src/HOL/Library/Multiset.thy
 changeset 51548 757fa47af981 parent 51161 6ed12ae3b3e1 child 51599 1559e9266280
```     1.1 --- a/src/HOL/Library/Multiset.thy	Tue Mar 26 22:09:39 2013 +0100
1.2 +++ b/src/HOL/Library/Multiset.thy	Wed Mar 27 10:55:05 2013 +0100
1.3 @@ -702,7 +702,7 @@
1.4    then show ?thesis by simp
1.5  qed
1.6
1.7 -lemma fold_mset_fun_comm:
1.8 +lemma fold_mset_fun_left_comm:
1.9    "f x (fold f s M) = fold f (f x s) M"
1.10    by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
1.11
1.12 @@ -714,7 +714,7 @@
1.14    have "M + {#x#} + N = (M + N) + {#x#}"
1.18  qed
1.19
1.20  lemma fold_mset_fusion:
1.21 @@ -821,9 +821,7 @@
1.22  declare image_mset.identity [simp]
1.23
1.24
1.25 -subsection {* Alternative representations *}
1.26 -
1.27 -subsubsection {* Lists *}
1.28 +subsection {* Further conversions *}
1.29
1.30  primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
1.31    "multiset_of [] = {#}" |
1.32 @@ -950,6 +948,257 @@
1.33    ultimately show ?case by simp
1.34  qed
1.35
1.36 +lemma multiset_of_insort [simp]:
1.37 +  "multiset_of (insort x xs) = multiset_of xs + {#x#}"
1.38 +  by (induct xs) (simp_all add: ac_simps)
1.39 +
1.40 +definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
1.41 +where
1.42 +  "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
1.43 +
1.44 +interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
1.45 +where
1.46 +  "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
1.47 +proof -
1.48 +  interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
1.49 +  show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
1.50 +  from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
1.51 +qed
1.52 +
1.53 +context linorder
1.54 +begin
1.55 +
1.56 +definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
1.57 +where
1.58 +  "sorted_list_of_multiset M = fold insort [] M"
1.59 +
1.60 +lemma sorted_list_of_multiset_empty [simp]:
1.61 +  "sorted_list_of_multiset {#} = []"
1.62 +  by (simp add: sorted_list_of_multiset_def)
1.63 +
1.64 +lemma sorted_list_of_multiset_singleton [simp]:
1.65 +  "sorted_list_of_multiset {#x#} = [x]"
1.66 +proof -
1.67 +  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1.68 +  show ?thesis by (simp add: sorted_list_of_multiset_def)
1.69 +qed
1.70 +
1.71 +lemma sorted_list_of_multiset_insert [simp]:
1.72 +  "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
1.73 +proof -
1.74 +  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
1.75 +  show ?thesis by (simp add: sorted_list_of_multiset_def)
1.76 +qed
1.77 +
1.78 +end
1.79 +
1.80 +lemma multiset_of_sorted_list_of_multiset [simp]:
1.81 +  "multiset_of (sorted_list_of_multiset M) = M"
1.82 +  by (induct M) simp_all
1.83 +
1.84 +lemma sorted_list_of_multiset_multiset_of [simp]:
1.85 +  "sorted_list_of_multiset (multiset_of xs) = sort xs"
1.86 +  by (induct xs) simp_all
1.87 +
1.88 +lemma finite_set_of_multiset_of_set:
1.89 +  assumes "finite A"
1.90 +  shows "set_of (multiset_of_set A) = A"
1.91 +  using assms by (induct A) simp_all
1.92 +
1.93 +lemma infinite_set_of_multiset_of_set:
1.94 +  assumes "\<not> finite A"
1.95 +  shows "set_of (multiset_of_set A) = {}"
1.96 +  using assms by simp
1.97 +
1.98 +lemma set_sorted_list_of_multiset [simp]:
1.99 +  "set (sorted_list_of_multiset M) = set_of M"
1.100 +  by (induct M) (simp_all add: set_insort)
1.101 +
1.102 +lemma sorted_list_of_multiset_of_set [simp]:
1.103 +  "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
1.104 +  by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
1.105 +
1.106 +
1.107 +subsection {* Big operators *}
1.108 +
1.109 +no_notation times (infixl "*" 70)
1.110 +no_notation Groups.one ("1")
1.111 +
1.112 +locale comm_monoid_mset = comm_monoid
1.113 +begin
1.114 +
1.115 +definition F :: "'a multiset \<Rightarrow> 'a"
1.116 +where
1.117 +  eq_fold: "F M = Multiset.fold f 1 M"
1.118 +
1.119 +lemma empty [simp]:
1.120 +  "F {#} = 1"
1.121 +  by (simp add: eq_fold)
1.122 +
1.123 +lemma singleton [simp]:
1.124 +  "F {#x#} = x"
1.125 +proof -
1.126 +  interpret comp_fun_commute
1.127 +    by default (simp add: fun_eq_iff left_commute)
1.128 +  show ?thesis by (simp add: eq_fold)
1.129 +qed
1.130 +
1.131 +lemma union [simp]:
1.132 +  "F (M + N) = F M * F N"
1.133 +proof -
1.134 +  interpret comp_fun_commute f
1.135 +    by default (simp add: fun_eq_iff left_commute)
1.136 +  show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
1.137 +qed
1.138 +
1.139 +end
1.140 +
1.141 +notation times (infixl "*" 70)
1.142 +notation Groups.one ("1")
1.143 +
1.144 +definition (in comm_monoid_add) msetsum :: "'a multiset \<Rightarrow> 'a"
1.145 +where
1.146 +  "msetsum = comm_monoid_mset.F plus 0"
1.147 +
1.148 +definition (in comm_monoid_mult) msetprod :: "'a multiset \<Rightarrow> 'a"
1.149 +where
1.150 +  "msetprod = comm_monoid_mset.F times 1"
1.151 +
1.152 +sublocale comm_monoid_add < msetsum!: comm_monoid_mset plus 0
1.153 +where
1.154 +  "comm_monoid_mset.F plus 0 = msetsum"
1.155 +proof -
1.156 +  show "comm_monoid_mset plus 0" ..
