src/HOL/Library/Multiset_Permutations.thy
 author Manuel Eberl Mon Mar 26 16:14:16 2018 +0200 (18 months ago) changeset 67951 655aa11359dc parent 67399 eab6ce8368fa child 68406 6beb45f6cf67 permissions -rw-r--r--
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
1 (*  Title:      HOL/Library/Multiset_Permutations.thy
2     Author:     Manuel Eberl (TU München)
4 Defines the set of permutations of a given multiset (or set), i.e. the set of all lists whose
5 entries correspond to the multiset (resp. set).
6 *)
8 section \<open>Permutations of a Multiset\<close>
10 theory Multiset_Permutations
11 imports
12   Complex_Main
13   Multiset
14   Permutations
15 begin
17 (* TODO Move *)
18 lemma mset_tl: "xs \<noteq> [] \<Longrightarrow> mset (tl xs) = mset xs - {#hd xs#}"
19   by (cases xs) simp_all
21 lemma mset_set_image_inj:
22   assumes "inj_on f A"
23   shows   "mset_set (f ` A) = image_mset f (mset_set A)"
24 proof (cases "finite A")
25   case True
26   from this and assms show ?thesis by (induction A) auto
27 qed (insert assms, simp add: finite_image_iff)
29 lemma multiset_remove_induct [case_names empty remove]:
30   assumes "P {#}" "\<And>A. A \<noteq> {#} \<Longrightarrow> (\<And>x. x \<in># A \<Longrightarrow> P (A - {#x#})) \<Longrightarrow> P A"
31   shows   "P A"
32 proof (induction A rule: full_multiset_induct)
33   case (less A)
34   hence IH: "P B" if "B \<subset># A" for B using that by blast
35   show ?case
36   proof (cases "A = {#}")
37     case True
38     thus ?thesis by (simp add: assms)
39   next
40     case False
41     hence "P (A - {#x#})" if "x \<in># A" for x
42       using that by (intro IH) (simp add: mset_subset_diff_self)
43     from False and this show "P A" by (rule assms)
44   qed
45 qed
47 lemma map_list_bind: "map g (List.bind xs f) = List.bind xs (map g \<circ> f)"
48   by (simp add: List.bind_def map_concat)
50 lemma mset_eq_mset_set_imp_distinct:
51   "finite A \<Longrightarrow> mset_set A = mset xs \<Longrightarrow> distinct xs"
52 proof (induction xs arbitrary: A)
53   case (Cons x xs A)
54   from Cons.prems(2) have "x \<in># mset_set A" by simp
55   with Cons.prems(1) have [simp]: "x \<in> A" by simp
56   from Cons.prems have "x \<notin># mset_set (A - {x})" by simp
57   also from Cons.prems have "mset_set (A - {x}) = mset_set A - {#x#}"
58     by (subst mset_set_Diff) simp_all
59   also have "mset_set A = mset (x#xs)" by (simp add: Cons.prems)
60   also have "\<dots> - {#x#} = mset xs" by simp
61   finally have [simp]: "x \<notin> set xs" by (simp add: in_multiset_in_set)
62   from Cons.prems show ?case by (auto intro!: Cons.IH[of "A - {x}"] simp: mset_set_Diff)
63 qed simp_all
64 (* END TODO *)
67 subsection \<open>Permutations of a multiset\<close>
69 definition permutations_of_multiset :: "'a multiset \<Rightarrow> 'a list set" where
70   "permutations_of_multiset A = {xs. mset xs = A}"
72 lemma permutations_of_multisetI: "mset xs = A \<Longrightarrow> xs \<in> permutations_of_multiset A"
73   by (simp add: permutations_of_multiset_def)
75 lemma permutations_of_multisetD: "xs \<in> permutations_of_multiset A \<Longrightarrow> mset xs = A"
76   by (simp add: permutations_of_multiset_def)
78 lemma permutations_of_multiset_Cons_iff:
