src/HOL/Library/Multiset.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 15869 3aca7f05cd12 child 17161 57c69627d71a permissions -rw-r--r--
Constant "If" is now local
1 (*  Title:      HOL/Library/Multiset.thy
2     ID:         \$Id\$
3     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
4 *)
6 header {* Multisets *}
8 theory Multiset
9 imports Accessible_Part
10 begin
12 subsection {* The type of multisets *}
14 typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
15 proof
16   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
17 qed
19 lemmas multiset_typedef [simp] =
20     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
21   and [simp] = Rep_multiset_inject [symmetric]
23 constdefs
24   Mempty :: "'a multiset"    ("{#}")
25   "{#} == Abs_multiset (\<lambda>a. 0)"
27   single :: "'a => 'a multiset"    ("{#_#}")
28   "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
30   count :: "'a multiset => 'a => nat"
31   "count == Rep_multiset"
33   MCollect :: "'a multiset => ('a => bool) => 'a multiset"
34   "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
36 syntax
37   "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
38   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
39 translations
40   "a :# M" == "0 < count M a"
41   "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
43 constdefs
44   set_of :: "'a multiset => 'a set"
45   "set_of M == {x. x :# M}"
47 instance multiset :: (type) "{plus, minus, zero}" ..
50   union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
51   diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
52   Zero_multiset_def [simp]: "0 == {#}"
53   size_def: "size M == setsum (count M) (set_of M)"
55 constdefs
56  multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70)
57  "multiset_inter A B \<equiv> A - (A - B)"
60 text {*
61  \medskip Preservation of the representing set @{term multiset}.
62 *}
64 lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
65 by (simp add: multiset_def)
67 lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
68 by (simp add: multiset_def)
70 lemma union_preserves_multiset [simp]:
71     "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
72   apply (unfold multiset_def, simp)
73   apply (drule finite_UnI, assumption)
74   apply (simp del: finite_Un add: Un_def)
75   done
77 lemma diff_preserves_multiset [simp]:
78     "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
79   apply (unfold multiset_def, simp)
80   apply (rule finite_subset)
81    prefer 2
82    apply assumption
83   apply auto
84   done
87 subsection {* Algebraic properties of multisets *}
89 subsubsection {* Union *}
91 theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
92 by (simp add: union_def Mempty_def)
94 theorem union_commute: "M + N = N + (M::'a multiset)"
97 theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
100 theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
101   apply (rule union_commute [THEN trans])
102   apply (rule union_assoc [THEN trans])
103   apply (rule union_commute [THEN arg_cong])
104   done
106 theorems union_ac = union_assoc union_commute union_lcomm
108 instance multiset :: (type) comm_monoid_add
109 proof
110   fix a b c :: "'a multiset"
111   show "(a + b) + c = a + (b + c)" by (rule union_assoc)
112   show "a + b = b + a" by (rule union_commute)
113   show "0 + a = a" by simp
114 qed
117 subsubsection {* Difference *}
119 theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
120 by (simp add: Mempty_def diff_def)
122 theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
123 by (simp add: union_def diff_def)
126 subsubsection {* Count of elements *}
128 theorem count_empty [simp]: "count {#} a = 0"
129 by (simp add: count_def Mempty_def)
131 theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
132 by (simp add: count_def single_def)
134 theorem count_union [simp]: "count (M + N) a = count M a + count N a"
135 by (simp add: count_def union_def)
137 theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
138 by (simp add: count_def diff_def)
141 subsubsection {* Set of elements *}
143 theorem set_of_empty [simp]: "set_of {#} = {}"
144 by (simp add: set_of_def)
146 theorem set_of_single [simp]: "set_of {#b#} = {b}"
147 by (simp add: set_of_def)
149 theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
150 by (auto simp add: set_of_def)
152 theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
153 by (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
155 theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
156 by (auto simp add: set_of_def)
159 subsubsection {* Size *}
161 theorem size_empty [simp]: "size {#} = 0"
162 by (simp add: size_def)
164 theorem size_single [simp]: "size {#b#} = 1"
165 by (simp add: size_def)
