src/HOL/Library/Multiset.thy
 changeset 60607 d440af2e584f parent 60606 e5cb9271e339 child 60608 c5cbd90bd94e
```     1.1 --- a/src/HOL/Library/Multiset.thy	Mon Jun 29 15:36:29 2015 +0200
1.2 +++ b/src/HOL/Library/Multiset.thy	Mon Jun 29 15:41:16 2015 +0200
1.3 @@ -305,7 +305,7 @@
1.4  lemma mset_le_add_right [simp]: "B \<le># (A::'a multiset) + B"
1.5    unfolding subseteq_mset_def by auto
1.6
1.7 -lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
1.8 +lemma mset_le_single: "a \<in># B \<Longrightarrow> {#a#} \<le># B"
1.10
1.11  lemma multiset_diff_union_assoc:
1.12 @@ -510,7 +510,7 @@
1.13  subsubsection \<open>Set of elements\<close>
1.14
1.15  definition set_mset :: "'a multiset \<Rightarrow> 'a set"
1.16 -  where "set_mset M = {x. x :# M}"
1.17 +  where "set_mset M = {x. x \<in># M}"
1.18
1.19  lemma set_mset_empty [simp]: "set_mset {#} = {}"
1.21 @@ -524,16 +524,16 @@
1.22  lemma set_mset_eq_empty_iff [simp]: "(set_mset M = {}) = (M = {#})"
1.23  by (auto simp add: set_mset_def multiset_eq_iff)
1.24
1.25 -lemma mem_set_mset_iff [simp]: "(x \<in> set_mset M) = (x :# M)"
1.26 +lemma mem_set_mset_iff [simp]: "(x \<in> set_mset M) = (x \<in># M)"
1.27  by (auto simp add: set_mset_def)
1.28
1.29 -lemma set_mset_filter [simp]: "set_mset {# x:#M. P x #} = set_mset M \<inter> {x. P x}"
1.30 +lemma set_mset_filter [simp]: "set_mset {# x\<in>#M. P x #} = set_mset M \<inter> {x. P x}"
1.31  by (auto simp add: set_mset_def)
1.32
1.33  lemma finite_set_mset [iff]: "finite (set_mset M)"
1.34    using count [of M] by (simp add: multiset_def set_mset_def)
1.35
1.36 -lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
1.37 +lemma finite_Collect_mem [iff]: "finite {x. x \<in># M}"
1.38    unfolding set_mset_def[symmetric] by simp
1.39
1.40  lemma set_mset_mono: "A \<le># B \<Longrightarrow> set_mset A \<subseteq> set_mset B"
1.41 @@ -605,7 +605,7 @@
1.42  lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
1.43  by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
1.44
1.45 -lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a :# M"
1.46 +lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a \<in># M"
1.48  apply (drule setsum_SucD)
1.49  apply auto
1.50 @@ -665,7 +665,7 @@
1.51  lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
1.52  by (cases "B = {#}") (auto dest: multi_member_split)
1.53
1.54 -lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
1.55 +lemma multiset_partition: "M = {# x\<in>#M. P x #} + {# x\<in>#M. \<not> P x #}"
1.56  apply (subst multiset_eq_iff)
1.57  apply auto
1.58  done
1.59 @@ -816,9 +816,9 @@
1.60  subsection \<open>Image\<close>
1.61
1.62  definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
1.63 -  "image_mset f = fold_mset (plus o single o f) {#}"
1.64 -
1.65 -lemma comp_fun_commute_mset_image: "comp_fun_commute (plus o single o f)"
1.66 +  "image_mset f = fold_mset (plus \<circ> single \<circ> f) {#}"
1.67 +
1.68 +lemma comp_fun_commute_mset_image: "comp_fun_commute (plus \<circ> single \<circ> f)"
1.69  proof
1.70  qed (simp add: ac_simps fun_eq_iff)
1.71
1.72 @@ -827,14 +827,14 @@
1.73
1.74  lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
1.75  proof -
1.76 -  interpret comp_fun_commute "plus o single o f"
1.77 +  interpret comp_fun_commute "plus \<circ> single \<circ> f"
1.78      by (fact comp_fun_commute_mset_image)
1.79    show ?thesis by (simp add: image_mset_def)
1.80  qed
1.81
1.82  lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
1.83  proof -
1.84 -  interpret comp_fun_commute "plus o single o f"
1.85 +  interpret comp_fun_commute "plus \<circ> single \<circ> f"
1.86      by (fact comp_fun_commute_mset_image)
1.87    show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
1.88  qed
1.89 @@ -876,10 +876,10 @@
1.90    "{#e | x\<in>#M. P#}" \<rightharpoonup> "{#e. x \<in># {# x\<in>#M. P#}#}"
1.91
1.92  text \<open>
1.93 -  This allows to write not just filters like @{term "{#x:#M. x<c#}"}
1.94 -  but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
1.95 -  "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
1.96 -  @{term "{#x+x|x:#M. x<c#}"}.
1.97 +  This allows to write not just filters like @{term "{#x\<in>#M. x<c#}"}
1.98 +  but also images like @{term "{#x+x. x\<in>#M #}"} and @{term [source]
1.99 +  "{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as
1.100 +  @{term "{#x+x|x\<in>#M. x<c#}"}.