79   "x # xs \<in> permutations_of_multiset A \<longleftrightarrow> x \<in># A \<and> xs \<in> permutations_of_multiset (A - {#x#})"
80   by (auto simp: permutations_of_multiset_def)
82 lemma permutations_of_multiset_empty [simp]: "permutations_of_multiset {#} = {[]}"
83   unfolding permutations_of_multiset_def by simp
85 lemma permutations_of_multiset_nonempty:
86   assumes nonempty: "A \<noteq> {#}"
87   shows   "permutations_of_multiset A =
88              (\<Union>x\<in>set_mset A. ((#) x) ` permutations_of_multiset (A - {#x#}))" (is "_ = ?rhs")
89 proof safe
90   fix xs assume "xs \<in> permutations_of_multiset A"
91   hence mset_xs: "mset xs = A" by (simp add: permutations_of_multiset_def)
92   hence "xs \<noteq> []" by (auto simp: nonempty)
93   then obtain x xs' where xs: "xs = x # xs'" by (cases xs) simp_all
94   with mset_xs have "x \<in> set_mset A" "xs' \<in> permutations_of_multiset (A - {#x#})"
95     by (auto simp: permutations_of_multiset_def)
96   with xs show "xs \<in> ?rhs" by auto
97 qed (auto simp: permutations_of_multiset_def)
99 lemma permutations_of_multiset_singleton [simp]: "permutations_of_multiset {#x#} = {[x]}"
100   by (simp add: permutations_of_multiset_nonempty)
102 lemma permutations_of_multiset_doubleton:
103   "permutations_of_multiset {#x,y#} = {[x,y], [y,x]}"
104   by (simp add: permutations_of_multiset_nonempty insert_commute)
106 lemma rev_permutations_of_multiset [simp]:
107   "rev ` permutations_of_multiset A = permutations_of_multiset A"
108 proof
109   have "rev ` rev ` permutations_of_multiset A \<subseteq> rev ` permutations_of_multiset A"
110     unfolding permutations_of_multiset_def by auto
111   also have "rev ` rev ` permutations_of_multiset A = permutations_of_multiset A"
112     by (simp add: image_image)
113   finally show "permutations_of_multiset A \<subseteq> rev ` permutations_of_multiset A" .
114 next
115   show "rev ` permutations_of_multiset A \<subseteq> permutations_of_multiset A"
116     unfolding permutations_of_multiset_def by auto
117 qed
119 lemma length_finite_permutations_of_multiset:
120   "xs \<in> permutations_of_multiset A \<Longrightarrow> length xs = size A"
121   by (auto simp: permutations_of_multiset_def)
123 lemma permutations_of_multiset_lists: "permutations_of_multiset A \<subseteq> lists (set_mset A)"
124   by (auto simp: permutations_of_multiset_def)
126 lemma finite_permutations_of_multiset [simp]: "finite (permutations_of_multiset A)"
127 proof (rule finite_subset)
128   show "permutations_of_multiset A \<subseteq> {xs. set xs \<subseteq> set_mset A \<and> length xs = size A}"
129     by (auto simp: permutations_of_multiset_def)
130   show "finite {xs. set xs \<subseteq> set_mset A \<and> length xs = size A}"
131     by (rule finite_lists_length_eq) simp_all
132 qed
134 lemma permutations_of_multiset_not_empty [simp]: "permutations_of_multiset A \<noteq> {}"
135 proof -
136   from ex_mset[of A] guess xs ..
137   thus ?thesis by (auto simp: permutations_of_multiset_def)
138 qed
140 lemma permutations_of_multiset_image:
141   "permutations_of_multiset (image_mset f A) = map f ` permutations_of_multiset A"
142 proof safe
143   fix xs assume A: "xs \<in> permutations_of_multiset (image_mset f A)"
144   from ex_mset[of A] obtain ys where ys: "mset ys = A" ..