167 theorem finite_set_of [iff]: "finite (set_of M)"
168   apply (cut_tac x = M in Rep_multiset)
169   apply (simp add: multiset_def set_of_def count_def)
170   done
172 theorem setsum_count_Int:
173     "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
174   apply (erule finite_induct, simp)
175   apply (simp add: Int_insert_left set_of_def)
176   done
178 theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
179   apply (unfold size_def)
180   apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
181    prefer 2
182    apply (rule ext, simp)
183   apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
184   apply (subst Int_commute)
185   apply (simp (no_asm_simp) add: setsum_count_Int)
186   done
188 theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
189   apply (unfold size_def Mempty_def count_def, auto)
190   apply (simp add: set_of_def count_def expand_fun_eq)
191   done
193 theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
194   apply (unfold size_def)
195   apply (drule setsum_SucD, auto)
196   done
199 subsubsection {* Equality of multisets *}
201 theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
202 by (simp add: count_def expand_fun_eq)
204 theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
205 by (simp add: single_def Mempty_def expand_fun_eq)
207 theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
208 by (auto simp add: single_def expand_fun_eq)
210 theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
211 by (auto simp add: union_def Mempty_def expand_fun_eq)
213 theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
214 by (auto simp add: union_def Mempty_def expand_fun_eq)
216 theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
217 by (simp add: union_def expand_fun_eq)
219 theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
220 by (simp add: union_def expand_fun_eq)
222 theorem union_is_single:
223     "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
224   apply (simp add: Mempty_def single_def union_def add_is_1 expand_fun_eq)
225   apply blast
226   done
228 theorem single_is_union:
229      "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
230   apply (unfold Mempty_def single_def union_def)
232   apply (blast dest: sym)
233   done
236   "(M + {#a#} = N + {#b#}) =
237    (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
238   apply (unfold single_def union_def diff_def)
239   apply (simp (no_asm) add: expand_fun_eq)
240   apply (rule conjI, force, safe, simp_all)
241   apply (simp add: eq_sym_conv)
242   done
244 (*
245 val prems = Goal
246  "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
247 by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
248      measure_induct 1);
249 by (Clarify_tac 1)
250 by (resolve_tac prems 1)
251  by (assume_tac 1)
252 by (Clarify_tac 1)
253 by (subgoal_tac "finite G" 1)
254  by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
255 by (etac allE 1)
256 by (etac impE 1)
257  by (Blast_tac 2)
258 by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
259 no_qed();
260 val lemma = result();
262 val prems = Goal
263  "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
264 by (rtac (lemma RS mp) 1);
265 by (REPEAT(ares_tac prems 1));
266 qed "finite_psubset_induct";
268 Better: use wf_finite_psubset in WF_Rel
269 *)
271 declare Rep_multiset_inject [symmetric, simp del]
274 subsubsection {* Intersection *}
276 lemma multiset_inter_count:
277   "count (A #\<inter> B) x = min (count A x) (count B x)"
278   by (simp add:multiset_inter_def min_def)
280 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
281   by (simp add: multiset_eq_conv_count_eq multiset_inter_count
282                 min_max.below_inf.inf_commute)
284 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
285   by (simp add: multiset_eq_conv_count_eq multiset_inter_count
286                 min_max.below_inf.inf_assoc)
288 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
289   by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
291 lemmas multiset_inter_ac = multiset_inter_commute multiset_inter_assoc
292                            multiset_inter_left_commute
294 lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
295   apply (simp add:multiset_eq_conv_count_eq multiset_inter_count min_def
296               split:split_if_asm)
297   apply clarsimp
298   apply (erule_tac x="a" in allE)
299   apply auto
300   done
303 subsection {* Induction over multisets *}
305 lemma setsum_decr:
306   "finite F ==> (0::nat) < f a ==>
307     setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
308   apply (erule finite_induct, auto)
309   apply (drule_tac a = a in mk_disjoint_insert, auto)
310   done
312 lemma rep_multiset_induct_aux:
313   "P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
