cases)
1334     qed
1335   qed
1336 qed
1338 end
1340 hide_const (open) part
1342 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
1343   by (induct xs) (auto intro: order_trans)
1345 lemma multiset_of_update:
1346   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
1347 proof (induct ls arbitrary: i)
1348   case Nil then show ?case by simp
1349 next
1350   case (Cons x xs)
1351   show ?case
1352   proof (cases i)
1353     case 0 then show ?thesis by simp
1354   next
1355     case (Suc i')
1356     with Cons show ?thesis
1357       apply simp
1358       apply (subst add_assoc)
1359       apply (subst add_commute [of "{#v#}" "{#x#}"])
1360       apply (subst add_assoc [symmetric])
1361       apply simp
1362       apply (rule mset_le_multiset_union_diff_commute)
1363       apply (simp add: mset_le_single nth_mem_multiset_of)
1364       done
1365   qed
1366 qed
1368 lemma multiset_of_swap:
1369   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
1370     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
1371   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
1374 subsubsection {* Association lists -- including code generation *}
1376 text {* Preliminaries *}
1378 text {* Raw operations on lists *}
1380 definition join_raw :: "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
1381 where
1382   "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
1384 lemma join_raw_Nil [simp]:
1385   "join_raw f xs [] = xs"
1386 by (simp add: join_raw_def)
1388 lemma join_raw_Cons [simp]:
1389   "join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
1390 by (simp add: join_raw_def)
1392 lemma map_of_join_raw:
1393   assumes "distinct (map fst ys)"
1394   shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v =>
1395     (case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
1396 using assms
1397 apply (induct ys)
1398 apply (auto simp add: map_of_map_default split: option.split)
1399 apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
1400 by (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
1402 lemma distinct_join_raw:
1403   assumes "distinct (map fst xs)"
1404   shows "distinct (map fst (join_raw f xs ys))"
1405 using assms
1406 proof (induct ys)
1407   case (Cons y ys)
1408   thus ?case by (cases y) (simp add: distinct_map_default)
1409 qed auto
1411 definition
1412   "subtract_entries_raw xs ys = foldr (%(k, v). AList.map_entry k (%v'. v' - v)) ys xs"
1414 lemma map_of_subtract_entries_raw:
1415   assumes "distinct (map fst ys)"
1416   shows "map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None => None | Some v =>
1417     (case map_of ys x of None => Some v | Some v' => Some (v - v')))"
1418 using assms unfolding subtract_entries_raw_def
1419 apply (induct ys)
1420 apply auto
1421 apply (simp split: option.split)
1422 apply (simp add: map_of_map_entry)
1423 apply (auto split: option.split)
1424 apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
1425 by (metis map_of_eq_None_iff option.simps(4) option.simps(5))
1427 lemma distinct_subtract_entries_raw:
1428   assumes "distinct (map fst xs)"
1429   shows "distinct (map fst (subtract_entries_raw xs ys))"
1430 using assms
1431 unfolding subtract_entries_raw_def by (induct ys) (auto simp add: distinct_map_entry)
1433 text {* Operations on alists with distinct keys *}
1435 lift_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
1436 is join_raw
1437 by (simp add: distinct_join_raw)
1439 lift_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
1440 is subtract_entries_raw
1441 by (simp add: distinct_subtract_entries_raw)
1443 text {* Implementing multisets by means of association lists *}
1445 definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
1446   "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
1448 lemma count_of_multiset:
1449   "count_of xs \<in> multiset"
1450 proof -
1451   let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
1452   have "?A \<subseteq> dom (map_of xs)"
1453   proof
1454     fix x
1455     assume "x \<in> ?A"
1456     then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
1457     then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
1458     then show "x \<in> dom (map_of xs)" by auto
1459   qed
1460   with finite_dom_map_of [of xs] have "finite ?A"
1461     by (auto intro: finite_subset)
1462   then show ?thesis
1463     by (simp add: count_of_def fun_eq_iff multiset_def)
1464 qed
1466 lemma count_simps [simp]:
1467   "count_of [] = (\<lambda>_. 0)"
1468   "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
1469   by (simp_all add: count_of_def fun_eq_iff)
1471 lemma count_of_empty:
1472   "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
1473   by (induct xs) (simp_all add: count_of_def)
1475 lemma count_of_filter:
1476   "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
1477   by (induct xs) auto
1479 lemma count_of_map_default [simp]:
1480   "count_of (map_default x b (%x. x + b) xs) y = (if x = y then count_of xs x + b else count_of xs y)"
1481 unfolding count_of_def by (simp add: map_of_map_default split: option.split)
1483 lemma count_of_join_raw:
1484   "distinct (map fst ys) ==> count_of xs x + count_of ys x = count_of (join_raw (%x (x, y). x + y) xs ys) x"
1485 unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