145   with A have "mset xs = mset (map f ys)"
146     by (simp add: permutations_of_multiset_def)
147   from mset_eq_permutation[OF this] guess \<sigma> . note \<sigma> = this
148   with ys have "xs = map f (permute_list \<sigma> ys)"
149     by (simp add: permute_list_map)
150   moreover from \<sigma> ys have "permute_list \<sigma> ys \<in> permutations_of_multiset A"
151     by (simp add: permutations_of_multiset_def)
152   ultimately show "xs \<in> map f ` permutations_of_multiset A" by blast
153 qed (auto simp: permutations_of_multiset_def)
156 subsection \<open>Cardinality of permutations\<close>
158 text \<open>
159   In this section, we prove some basic facts about the number of permutations of a multiset.
160 \<close>
162 context
163 begin
165 private lemma multiset_prod_fact_insert:
166   "(\<Prod>y\<in>set_mset (A+{#x#}). fact (count (A+{#x#}) y)) =
167      (count A x + 1) * (\<Prod>y\<in>set_mset A. fact (count A y))"
168 proof -
169   have "(\<Prod>y\<in>set_mset (A+{#x#}). fact (count (A+{#x#}) y)) =
170           (\<Prod>y\<in>set_mset (A+{#x#}). (if y = x then count A x + 1 else 1) * fact (count A y))"
171     by (intro prod.cong) simp_all
172   also have "\<dots> = (count A x + 1) * (\<Prod>y\<in>set_mset (A+{#x#}). fact (count A y))"
173     by (simp add: prod.distrib prod.delta)
174   also have "(\<Prod>y\<in>set_mset (A+{#x#}). fact (count A y)) = (\<Prod>y\<in>set_mset A. fact (count A y))"
175     by (intro prod.mono_neutral_right) (auto simp: not_in_iff)
176   finally show ?thesis .
177 qed
179 private lemma multiset_prod_fact_remove:
180   "x \<in># A \<Longrightarrow> (\<Prod>y\<in>set_mset A. fact (count A y)) =
181                    count A x * (\<Prod>y\<in>set_mset (A-{#x#}). fact (count (A-{#x#}) y))"
182   using multiset_prod_fact_insert[of "A - {#x#}" x] by simp
184 lemma card_permutations_of_multiset_aux:
185   "card (permutations_of_multiset A) * (\<Prod>x\<in>set_mset A. fact (count A x)) = fact (size A)"
186 proof (induction A rule: multiset_remove_induct)
187   case (remove A)
188   have "card (permutations_of_multiset A) =
189           card (\<Union>x\<in>set_mset A. (#) x ` permutations_of_multiset (A - {#x#}))"
190     by (simp add: permutations_of_multiset_nonempty remove.hyps)
191   also have "\<dots> = (\<Sum>x\<in>set_mset A. card (permutations_of_multiset (A - {#x#})))"
192     by (subst card_UN_disjoint) (auto simp: card_image)
193   also have "\<dots> * (\<Prod>x\<in>set_mset A. fact (count A x)) =
194                (\<Sum>x\<in>set_mset A. card (permutations_of_multiset (A - {#x#})) *
195                  (\<Prod>y\<in>set_mset A. fact (count A y)))"
196     by (subst sum_distrib_right) simp_all
197   also have "\<dots> = (\<Sum>x\<in>set_mset A. count A x * fact (size A - 1))"
198   proof (intro sum.cong refl)
199     fix x assume x: "x \<in># A"
200     have "card (permutations_of_multiset (A - {#x#})) * (\<Prod>y\<in>set_mset A. fact (count A y)) =
201             count A x * (card (permutations_of_multiset (A - {#x#})) *
202               (\<Prod>y\<in>set_mset (A - {#x#}). fact (count (A - {#x#}) y)))" (is "?lhs = _")
203       by (subst multiset_prod_fact_remove[OF x]) simp_all
204     also note remove.IH[OF x]
205     also from x have "size (A - {#x#}) = size A - 1" by (simp add: size_Diff_submset)
206     finally show "?lhs = count A x * fact (size A - 1)" .