314     ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
315 proof -
316   case rule_context
317   note premises = this [unfolded multiset_def]
318   show ?thesis
319     apply (unfold multiset_def)
320     apply (induct_tac n, simp, clarify)
321      apply (subgoal_tac "f = (\<lambda>a.0)")
322       apply simp
323       apply (rule premises)
324      apply (rule ext, force, clarify)
325     apply (frule setsum_SucD, clarify)
326     apply (rename_tac a)
327     apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
328      prefer 2
329      apply (rule finite_subset)
330       prefer 2
331       apply assumption
332      apply simp
333      apply blast
334     apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
335      prefer 2
336      apply (rule ext)
337      apply (simp (no_asm_simp))
338      apply (erule ssubst, rule premises, blast)
339     apply (erule allE, erule impE, erule_tac  mp, blast)
340     apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
341     apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
342      prefer 2
343      apply blast
344     apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
345      prefer 2
346      apply blast
347     apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
348     done
349 qed
351 theorem rep_multiset_induct:
352   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
353     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
354   by (insert rep_multiset_induct_aux, blast)
356 theorem multiset_induct [induct type: multiset]:
357   "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
358 proof -
359   note defns = union_def single_def Mempty_def
360   assume prem1 [unfolded defns]: "P {#}"
361   assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
362   show ?thesis
363     apply (rule Rep_multiset_inverse [THEN subst])
364     apply (rule Rep_multiset [THEN rep_multiset_induct])
365      apply (rule prem1)
366     apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
367      prefer 2
368      apply (simp add: expand_fun_eq)
369     apply (erule ssubst)
370     apply (erule Abs_multiset_inverse [THEN subst])
371     apply (erule prem2 [simplified])
372     done
373 qed
376 lemma MCollect_preserves_multiset:
377     "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
378   apply (simp add: multiset_def)
379   apply (rule finite_subset, auto)
380   done
382 theorem count_MCollect [simp]:
383     "count {# x:M. P x #} a = (if P a then count M a else 0)"
384   by (simp add: count_def MCollect_def MCollect_preserves_multiset)
386 theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
387 by (auto simp add: set_of_def)
389 theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
390 by (subst multiset_eq_conv_count_eq, auto)
393       "(M + {#a#} = N + {#b#}) =
394        (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
397 declare multiset_typedef [simp del]
399 subsection {* Multiset orderings *}
401 subsubsection {* Well-foundedness *}
403 constdefs
404   mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
405   "mult1 r ==
406     {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
407       (\<forall>b. b :# K --> (b, a) \<in> r)}"
409   mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
410   "mult r == (mult1 r)\<^sup>+"
412 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
413   by (simp add: mult1_def)
415 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
416     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
417     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
418   (concl is "?case1 (mult1 r) \<or> ?case2")
419 proof (unfold mult1_def)
420   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
421   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
422   let ?case1 = "?case1 {(N, M). ?R N M}"
424   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
425   hence "\<exists>a' M0' K.
426       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
427   thus "?case1 \<or> ?case2"
428   proof (elim exE conjE)
429     fix a' M0' K
430     assume N: "N = M0' + K" and r: "?r K a'"
431     assume "M0 + {#a#} = M0' + {#a'#}"
432     hence "M0 = M0' \<and> a = a' \<or>
433         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
434       by (simp only: add_eq_conv_ex)
435     thus ?thesis
436     proof (elim disjE conjE exE)
437       assume "M0 = M0'" "a = a'"
438       with N r have "?r K a \<and> N = M0 + K" by simp
439       hence ?case2 .. thus ?thesis ..
440     next
441       fix K'
442       assume "M0' = K' + {#a#}"
443       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
445       assume "M0 = K' + {#a'#}"
446       with r have "?R (K' + K) M0" by blast
447       with n have ?case1 by simp thus ?thesis ..