1487 lemma count_of_subtract_entries_raw:
1488   "distinct (map fst ys) ==> count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
1489 unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split)
1491 text {* Code equations for multiset operations *}
1493 definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset" where
1494   "Bag xs = Abs_multiset (count_of (DAList.impl_of xs))"
1496 code_datatype Bag
1498 lemma count_Bag [simp, code]:
1499   "count (Bag xs) = count_of (DAList.impl_of xs)"
1500   by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
1502 lemma Mempty_Bag [code]:
1503   "{#} = Bag (DAList.empty)"
1504   by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)
1506 lemma single_Bag [code]:
1507   "{#x#} = Bag (DAList.update x 1 DAList.empty)"
1508   by (simp add: multiset_eq_iff alist.Alist_inverse update.rep_eq empty.rep_eq)
1510 lemma union_Bag [code]:
1511   "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
1512 by (rule multiset_eqI) (simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def)
1514 lemma minus_Bag [code]:
1515   "Bag xs - Bag ys = Bag (subtract_entries xs ys)"
1516 by (rule multiset_eqI)
1517   (simp add: count_of_subtract_entries_raw alist.Alist_inverse distinct_subtract_entries_raw subtract_entries_def)
1519 lemma filter_Bag [code]:
1520   "Multiset.filter P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)"
1521 by (rule multiset_eqI) (simp add: count_of_filter DAList.filter.rep_eq)
1523 lemma mset_less_eq_Bag [code]:
1524   "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). count_of (DAList.impl_of xs) x \<le> count A x)"
1525     (is "?lhs \<longleftrightarrow> ?rhs")
1526 proof
1527   assume ?lhs then show ?rhs
1528     by (auto simp add: mset_le_def)
1529 next
1530   assume ?rhs
1531   show ?lhs
1532   proof (rule mset_less_eqI)
1533     fix x
1534     from `?rhs` have "count_of (DAList.impl_of xs) x \<le> count A x"
1535       by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
1536     then show "count (Bag xs) x \<le> count A x"
1537       by (simp add: mset_le_def)
1538   qed
1539 qed
1541 instantiation multiset :: (equal) equal
1542 begin
1544 definition
1545   [code]: "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"
1547 instance
1548   by default (simp add: equal_multiset_def eq_iff)
1550 end
1552 text {* Quickcheck generators *}
1554 definition (in term_syntax)
1555   bagify :: "('a\<Colon>typerep, nat) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)
1556     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
1557   [code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs"
1559 notation fcomp (infixl "\<circ>>" 60)
1560 notation scomp (infixl "\<circ>\<rightarrow>" 60)
1562 instantiation multiset :: (random) random
1563 begin
1565 definition
1566   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (bagify xs))"
1568 instance ..
1570 end
1572 no_notation fcomp (infixl "\<circ>>" 60)
1573 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
1575 instantiation multiset :: (exhaustive) exhaustive
1576 begin
1578 definition exhaustive_multiset :: "('a multiset => (bool * term list) option) => natural => (bool * term list) option"
1579 where
1580   "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (%xs. f (Bag xs)) i"
1582 instance ..
1584 end
1586 instantiation multiset :: (full_exhaustive) full_exhaustive
1587 begin
1589 definition full_exhaustive_multiset :: "('a multiset * (unit => term) => (bool * term list) option) => natural => (bool * term list) option"
1590 where
1591   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (%xs. f (bagify xs)) i"
1593 instance ..
1595 end
1597 hide_const (open) bagify
1600 subsection {* The multiset order *}
1602 subsubsection {* Well-foundedness *}
1604 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1605   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1606       (\<forall>b. b :# K --> (b, a) \<in> r)}"
1608 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
1609   "mult r = (mult1 r)\<^sup>+"
1611 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1612 by (simp add: mult1_def)
1614 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
1615     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1616     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
1617   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
1618 proof (unfold mult1_def)
1619   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
1620   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1621   let ?case1 = "?case1 {(N, M). ?R N M}"
1623   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
1624   then have "\<exists>a' M0' K.
1625       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
1626   then show "?case1 \<or> ?case2"
1627   proof (elim exE conjE)
1628     fix a' M0' K
1629     assume N: "N = M0' + K" and r: "?r K a'"
1630     assume "M0 + {#a#} = M0' + {#a'#}"
1631     then have "M0 = M0' \<and> a = a' \<or>
1632         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
1633       by (simp only: add_eq_conv_ex)
1634     then show ?thesis
1635     proof (elim disjE conjE exE)
1636       assume "M0 = M0'" "a = a'"
1637       with N r have "?r K a \<and> N = M0 + K" by simp
1638       then have ?case2 .. then show ?thesis ..