1.101  \<close>
1.102
1.103  lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M"
1.104 @@ -939,7 +939,7 @@
1.105  lemma set_mset_mset[simp]: "set_mset (mset x) = set x"
1.106  by (induct x) auto
1.107
1.108 -lemma mem_set_multiset_eq: "x \<in> set xs = (x :# mset xs)"
1.109 +lemma mem_set_multiset_eq: "x \<in> set xs = (x \<in># mset xs)"
1.110  by (induct xs) auto
1.111
1.112  lemma size_mset [simp]: "size (mset xs) = length xs"
1.113 @@ -948,7 +948,7 @@
1.114  lemma mset_append [simp]: "mset (xs @ ys) = mset xs + mset ys"
1.115    by (induct xs arbitrary: ys) (auto simp: ac_simps)
1.116
1.117 -lemma mset_filter: "mset (filter P xs) = {#x :# mset xs. P x #}"
1.118 +lemma mset_filter: "mset (filter P xs) = {#x \<in># mset xs. P x #}"
1.119    by (induct xs) simp_all
1.120
1.121  lemma mset_rev [simp]:
1.122 @@ -997,7 +997,7 @@
1.123  lemma mset_compl_union [simp]: "mset [x\<leftarrow>xs. P x] + mset [x\<leftarrow>xs. \<not>P x] = mset xs"
1.124    by (induct xs) (auto simp: ac_simps)
1.125
1.126 -lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) :# mset ls"
1.127 +lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls"
1.128  apply (induct ls arbitrary: i)
1.129   apply simp
1.130  apply (case_tac i)
1.131 @@ -1501,7 +1501,7 @@
1.132
1.133  definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
1.134    "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1.135 -      (\<forall>b. b :# K \<longrightarrow> (b, a) \<in> r)}"
1.136 +      (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
1.137
1.138  definition mult :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
1.139    "mult r = (mult1 r)\<^sup>+"
1.140 @@ -1511,10 +1511,10 @@
1.141
1.142  lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r \<Longrightarrow>
1.143      (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1.144 -    (\<exists>K. (\<forall>b. b :# K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
1.145 +    (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
1.146    (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
1.147  proof (unfold mult1_def)
1.148 -  let ?r = "\<lambda>K a. \<forall>b. b :# K \<longrightarrow> (b, a) \<in> r"
1.149 +  let ?r = "\<lambda>K a. \<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r"
1.150    let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1.151    let ?case1 = "?case1 {(N, M). ?R N M}"
1.152
1.153 @@ -1556,7 +1556,7 @@
1.154        fix N
1.155        assume "(N, M0 + {#a#}) \<in> ?R"
1.156        then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
1.157 -          (\<exists>K. (\<forall>b. b :# K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K))"
1.158 +          (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K))"
1.160        then show "N \<in> ?W"
1.161        proof (elim exE disjE conjE)
1.162 @@ -1567,7 +1567,7 @@
1.163        next
1.164          fix K
1.165          assume N: "N = M0 + K"
1.166 -        assume "\<forall>b. b :# K \<longrightarrow> (b, a) \<in> r"
1.167 +        assume "\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r"
1.168          then have "M0 + K \<in> ?W"
1.169          proof (induct K)
1.170            case empty
1.171 @@ -1629,7 +1629,7 @@
1.172  apply (unfold mult_def mult1_def set_mset_def)
1.173  apply (erule converse_trancl_induct, clarify)
1.174   apply (rule_tac x = M0 in exI, simp, clarify)
1.175 -apply (case_tac "a :# K")
1.176 +apply (case_tac "a \<in># K")
1.177   apply (rule_tac x = I in exI)
1.178   apply (simp (no_asm))
1.179   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1.180 @@ -1638,7 +1638,7 @@
1.182   apply (simp (no_asm_use) add: trans_def)
1.183   apply blast
1.184 -apply (subgoal_tac "a :# I")
1.185 +apply (subgoal_tac "a \<in># I")
1.186   apply (rule_tac x = "I - {#a#}" in exI)
1.187   apply (rule_tac x = "J + {#a#}" in exI)
1.188   apply (rule_tac x = "K + Ka" in exI)
1.189 @@ -1649,7 +1649,7 @@
1.190    apply (simp add: multiset_eq_iff split: nat_diff_split)
1.191   apply (simp (no_asm_use) add: trans_def)
1.192   apply blast
1.193 -apply (subgoal_tac "a :# (M0 + {#a#})")
1.194 +apply (subgoal_tac "a \<in># (M0 + {#a#})")
1.195   apply simp
1.196  apply (simp (no_asm))
1.197  done
1.198 @@ -1672,8 +1672,8 @@
1.199  apply (erule ssubst)
1.200  apply (simp add: Ball_def, auto)
1.201  apply (subgoal_tac
1.202 -  "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
1.203 -    (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
1.204 +  "((I + {# x \<in># K. (x, a) \<in> r #}) + {# x \<in># K. (x, a) \<notin> r #},
1.205 +    (I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r")
1.206   prefer 2
1.207   apply force