448     qed
449   qed
450 qed
452 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
453 proof
454   let ?R = "mult1 r"
455   let ?W = "acc ?R"
456   {
457     fix M M0 a
458     assume M0: "M0 \<in> ?W"
459       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
460       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
461     have "M0 + {#a#} \<in> ?W"
462     proof (rule accI [of "M0 + {#a#}"])
463       fix N
464       assume "(N, M0 + {#a#}) \<in> ?R"
465       hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
466           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
467         by (rule less_add)
468       thus "N \<in> ?W"
469       proof (elim exE disjE conjE)
470         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
471         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
472         hence "M + {#a#} \<in> ?W" ..
473         thus "N \<in> ?W" by (simp only: N)
474       next
475         fix K
476         assume N: "N = M0 + K"
477         assume "\<forall>b. b :# K --> (b, a) \<in> r"
478         have "?this --> M0 + K \<in> ?W" (is "?P K")
479         proof (induct K)
480           from M0 have "M0 + {#} \<in> ?W" by simp
481           thus "?P {#}" ..
483           fix K x assume hyp: "?P K"
484           show "?P (K + {#x#})"
485           proof
486             assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
487             hence "(x, a) \<in> r" by simp
488             with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
490             from a hyp have "M0 + K \<in> ?W" by simp
491             with b have "(M0 + K) + {#x#} \<in> ?W" ..
492             thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
493           qed
494         qed
495         hence "M0 + K \<in> ?W" ..
496         thus "N \<in> ?W" by (simp only: N)
497       qed
498     qed
499   } note tedious_reasoning = this
501   assume wf: "wf r"
502   fix M
503   show "M \<in> ?W"
504   proof (induct M)
505     show "{#} \<in> ?W"
506     proof (rule accI)
507       fix b assume "(b, {#}) \<in> ?R"
508       with not_less_empty show "b \<in> ?W" by contradiction
509     qed
511     fix M a assume "M \<in> ?W"
512     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
513     proof induct
514       fix a
515       assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
516       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
517       proof
518         fix M assume "M \<in> ?W"
519         thus "M + {#a#} \<in> ?W"
520           by (rule acc_induct) (rule tedious_reasoning)
521       qed
522     qed
523     thus "M + {#a#} \<in> ?W" ..
524   qed
525 qed
527 theorem wf_mult1: "wf r ==> wf (mult1 r)"
528   by (rule acc_wfI, rule all_accessible)
530 theorem wf_mult: "wf r ==> wf (mult r)"
531   by (unfold mult_def, rule wf_trancl, rule wf_mult1)
534 subsubsection {* Closure-free presentation *}
536 (*Badly needed: a linear arithmetic procedure for multisets*)
538 lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
539 by (simp add: multiset_eq_conv_count_eq)
541 text {* One direction. *}
543 lemma mult_implies_one_step:
544   "trans r ==> (M, N) \<in> mult r ==>
545     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
546     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
547   apply (unfold mult_def mult1_def set_of_def)
548   apply (erule converse_trancl_induct, clarify)
549    apply (rule_tac x = M0 in exI, simp, clarify)
550   apply (case_tac "a :# K")
551    apply (rule_tac x = I in exI)
552    apply (simp (no_asm))
553    apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
554    apply (simp (no_asm_simp) add: union_assoc [symmetric])
555    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
556    apply (simp add: diff_union_single_conv)
557    apply (simp (no_asm_use) add: trans_def)
558    apply blast
559   apply (subgoal_tac "a :# I")
560    apply (rule_tac x = "I - {#a#}" in exI)
561    apply (rule_tac x = "J + {#a#}" in exI)
562    apply (rule_tac x = "K + Ka" in exI)
563    apply (rule conjI)
564     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
565    apply (rule conjI)
566     apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
567     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
568    apply (simp (no_asm_use) add: trans_def)
569    apply blast
570   apply (subgoal_tac "a :# (M0 + {#a#})")
571    apply simp
572   apply (simp (no_asm))
573   done
575 lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
576 by (simp add: multiset_eq_conv_count_eq)
578 lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
579   apply (erule size_eq_Suc_imp_elem [THEN exE])
580   apply (drule elem_imp_eq_diff_union, auto)
581   done
583 lemma one_step_implies_mult_aux:
584   "trans r ==>