1639     next
1640       fix K'
1641       assume "M0' = K' + {#a#}"
1642       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
1644       assume "M0 = K' + {#a'#}"
1645       with r have "?R (K' + K) M0" by blast
1646       with n have ?case1 by simp then show ?thesis ..
1647     qed
1648   qed
1649 qed
1651 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
1652 proof
1653   let ?R = "mult1 r"
1654   let ?W = "acc ?R"
1655   {
1656     fix M M0 a
1657     assume M0: "M0 \<in> ?W"
1658       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1659       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
1660     have "M0 + {#a#} \<in> ?W"
1661     proof (rule accI [of "M0 + {#a#}"])
1662       fix N
1663       assume "(N, M0 + {#a#}) \<in> ?R"
1664       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
1665           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
1666         by (rule less_add)
1667       then show "N \<in> ?W"
1668       proof (elim exE disjE conjE)
1669         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
1670         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
1671         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
1672         then show "N \<in> ?W" by (simp only: N)
1673       next
1674         fix K
1675         assume N: "N = M0 + K"
1676         assume "\<forall>b. b :# K --> (b, a) \<in> r"
1677         then have "M0 + K \<in> ?W"
1678         proof (induct K)
1679           case empty
1680           from M0 show "M0 + {#} \<in> ?W" by simp
1681         next
1682           case (add K x)
1683           from add.prems have "(x, a) \<in> r" by simp
1684           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
1685           moreover from add have "M0 + K \<in> ?W" by simp
1686           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
1687           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
1688         qed
1689         then show "N \<in> ?W" by (simp only: N)
1690       qed
1691     qed
1692   } note tedious_reasoning = this
1694   assume wf: "wf r"
1695   fix M
1696   show "M \<in> ?W"
1697   proof (induct M)
1698     show "{#} \<in> ?W"
1699     proof (rule accI)
1700       fix b assume "(b, {#}) \<in> ?R"
1701       with not_less_empty show "b \<in> ?W" by contradiction
1702     qed
1704     fix M a assume "M \<in> ?W"
1705     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1706     proof induct
1707       fix a
1708       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1709       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1710       proof
1711         fix M assume "M \<in> ?W"
1712         then show "M + {#a#} \<in> ?W"
1713           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
1714       qed
1715     qed
1716     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
1717   qed
1718 qed
1720 theorem wf_mult1: "wf r ==> wf (mult1 r)"
1721 by (rule acc_wfI) (rule all_accessible)
1723 theorem wf_mult: "wf r ==> wf (mult r)"
1724 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
1727 subsubsection {* Closure-free presentation *}
1729 text {* One direction. *}
1731 lemma mult_implies_one_step:
1732   "trans r ==> (M, N) \<in> mult r ==>
1733     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
1734     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
1735 apply (unfold mult_def mult1_def set_of_def)
1736 apply (erule converse_trancl_induct, clarify)
1737  apply (rule_tac x = M0 in exI, simp, clarify)
1738 apply (case_tac "a :# K")
1739  apply (rule_tac x = I in exI)
1740  apply (simp (no_asm))
1741  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1742  apply (simp (no_asm_simp) add: add_assoc [symmetric])
1743  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
1744  apply (simp add: diff_union_single_conv)
1745  apply (simp (no_asm_use) add: trans_def)
1746  apply blast
1747 apply (subgoal_tac "a :# I")
1748  apply (rule_tac x = "I - {#a#}" in exI)
1749  apply (rule_tac x = "J + {#a#}" in exI)
1750  apply (rule_tac x = "K + Ka" in exI)
1751  apply (rule conjI)
1752   apply (simp add: multiset_eq_iff split: nat_diff_split)
1753  apply (rule conjI)
1754   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
1755   apply (simp add: multiset_eq_iff split: nat_diff_split)
1756  apply (simp (no_asm_use) add: trans_def)
1757  apply blast
1758 apply (subgoal_tac "a :# (M0 + {#a#})")
1759  apply simp
1760 apply (simp (no_asm))
1761 done
1763 lemma one_step_implies_mult_aux:
1764   "trans r ==>
1765     \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
1766       --> (I + K, I + J) \<in> mult r"
1767 apply (induct_tac n, auto)
1768 apply (frule size_eq_Suc_imp_eq_union, clarify)
1769 apply (rename_tac "J'", simp)
1770 apply (erule notE, auto)
1771 apply (case_tac "J' = {#}")
1772  apply (simp add: mult_def)
1773  apply (rule r_into_trancl)
1774  apply (simp add: mult1_def set_of_def, blast)
1775 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
1776 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
1777 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
1778 apply (erule ssubst)
1779 apply (simp add: Ball_def, auto)
1780 apply (subgoal_tac
1781   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
1782     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
1783  prefer 2
1784  apply force
1785 apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
1786 apply (erule trancl_trans)
1787 apply (rule r_into_trancl)
1788 apply (simp add: mult1_def set_of_def)
1789 apply (rule_tac x = a in exI)
1790 apply (rule_tac x = "I + J'" in exI)
1792 done