207   qed
208   also have "(\<Sum>x\<in>set_mset A. count A x * fact (size A - 1)) =
209                 size A * fact (size A - 1)"
211   also from remove.hyps have "\<dots> = fact (size A)"
212     by (cases "size A") auto
213   finally show ?case .
214 qed simp_all
216 theorem card_permutations_of_multiset:
217   "card (permutations_of_multiset A) = fact (size A) div (\<Prod>x\<in>set_mset A. fact (count A x))"
218   "(\<Prod>x\<in>set_mset A. fact (count A x) :: nat) dvd fact (size A)"
219   by (simp_all add: card_permutations_of_multiset_aux[of A, symmetric])
221 lemma card_permutations_of_multiset_insert_aux:
222   "card (permutations_of_multiset (A + {#x#})) * (count A x + 1) =
223       (size A + 1) * card (permutations_of_multiset A)"
224 proof -
225   note card_permutations_of_multiset_aux[of "A + {#x#}"]
226   also have "fact (size (A + {#x#})) = (size A + 1) * fact (size A)" by simp
227   also note multiset_prod_fact_insert[of A x]
228   also note card_permutations_of_multiset_aux[of A, symmetric]
229   finally have "card (permutations_of_multiset (A + {#x#})) * (count A x + 1) *
230                     (\<Prod>y\<in>set_mset A. fact (count A y)) =
231                 (size A + 1) * card (permutations_of_multiset A) *
232                     (\<Prod>x\<in>set_mset A. fact (count A x))" by (simp only: mult_ac)
233   thus ?thesis by (subst (asm) mult_right_cancel) simp_all
234 qed
236 lemma card_permutations_of_multiset_remove_aux:
237   assumes "x \<in># A"
238   shows   "card (permutations_of_multiset A) * count A x =
239              size A * card (permutations_of_multiset (A - {#x#}))"
240 proof -
241   from assms have A: "A - {#x#} + {#x#} = A" by simp
242   from assms have B: "size A = size (A - {#x#}) + 1"
243     by (subst A [symmetric], subst size_union) simp
244   show ?thesis
245     using card_permutations_of_multiset_insert_aux[of "A - {#x#}" x, unfolded A] assms
246     by (simp add: B)
247 qed
249 lemma real_card_permutations_of_multiset_remove:
250   assumes "x \<in># A"
251   shows   "real (card (permutations_of_multiset (A - {#x#}))) =
252              real (card (permutations_of_multiset A) * count A x) / real (size A)"
253   using assms by (subst card_permutations_of_multiset_remove_aux[OF assms]) auto
255 lemma real_card_permutations_of_multiset_remove':
256   assumes "x \<in># A"
257   shows   "real (card (permutations_of_multiset A)) =
258              real (size A * card (permutations_of_multiset (A - {#x#}))) / real (count A x)"
259   using assms by (subst card_permutations_of_multiset_remove_aux[OF assms, symmetric]) simp
261 end
265 subsection \<open>Permutations of a set\<close>
267 definition permutations_of_set :: "'a set \<Rightarrow> 'a list set" where
268   "permutations_of_set A = {xs. set xs = A \<and> distinct xs}"
270 lemma permutations_of_set_altdef:
271   "finite A \<Longrightarrow> permutations_of_set A = permutations_of_multiset (mset_set A)"
272   by (auto simp add: permutations_of_set_def permutations_of_multiset_def mset_set_set
273         in_multiset_in_set [symmetric] mset_eq_mset_set_imp_distinct)
275 lemma permutations_of_setI [intro]:
276   assumes "set xs = A" "distinct xs"
277   shows   "xs \<in> permutations_of_set A"
