1.1 --- a/src/HOL/Library/RBT.thy Tue Jul 31 13:55:39 2012 +0200
1.2 +++ b/src/HOL/Library/RBT.thy Tue Jul 31 13:55:39 2012 +0200
1.3 @@ -1,9 +1,10 @@
1.4 -(* Author: Florian Haftmann, TU Muenchen *)
1.5 +(* Title: HOL/Library/RBT.thy
1.6 + Author: Lukas Bulwahn and Ondrej Kuncar
1.7 +*)
1.8
1.9 -header {* Abstract type of Red-Black Trees *}
1.10 +header {* Abstract type of RBT trees *}
1.11
1.12 -(*<*)
1.13 -theory RBT
1.14 +theory RBT
1.15 imports Main RBT_Impl
1.16 begin
1.17
1.18 @@ -11,8 +12,9 @@
1.19
1.20 typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}"
1.21 morphisms impl_of RBT
1.22 -proof
1.23 - show "RBT_Impl.Empty \<in> {t. is_rbt t}" by simp
1.24 +proof -
1.25 + have "RBT_Impl.Empty \<in> ?rbt" by simp
1.26 + then show ?thesis ..
1.27 qed
1.28
1.29 lemma rbt_eq_iff:
1.30 @@ -31,63 +33,45 @@
1.31 "RBT (impl_of t) = t"
1.32 by (simp add: impl_of_inverse)
1.33
1.34 -
1.35 subsection {* Primitive operations *}
1.36
1.37 -definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" where
1.38 - [code]: "lookup t = rbt_lookup (impl_of t)"
1.39 +setup_lifting type_definition_rbt
1.40
1.41 -definition empty :: "('a\<Colon>linorder, 'b) rbt" where
1.42 - "empty = RBT RBT_Impl.Empty"
1.43 +lift_definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" is "rbt_lookup"
1.44 +by simp
1.45
1.46 -lemma impl_of_empty [code abstract]:
1.47 - "impl_of empty = RBT_Impl.Empty"
1.48 - by (simp add: empty_def RBT_inverse)
1.49 +lift_definition empty :: "('a\<Colon>linorder, 'b) rbt" is RBT_Impl.Empty
1.50 +by (simp add: empty_def)
1.51
1.52 -definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
1.53 - "insert k v t = RBT (rbt_insert k v (impl_of t))"
1.54 +lift_definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_insert"
1.55 +by simp
1.56
1.57 -lemma impl_of_insert [code abstract]:
1.58 - "impl_of (insert k v t) = rbt_insert k v (impl_of t)"
1.59 - by (simp add: insert_def RBT_inverse)
1.60 +lift_definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_delete"
1.61 +by simp
1.62
1.63 -definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
1.64 - "delete k t = RBT (rbt_delete k (impl_of t))"
1.65 +lift_definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" is RBT_Impl.entries
1.66 +by simp
1.67
1.68 -lemma impl_of_delete [code abstract]:
1.69 - "impl_of (delete k t) = rbt_delete k (impl_of t)"
1.70 - by (simp add: delete_def RBT_inverse)
1.71 +lift_definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" is RBT_Impl.keys
1.72 +by simp
1.73
1.74 -definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" where
1.75 - [code]: "entries t = RBT_Impl.entries (impl_of t)"
1.76 +lift_definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" is "rbt_bulkload"
1.77 +by simp
1.78
1.79 -definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" where
1.80 - [code]: "keys t = RBT_Impl.keys (impl_of t)"
1.81 +lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is rbt_map_entry
1.82 +by simp
1.83
1.84 -definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where
1.85 - "bulkload xs = RBT (rbt_bulkload xs)"
1.86 +lift_definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is RBT_Impl.map
1.87 +by simp
1.88
1.89 -lemma impl_of_bulkload [code abstract]:
1.90 - "impl_of (bulkload xs) = rbt_bulkload xs"
1.91 - by (simp add: bulkload_def RBT_inverse)
1.92 +lift_definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" is RBT_Impl.fold
1.93 +by simp
1.94
1.95 -definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
1.96 - "map_entry k f t = RBT (rbt_map_entry k f (impl_of t))"
1.97 +lift_definition union :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_union"
1.98 +by (simp add: rbt_union_is_rbt)
1.99
1.100 -lemma impl_of_map_entry [code abstract]:
1.101 - "impl_of (map_entry k f t) = rbt_map_entry k f (impl_of t)"
1.102 - by (simp add: map_entry_def RBT_inverse)
1.103 -
1.104 -definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
1.105 - "map f t = RBT (RBT_Impl.map f (impl_of t))"
1.106 -
1.107 -lemma impl_of_map [code abstract]:
1.108 - "impl_of (map f t) = RBT_Impl.map f (impl_of t)"
1.109 - by (simp add: map_def RBT_inverse)
1.110 -
1.111 -definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
1.112 - [code]: "fold f t = RBT_Impl.fold f (impl_of t)"
1.113 -
1.114 +lift_definition foldi :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a :: linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"
1.115 + is RBT_Impl.foldi by simp
1.116
1.117 subsection {* Derived operations *}
1.118