1794 lemma one_step_implies_mult:
1795   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
1796     ==> (I + K, I + J) \<in> mult r"
1797 using one_step_implies_mult_aux by blast
1800 subsubsection {* Partial-order properties *}
1802 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
1803   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
1805 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
1806   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
1808 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
1809 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
1811 interpretation multiset_order: order le_multiset less_multiset
1812 proof -
1813   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
1814   proof
1815     fix M :: "'a multiset"
1816     assume "M \<subset># M"
1817     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
1818     have "trans {(x'::'a, x). x' < x}"
1819       by (rule transI) simp
1820     moreover note MM
1821     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
1822       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
1823       by (rule mult_implies_one_step)
1824     then obtain I J K where "M = I + J" and "M = I + K"
1825       and "J \<noteq> {#}" and "(\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})" by blast
1826     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
1827     have "finite (set_of K)" by simp
1828     moreover note aux2
1829     ultimately have "set_of K = {}"
1830       by (induct rule: finite_induct) (auto intro: order_less_trans)
1831     with aux1 show False by simp
1832   qed
1833   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
1834     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
1835   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
1836     by default (auto simp add: le_multiset_def irrefl dest: trans)
1837 qed
1839 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
1840   by simp
1843 subsubsection {* Monotonicity of multiset union *}
1845 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
1846 apply (unfold mult1_def)
1847 apply auto
1848 apply (rule_tac x = a in exI)
1849 apply (rule_tac x = "C + M0" in exI)
1851 done
1853 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
1854 apply (unfold less_multiset_def mult_def)
1855 apply (erule trancl_induct)
1856  apply (blast intro: mult1_union)
1857 apply (blast intro: mult1_union trancl_trans)
1858 done
1860 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
1861 apply (subst add_commute [of B C])
1862 apply (subst add_commute [of D C])
1863 apply (erule union_less_mono2)
1864 done
1866 lemma union_less_mono:
1867   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
1868   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
1870 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
1871 proof
1872 qed (auto simp add: le_multiset_def intro: union_less_mono2)
1875 subsection {* Termination proofs with multiset orders *}
1877 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
1878   and multi_member_this: "x \<in># {# x #} + XS"
1879   and multi_member_last: "x \<in># {# x #}"
1880   by auto
1882 definition "ms_strict = mult pair_less"
1883 definition "ms_weak = ms_strict \<union> Id"
1885 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
1886 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
1887 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
1889 lemma smsI:
1890   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
1891   unfolding ms_strict_def
1892 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
1894 lemma wmsI:
1895   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
1896   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
1897 unfolding ms_weak_def ms_strict_def
1898 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
1900 inductive pw_leq
1901 where
1902   pw_leq_empty: "pw_leq {#} {#}"
1903 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
1905 lemma pw_leq_lstep:
1906   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
1907 by (drule pw_leq_step) (rule pw_leq_empty, simp)
1909 lemma pw_leq_split:
1910   assumes "pw_leq X Y"
1911   shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1912   using assms
1913 proof (induct)
1914   case pw_leq_empty thus ?case by auto
1915 next
1916   case (pw_leq_step x y X Y)
1917   then obtain A B Z where
1918     [simp]: "X = A + Z" "Y = B + Z"
1919       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
1920     by auto
1921   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
1922     unfolding pair_leq_def by auto
1923   thus ?case
1924   proof
1925     assume [simp]: "x = y"
1926     have
1927       "{#x#} + X = A + ({#y#}+Z)
1928       \<and> {#y#} + Y = B + ({#y#}+Z)
1929       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
1930       by (auto simp: add_ac)
1931     thus ?case by (intro exI)
1932   next
1933     assume A: "(x, y) \<in> pair_less"
1934     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
1935     have "{#x#} + X = ?A' + Z"
1936       "{#y#} + Y = ?B' + Z"
1938     moreover have
1939       "(set_of ?A', set_of ?B') \<in> max_strict"
1940       using 1 A unfolding max_strict_def
1941       by (auto elim!: max_ext.cases)
1942     ultimately show ?thesis by blast
1943   qed
1944 qed
1946 lemma
1947   assumes pwleq: "pw_leq Z Z'"
1948   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
1949   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
1950   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
1951 proof -
1952   from pw_leq_split[OF pwleq]
1953   obtain A' B' Z''
1954     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