1.157 +  from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
1.158 +qed
1.159 +
1.161 +begin
1.162 +
1.163 +lemma setsum_unfold_msetsum:
1.164 +  "setsum f A = msetsum (image_mset f (multiset_of_set A))"
1.165 +  by (cases "finite A") (induct A rule: finite_induct, simp_all)
1.166 +
1.167 +abbreviation msetsum_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1.168 +where
1.169 +  "msetsum_image f M \<equiv> msetsum (image_mset f M)"
1.170 +
1.171 +end
1.172 +
1.173 +syntax
1.174 +  "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1.175 +      ("(3SUM _:#_. _)" [0, 51, 10] 10)
1.176 +
1.177 +syntax (xsymbols)
1.178 +  "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1.179 +      ("(3\<Sum>_:#_. _)" [0, 51, 10] 10)
1.180 +
1.181 +syntax (HTML output)
1.182 +  "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
1.183 +      ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
1.184 +
1.185 +translations
1.186 +  "SUM i :# A. b" == "CONST msetsum_image (\<lambda>i. b) A"
1.187 +
1.188 +sublocale comm_monoid_mult < msetprod!: comm_monoid_mset times 1
1.189 +where
1.190 +  "comm_monoid_mset.F times 1 = msetprod"
1.191 +proof -
1.192 +  show "comm_monoid_mset times 1" ..
1.193 +  from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
1.194 +qed
1.195 +
1.196 +context comm_monoid_mult
1.197 +begin
1.198 +
1.199 +lemma msetprod_empty:
1.200 +  "msetprod {#} = 1"
1.201 +  by (fact msetprod.empty)
1.202 +
1.203 +lemma msetprod_singleton:
1.204 +  "msetprod {#x#} = x"
1.205 +  by (fact msetprod.singleton)
1.206 +
1.207 +lemma msetprod_Un:
1.208 +  "msetprod (A + B) = msetprod A * msetprod B"
1.209 +  by (fact msetprod.union)
1.210 +
1.211 +lemma setprod_unfold_msetprod:
1.212 +  "setprod f A = msetprod (image_mset f (multiset_of_set A))"
1.213 +  by (cases "finite A") (induct A rule: finite_induct, simp_all)
1.214 +
1.215 +lemma msetprod_multiplicity:
1.216 +  "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
1.217 +  by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
1.218 +
1.219 +abbreviation msetprod_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
1.220 +where
1.221 +  "msetprod_image f M \<equiv> msetprod (image_mset f M)"
1.222 +
1.223 +end
1.224 +
1.225 +syntax
1.226 +  "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1.227 +      ("(3PROD _:#_. _)" [0, 51, 10] 10)
1.228 +
1.229 +syntax (xsymbols)
1.230 +  "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1.231 +      ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1.232 +
1.233 +syntax (HTML output)
1.234 +  "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
1.235 +      ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
1.236 +
1.237 +translations
1.238 +  "PROD i :# A. b" == "CONST msetprod_image (\<lambda>i. b) A"
1.239 +
1.240 +lemma (in comm_semiring_1) dvd_msetprod:
1.241 +  assumes "x \<in># A"
1.242 +  shows "x dvd msetprod A"
1.243 +proof -
1.244 +  from assms have "A = (A - {#x#}) + {#x#}" by simp
1.245 +  then obtain B where "A = B + {#x#}" ..
1.246 +  then show ?thesis by simp
1.247 +qed
1.248 +
1.249 +
1.250 +subsection {* Cardinality *}
1.251 +
1.252 +definition mcard :: "'a multiset \<Rightarrow> nat"
1.253 +where
1.254 +  "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
1.255 +
1.256 +lemma mcard_empty [simp]:
1.257 +  "mcard {#} = 0"
1.258 +  by (simp add: mcard_def)
1.259 +
1.260 +lemma mcard_singleton [simp]:
1.261 +  "mcard {#a#} = Suc 0"
1.262 +  by (simp add: mcard_def)
1.263 +
1.264 +lemma mcard_plus [simp]:
1.265 +  "mcard (M + N) = mcard M + mcard N"
1.266 +  by (simp add: mcard_def)
1.267 +
1.268 +lemma mcard_empty_iff [simp]:
1.269 +  "mcard M = 0 \<longleftrightarrow> M = {#}"
1.270 +  by (induct M) simp_all
1.271 +
1.272 +lemma mcard_unfold_setsum:
1.273 +  "mcard M = setsum (count M) (set_of M)"
1.274 +proof (induct M)
1.275 +  case empty then show ?case by simp
1.276 +next
1.277 +  case (add M x) then show ?case
1.278 +    by (cases "x \<in> set_of M")
1.279 +      (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
1.280 +qed
1.281 +
1.282 +
1.283 +subsection {* Alternative representations *}
1.284 +
1.285 +subsubsection {* Lists *}
1.286 +
1.287  context linorder
1.288  begin
1.289
```