278   using assms unfolding permutations_of_set_def by simp
280 lemma permutations_of_setD:
281   assumes "xs \<in> permutations_of_set A"
282   shows   "set xs = A" "distinct xs"
283   using assms unfolding permutations_of_set_def by simp_all
285 lemma permutations_of_set_lists: "permutations_of_set A \<subseteq> lists A"
286   unfolding permutations_of_set_def by auto
288 lemma permutations_of_set_empty [simp]: "permutations_of_set {} = {[]}"
289   by (auto simp: permutations_of_set_def)
291 lemma UN_set_permutations_of_set [simp]:
292   "finite A \<Longrightarrow> (\<Union>xs\<in>permutations_of_set A. set xs) = A"
293   using finite_distinct_list by (auto simp: permutations_of_set_def)
295 lemma permutations_of_set_infinite:
296   "\<not>finite A \<Longrightarrow> permutations_of_set A = {}"
297   by (auto simp: permutations_of_set_def)
299 lemma permutations_of_set_nonempty:
300   "A \<noteq> {} \<Longrightarrow> permutations_of_set A =
301                   (\<Union>x\<in>A. (\<lambda>xs. x # xs) ` permutations_of_set (A - {x}))"
302   by (cases "finite A")
303      (simp_all add: permutations_of_multiset_nonempty mset_set_empty_iff mset_set_Diff
304                     permutations_of_set_altdef permutations_of_set_infinite)
306 lemma permutations_of_set_singleton [simp]: "permutations_of_set {x} = {[x]}"
307   by (subst permutations_of_set_nonempty) auto
309 lemma permutations_of_set_doubleton:
310   "x \<noteq> y \<Longrightarrow> permutations_of_set {x,y} = {[x,y], [y,x]}"
311   by (subst permutations_of_set_nonempty)
312      (simp_all add: insert_Diff_if insert_commute)
314 lemma rev_permutations_of_set [simp]:
315   "rev ` permutations_of_set A = permutations_of_set A"
316   by (cases "finite A") (simp_all add: permutations_of_set_altdef permutations_of_set_infinite)
318 lemma length_finite_permutations_of_set:
319   "xs \<in> permutations_of_set A \<Longrightarrow> length xs = card A"
320   by (auto simp: permutations_of_set_def distinct_card)
322 lemma finite_permutations_of_set [simp]: "finite (permutations_of_set A)"
323   by (cases "finite A") (simp_all add: permutations_of_set_infinite permutations_of_set_altdef)
325 lemma permutations_of_set_empty_iff [simp]:
326   "permutations_of_set A = {} \<longleftrightarrow> \<not>finite A"
327   unfolding permutations_of_set_def using finite_distinct_list[of A] by auto
329 lemma card_permutations_of_set [simp]:
330   "finite A \<Longrightarrow> card (permutations_of_set A) = fact (card A)"
331   by (simp add: permutations_of_set_altdef card_permutations_of_multiset del: One_nat_def)
333 lemma permutations_of_set_image_inj:
334   assumes inj: "inj_on f A"
335   shows   "permutations_of_set (f ` A) = map f ` permutations_of_set A"
336   by (cases "finite A")
337      (simp_all add: permutations_of_set_infinite permutations_of_set_altdef
338                     permutations_of_multiset_image mset_set_image_inj inj finite_image_iff)
340 lemma permutations_of_set_image_permutes:
341   "\<sigma> permutes A \<Longrightarrow> map \<sigma> ` permutations_of_set A = permutations_of_set A"
342   by (subst permutations_of_set_image_inj [symmetric])
343      (simp_all add: permutes_inj_on permutes_image)