585     \<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))
586       --> (I + K, I + J) \<in> mult r"
587   apply (induct_tac n, auto)
588   apply (frule size_eq_Suc_imp_eq_union, clarify)
589   apply (rename_tac "J'", simp)
590   apply (erule notE, auto)
591   apply (case_tac "J' = {#}")
592    apply (simp add: mult_def)
593    apply (rule r_into_trancl)
594    apply (simp add: mult1_def set_of_def, blast)
595   txt {* Now we know @{term "J' \<noteq> {#}"}. *}
596   apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
597   apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
598   apply (erule ssubst)
599   apply (simp add: Ball_def, auto)
600   apply (subgoal_tac
601     "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
602       (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
603    prefer 2
604    apply force
605   apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
606   apply (erule trancl_trans)
607   apply (rule r_into_trancl)
608   apply (simp add: mult1_def set_of_def)
609   apply (rule_tac x = a in exI)
610   apply (rule_tac x = "I + J'" in exI)
611   apply (simp add: union_ac)
612   done
614 theorem one_step_implies_mult:
615   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
616     ==> (I + K, I + J) \<in> mult r"
617   apply (insert one_step_implies_mult_aux, blast)
618   done
621 subsubsection {* Partial-order properties *}
623 instance multiset :: (type) ord ..
626   less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
627   le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
629 lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
630   apply (unfold trans_def)
631   apply (blast intro: order_less_trans)
632   done
634 text {*
635  \medskip Irreflexivity.
636 *}
638 lemma mult_irrefl_aux:
639     "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
640   apply (erule finite_induct)
641    apply (auto intro: order_less_trans)
642   done
644 theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
645   apply (unfold less_multiset_def, auto)
646   apply (drule trans_base_order [THEN mult_implies_one_step], auto)
647   apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
648   apply (simp add: set_of_eq_empty_iff)
649   done
651 lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
652 by (insert mult_less_not_refl, fast)
655 text {* Transitivity. *}
657 theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
658   apply (unfold less_multiset_def mult_def)
659   apply (blast intro: trancl_trans)
660   done
662 text {* Asymmetry. *}
664 theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
665   apply auto
666   apply (rule mult_less_not_refl [THEN notE])
667   apply (erule mult_less_trans, assumption)
668   done
670 theorem mult_less_asym:
671     "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
672   by (insert mult_less_not_sym, blast)
674 theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
675 by (unfold le_multiset_def, auto)
677 text {* Anti-symmetry. *}
679 theorem mult_le_antisym:
680     "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
681   apply (unfold le_multiset_def)
682   apply (blast dest: mult_less_not_sym)
683   done
685 text {* Transitivity. *}
687 theorem mult_le_trans:
688     "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
689   apply (unfold le_multiset_def)
690   apply (blast intro: mult_less_trans)
691   done
693 theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
694 by (unfold le_multiset_def, auto)
696 text {* Partial order. *}
698 instance multiset :: (order) order
699   apply intro_classes
700      apply (rule mult_le_refl)
701     apply (erule mult_le_trans, assumption)
702    apply (erule mult_le_antisym, assumption)
703   apply (rule mult_less_le)
704   done
707 subsubsection {* Monotonicity of multiset union *}
709 theorem mult1_union:
710     "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
711   apply (unfold mult1_def, auto)
712   apply (rule_tac x = a in exI)
713   apply (rule_tac x = "C + M0" in exI)
714   apply (simp add: union_assoc)
715   done
717 lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
718   apply (unfold less_multiset_def mult_def)
719   apply (erule trancl_induct)
720    apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
721   apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
722   done
724 lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
725   apply (subst union_commute [of B C])
726   apply (subst union_commute [of D C])
727   apply (erule union_less_mono2)
728   done
730 theorem union_less_mono:
731     "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
732   apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
733   done
735 theorem union_le_mono:
736     "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