1955     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
1956     by blast
1957   {
1958     assume max: "(set_of A, set_of B) \<in> max_strict"
1959     from mx_or_empty
1960     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
1961     proof
1962       assume max': "(set_of A', set_of B') \<in> max_strict"
1963       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
1964         by (auto simp: max_strict_def intro: max_ext_additive)
1965       thus ?thesis by (rule smsI)
1966     next
1967       assume [simp]: "A' = {#} \<and> B' = {#}"
1968       show ?thesis by (rule smsI) (auto intro: max)
1969     qed
1970     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
1971     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
1972   }
1973   from mx_or_empty
1974   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
1975   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
1976 qed
1978 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
1979 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
1980 and nonempty_single: "{# x #} \<noteq> {#}"
1981 by auto
1983 setup {*
1984 let
1985   fun msetT T = Type (@{type_name multiset}, [T]);
1987   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
1988     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
1989     | mk_mset T (x :: xs) =
1990           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
1991                 mk_mset T [x] \$ mk_mset T xs
1993   fun mset_member_tac m i =
1994       (if m <= 0 then
1995            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
1996        else
1997            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
1999   val mset_nonempty_tac =
2000       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
2002   val regroup_munion_conv =
2003       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
2004         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))
2006   fun unfold_pwleq_tac i =
2007     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
2008       ORELSE (rtac @{thm pw_leq_lstep} i)
2009       ORELSE (rtac @{thm pw_leq_empty} i)
2011   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
2012                       @{thm Un_insert_left}, @{thm Un_empty_left}]
2013 in
2014   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
2015   {
2016     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
2017     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
2018     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
2019     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
2020     reduction_pair= @{thm ms_reduction_pair}
2021   })
2022 end
2023 *}
2026 subsection {* Legacy theorem bindings *}
2028 lemmas multi_count_eq = multiset_eq_iff [symmetric]
2030 lemma union_commute: "M + N = N + (M::'a multiset)"
2031   by (fact add_commute)
2033 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
2034   by (fact add_assoc)
2036 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
2037   by (fact add_left_commute)
2039 lemmas union_ac = union_assoc union_commute union_lcomm
2041 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
2042   by (fact add_right_cancel)
2044 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
2045   by (fact add_left_cancel)
2047 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
2048   by (fact add_imp_eq)
2050 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
2051   by (fact order_less_trans)
2053 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
2054   by (fact inf.commute)
2056 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
2057   by (fact inf.assoc [symmetric])
2059 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
2060   by (fact inf.left_commute)
2062 lemmas multiset_inter_ac =
2063   multiset_inter_commute
2064   multiset_inter_assoc
2065   multiset_inter_left_commute
2067 lemma mult_less_not_refl:
2068   "\<not> M \<subset># (M::'a::order multiset)"
2069   by (fact multiset_order.less_irrefl)
2071 lemma mult_less_trans:
2072   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
2073   by (fact multiset_order.less_trans)
2075 lemma mult_less_not_sym:
2076   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
2077   by (fact multiset_order.less_not_sym)
2079 lemma mult_less_asym:
2080   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
2081   by (fact multiset_order.less_asym)
2083 ML {*
2084 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
2085                       (Const _ \$ t') =
2086     let
2087       val (maybe_opt, ps) =
2088         Nitpick_Model.dest_plain_fun t' ||> op ~~
2089         ||> map (apsnd (snd o HOLogic.dest_number))
2090       fun elems_for t =
2091         case AList.lookup (op =) ps t of
2092           SOME n => replicate n t
2093         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
2094     in
2095       case maps elems_for (all_values elem_T) @
2096            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
2097             else []) of
2098         [] => Const (@{const_name zero_class.zero}, T)
2099       | ts => foldl1 (fn (t1, t2) =>
2100                          Const (@{const_name plus_class.plus}, T --> T --> T)
2101                          \$ t1 \$ t2)
2102                      (map (curry (op \$) (Const (@{const_name single},
2103                                                 elem_T --> T))) ts)
2104     end
2105   | multiset_postproc _ _ _ _ t = t
2106 *}
2108 declaration {*
2109 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
2110     multiset_postproc
2111 *}
2113 hide_const (open) fold
2115 end