346 subsection \<open>Code generation\<close>
348 text \<open>
349   First, we give code an implementation for permutations of lists.
350 \<close>
352 declare length_remove1 [termination_simp]
354 fun permutations_of_list_impl where
355   "permutations_of_list_impl xs = (if xs = [] then [[]] else
356      List.bind (remdups xs) (\<lambda>x. map ((#) x) (permutations_of_list_impl (remove1 x xs))))"
358 fun permutations_of_list_impl_aux where
359   "permutations_of_list_impl_aux acc xs = (if xs = [] then [acc] else
360      List.bind (remdups xs) (\<lambda>x. permutations_of_list_impl_aux (x#acc) (remove1 x xs)))"
362 declare permutations_of_list_impl_aux.simps [simp del]
363 declare permutations_of_list_impl.simps [simp del]
365 lemma permutations_of_list_impl_Nil [simp]:
366   "permutations_of_list_impl [] = [[]]"
367   by (simp add: permutations_of_list_impl.simps)
369 lemma permutations_of_list_impl_nonempty:
370   "xs \<noteq> [] \<Longrightarrow> permutations_of_list_impl xs =
371      List.bind (remdups xs) (\<lambda>x. map ((#) x) (permutations_of_list_impl (remove1 x xs)))"
372   by (subst permutations_of_list_impl.simps) simp_all
374 lemma set_permutations_of_list_impl:
375   "set (permutations_of_list_impl xs) = permutations_of_multiset (mset xs)"
376   by (induction xs rule: permutations_of_list_impl.induct)
377      (subst permutations_of_list_impl.simps,
378       simp_all add: permutations_of_multiset_nonempty set_list_bind)
380 lemma distinct_permutations_of_list_impl:
381   "distinct (permutations_of_list_impl xs)"
382   by (induction xs rule: permutations_of_list_impl.induct,
383       subst permutations_of_list_impl.simps)
384      (auto intro!: distinct_list_bind simp: distinct_map o_def disjoint_family_on_def)
386 lemma permutations_of_list_impl_aux_correct':
387   "permutations_of_list_impl_aux acc xs =
388      map (\<lambda>xs. rev xs @ acc) (permutations_of_list_impl xs)"
389   by (induction acc xs rule: permutations_of_list_impl_aux.induct,
390       subst permutations_of_list_impl_aux.simps, subst permutations_of_list_impl.simps)
391      (auto simp: map_list_bind intro!: list_bind_cong)
393 lemma permutations_of_list_impl_aux_correct:
394   "permutations_of_list_impl_aux [] xs = map rev (permutations_of_list_impl xs)"
395   by (simp add: permutations_of_list_impl_aux_correct')
397 lemma distinct_permutations_of_list_impl_aux:
398   "distinct (permutations_of_list_impl_aux acc xs)"
399   by (simp add: permutations_of_list_impl_aux_correct' distinct_map
400         distinct_permutations_of_list_impl inj_on_def)
402 lemma set_permutations_of_list_impl_aux:
403   "set (permutations_of_list_impl_aux [] xs) = permutations_of_multiset (mset xs)"
404   by (simp add: permutations_of_list_impl_aux_correct set_permutations_of_list_impl)
406 declare set_permutations_of_list_impl_aux [symmetric, code]
408 value [code] "permutations_of_multiset {#1,2,3,4::int#}"
412 text \<open>
413   Now we turn to permutations of sets. We define an auxiliary version with an
414   accumulator to avoid having to map over the results.