737   apply (unfold le_multiset_def)
738   apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
739   done
741 theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
742   apply (unfold le_multiset_def less_multiset_def)
743   apply (case_tac "M = {#}")
744    prefer 2
745    apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
746     prefer 2
747     apply (rule one_step_implies_mult)
748       apply (simp only: trans_def, auto)
749   done
751 theorem union_upper1: "A <= A + (B::'a::order multiset)"
752 proof -
753   have "A + {#} <= A + B" by (blast intro: union_le_mono)
754   thus ?thesis by simp
755 qed
757 theorem union_upper2: "B <= A + (B::'a::order multiset)"
758 by (subst union_commute, rule union_upper1)
761 subsection {* Link with lists *}
763 consts
764   multiset_of :: "'a list \<Rightarrow> 'a multiset"
765 primrec
766   "multiset_of [] = {#}"
767   "multiset_of (a # x) = multiset_of x + {# a #}"
769 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
770   by (induct_tac x, auto)
772 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
773   by (induct_tac x, auto)
775 lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
776   by (induct_tac x, auto)
778 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
779   by (induct xs) auto
781 lemma multiset_of_append[simp]:
782   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
783   by (rule_tac x=ys in spec, induct_tac xs, auto simp: union_ac)
785 lemma surj_multiset_of: "surj multiset_of"
786   apply (unfold surj_def, rule allI)
787   apply (rule_tac M=y in multiset_induct, auto)
788   apply (rule_tac x = "x # xa" in exI, auto)
789   done
791 lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}"
792   by (induct_tac x, auto)
794 lemma distinct_count_atmost_1:
795    "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
796    apply ( induct_tac x, simp, rule iffI, simp_all)
797    apply (rule conjI)
798    apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
799    apply (erule_tac x=a in allE, simp, clarify)
800    apply (erule_tac x=aa in allE, simp)
801    done
803 lemma multiset_of_eq_setD:
804   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
805   by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
807 lemma set_eq_iff_multiset_of_eq_distinct:
808   "\<lbrakk>distinct x; distinct y\<rbrakk>
809    \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
810   by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
812 lemma set_eq_iff_multiset_of_remdups_eq:
813    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
814   apply (rule iffI)
815   apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
816   apply (drule distinct_remdups[THEN distinct_remdups
817                       [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]])
818   apply simp
819   done
821 lemma multiset_of_compl_union[simp]:
822  "multiset_of [x\<in>xs. P x] + multiset_of [x\<in>xs. \<not>P x] = multiset_of xs"
823   by (induct xs) (auto simp: union_ac)
825 lemma count_filter:
826   "count (multiset_of xs) x = length [y \<in> xs. y = x]"
827   by (induct xs, auto)
830 subsection {* Pointwise ordering induced by count *}
832 consts
833   mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool"
835 syntax
836   "_mset_le" :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"   ("_ \<le># _"  [50,51] 50)
837 translations
838   "x \<le># y" == "mset_le x y"
840 defs
841   mset_le_def:   "xs \<le># ys  == (! a. count xs a \<le> count ys a)"
843 lemma mset_le_refl[simp]: "xs \<le># xs"
844   by (unfold mset_le_def, auto)
846 lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs"
847   by (unfold mset_le_def, fast intro: order_trans)
849 lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys"
850   apply (unfold mset_le_def)
851   apply (rule multiset_eq_conv_count_eq[THEN iffD2])
852   apply (blast intro: order_antisym)
853   done
855 lemma mset_le_exists_conv:
856   "(xs \<le># ys) = (? zs. ys = xs + zs)"
857   apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI)
858   apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
859   done
861 lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)"
862   by (unfold mset_le_def, auto)
864 lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)"
865   by (unfold mset_le_def, auto)
867 lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws"
868   apply (unfold mset_le_def, auto)
869   apply (erule_tac x=a in allE)+
870   apply auto
871   done
873 lemma mset_le_add_left[simp]: "xs \<le># xs + ys"
874   by (unfold mset_le_def, auto)
876 lemma mset_le_add_right[simp]: "ys \<le># xs + ys"
877   by (unfold mset_le_def, auto)
879 lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x"
880   by (induct_tac x, auto, rule mset_le_trans, auto)
882 end