415 \<close>
416 function permutations_of_set_aux where
417   "permutations_of_set_aux acc A =
418      (if \<not>finite A then {} else if A = {} then {acc} else
419         (\<Union>x\<in>A. permutations_of_set_aux (x#acc) (A - {x})))"
420 by auto
421 termination by (relation "Wellfounded.measure (card \<circ> snd)") (simp_all add: card_gt_0_iff)
423 lemma permutations_of_set_aux_altdef:
424   "permutations_of_set_aux acc A = (\<lambda>xs. rev xs @ acc) ` permutations_of_set A"
425 proof (cases "finite A")
426   assume "finite A"
427   thus ?thesis
428   proof (induction A arbitrary: acc rule: finite_psubset_induct)
429     case (psubset A acc)
430     show ?case
431     proof (cases "A = {}")
432       case False
433       note [simp del] = permutations_of_set_aux.simps
434       from psubset.hyps False
435         have "permutations_of_set_aux acc A =
436                 (\<Union>y\<in>A. permutations_of_set_aux (y#acc) (A - {y}))"
437         by (subst permutations_of_set_aux.simps) simp_all
438       also have "\<dots> = (\<Union>y\<in>A. (\<lambda>xs. rev xs @ acc) ` (\<lambda>xs. y # xs) ` permutations_of_set (A - {y}))"
439         by (intro SUP_cong refl, subst psubset) (auto simp: image_image)
440       also from False have "\<dots> = (\<lambda>xs. rev xs @ acc) ` permutations_of_set A"
441         by (subst (2) permutations_of_set_nonempty) (simp_all add: image_UN)
442       finally show ?thesis .
443     qed simp_all
444   qed
445 qed (simp_all add: permutations_of_set_infinite)
447 declare permutations_of_set_aux.simps [simp del]
449 lemma permutations_of_set_aux_correct:
450   "permutations_of_set_aux [] A = permutations_of_set A"
451   by (simp add: permutations_of_set_aux_altdef)
454 text \<open>
455   In another refinement step, we define a version on lists.
456 \<close>
457 declare length_remove1 [termination_simp]
459 fun permutations_of_set_aux_list where
460   "permutations_of_set_aux_list acc xs =
461      (if xs = [] then [acc] else
462         List.bind xs (\<lambda>x. permutations_of_set_aux_list (x#acc) (List.remove1 x xs)))"
464 definition permutations_of_set_list where
465   "permutations_of_set_list xs = permutations_of_set_aux_list [] xs"
467 declare permutations_of_set_aux_list.simps [simp del]
469 lemma permutations_of_set_aux_list_refine:
470   assumes "distinct xs"
471   shows   "set (permutations_of_set_aux_list acc xs) = permutations_of_set_aux acc (set xs)"
472   using assms
473   by (induction acc xs rule: permutations_of_set_aux_list.induct)
474      (subst permutations_of_set_aux_list.simps,
475       subst permutations_of_set_aux.simps,
476       simp_all add: set_list_bind cong: SUP_cong)
479 text \<open>
480   The permutation lists contain no duplicates if the inputs contain no duplicates.
481   Therefore, these functions can easily be used when working with a representation of
482   sets by distinct lists.
483   The same approach should generalise to any kind of set implementation that supports
484   a monadic bind operation, and since the results are disjoint, merging should be cheap.
485 \<close>
486 lemma distinct_permutations_of_set_aux_list:
487   "distinct xs \<Longrightarrow> distinct (permutations_of_set_aux_list acc xs)"
488   by (induction acc xs rule: permutations_of_set_aux_list.induct)
489      (subst permutations_of_set_aux_list.simps,
490       auto intro!: distinct_list_bind simp: disjoint_family_on_def
491          permutations_of_set_aux_list_refine permutations_of_set_aux_altdef)
493 lemma distinct_permutations_of_set_list:
494     "distinct xs \<Longrightarrow> distinct (permutations_of_set_list xs)"
495   by (simp add: permutations_of_set_list_def distinct_permutations_of_set_aux_list)
497 lemma permutations_of_list:
498     "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))"
499   by (simp add: permutations_of_set_aux_correct [symmetric]
500         permutations_of_set_aux_list_refine permutations_of_set_list_def)
502 lemma permutations_of_list_code [code]:
503   "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))"
504   "permutations_of_set (List.coset xs) =
505      Code.abort (STR ''Permutation of set complement not supported'')
506        (\<lambda>_. permutations_of_set (List.coset xs))"
507   by (simp_all add: permutations_of_list)
509 value [code] "permutations_of_set (set ''abcd'')"
511 end