1.119 @@ -103,15 +87,15 @@
1.120
1.121 lemma lookup_impl_of:
1.122 "rbt_lookup (impl_of t) = lookup t"
1.123 - by (simp add: lookup_def)
1.124 + by transfer (rule refl)
1.125
1.126 lemma entries_impl_of:
1.127 "RBT_Impl.entries (impl_of t) = entries t"
1.128 - by (simp add: entries_def)
1.129 + by transfer (rule refl)
1.130
1.131 lemma keys_impl_of:
1.132 "RBT_Impl.keys (impl_of t) = keys t"
1.133 - by (simp add: keys_def)
1.134 + by transfer (rule refl)
1.135
1.136 lemma lookup_empty [simp]:
1.137 "lookup empty = Map.empty"
1.138 @@ -119,39 +103,43 @@
1.139
1.140 lemma lookup_insert [simp]:
1.141 "lookup (insert k v t) = (lookup t)(k \<mapsto> v)"
1.142 - by (simp add: insert_def lookup_RBT rbt_lookup_rbt_insert lookup_impl_of)
1.143 + by transfer (rule rbt_lookup_rbt_insert)
1.144
1.145 lemma lookup_delete [simp]:
1.146 "lookup (delete k t) = (lookup t)(k := None)"
1.147 - by (simp add: delete_def lookup_RBT rbt_lookup_rbt_delete lookup_impl_of restrict_complement_singleton_eq)
1.148 + by transfer (simp add: rbt_lookup_rbt_delete restrict_complement_singleton_eq)
1.149
1.150 lemma map_of_entries [simp]:
1.151 "map_of (entries t) = lookup t"
1.152 - by (simp add: entries_def map_of_entries lookup_impl_of)
1.153 + by transfer (simp add: map_of_entries)
1.154
1.155 lemma entries_lookup:
1.156 "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
1.157 - by (simp add: entries_def lookup_def entries_rbt_lookup)
1.158 + by transfer (simp add: entries_rbt_lookup)
1.159
1.160 lemma lookup_bulkload [simp]:
1.161 "lookup (bulkload xs) = map_of xs"
1.162 - by (simp add: bulkload_def lookup_RBT rbt_lookup_rbt_bulkload)
1.163 + by transfer (rule rbt_lookup_rbt_bulkload)
1.164
1.165 lemma lookup_map_entry [simp]:
1.166 "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
1.167 - by (simp add: map_entry_def lookup_RBT rbt_lookup_rbt_map_entry lookup_impl_of)
1.168 + by transfer (rule rbt_lookup_rbt_map_entry)
1.169
1.170 lemma lookup_map [simp]:
1.171 "lookup (map f t) k = Option.map (f k) (lookup t k)"
1.172 - by (simp add: map_def lookup_RBT rbt_lookup_map lookup_impl_of)
1.173 + by transfer (rule rbt_lookup_map)
1.174
1.175 lemma fold_fold:
1.176 "fold f t = List.fold (prod_case f) (entries t)"
1.177 - by (simp add: fold_def fun_eq_iff RBT_Impl.fold_def entries_impl_of)
1.178 + by transfer (rule RBT_Impl.fold_def)
1.179 +
1.180 +lemma impl_of_empty:
1.181 + "impl_of empty = RBT_Impl.Empty"
1.182 + by transfer (rule refl)
1.183
1.184 lemma is_empty_empty [simp]:
1.185 "is_empty t \<longleftrightarrow> t = empty"
1.186 - by (simp add: rbt_eq_iff is_empty_def impl_of_empty split: rbt.split)
1.187 + unfolding is_empty_def by transfer (simp split: rbt.split)
1.188
1.189 lemma RBT_lookup_empty [simp]: (*FIXME*)
1.190 "rbt_lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
1.191 @@ -159,15 +147,41 @@
1.192
1.193 lemma lookup_empty_empty [simp]:
1.194 "lookup t = Map.empty \<longleftrightarrow> t = empty"
1.195 - by (cases t) (simp add: empty_def lookup_def RBT_inject RBT_inverse)
1.196 + by transfer (rule RBT_lookup_empty)
1.197
1.198 lemma sorted_keys [iff]:
1.199 "sorted (keys t)"
1.200 - by (simp add: keys_def RBT_Impl.keys_def rbt_sorted_entries)
1.201 + by transfer (simp add: RBT_Impl.keys_def rbt_sorted_entries)
1.202
1.203 lemma distinct_keys [iff]:
1.204 "distinct (keys t)"
1.205 - by (simp add: keys_def RBT_Impl.keys_def distinct_entries)
1.206 + by transfer (simp add: RBT_Impl.keys_def distinct_entries)
1.207 +
1.208 +lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
1.209 + by transfer simp
1.210 +
1.211 +lemma lookup_union: "lookup (union s t) = lookup s ++ lookup t"
1.212 + by transfer (simp add: rbt_lookup_rbt_union)
1.213 +
1.214 +lemma lookup_in_tree: "(lookup t k = Some v) = ((k, v) \<in> set (entries t))"
1.215 + by transfer (simp add: rbt_lookup_in_tree)
1.216 +
1.217 +lemma keys_entries: "(k \<in> set (keys t)) = (\<exists>v. (k, v) \<in> set (entries t))"
1.218 + by transfer (simp add: keys_entries)
1.219 +
1.220 +lemma fold_def_alt:
1.221 + "fold f t = List.fold (prod_case f) (entries t)"
1.222 + by transfer (auto simp: RBT_Impl.fold_def)
1.223 +
1.224 +lemma distinct_entries: "distinct (List.map fst (entries t))"
1.225 + by transfer (simp add: distinct_entries)
1.226 +
1.227 +lemma non_empty_keys: "t \<noteq> empty \<Longrightarrow> keys t \<noteq> []"
1.228 + by transfer (simp add: non_empty_rbt_keys)
1.229 +
1.230 +lemma keys_def_alt:
1.231 + "keys t = List.map fst (entries t)"
1.232 + by transfer (simp add: RBT_Impl.keys_def)
1.233
1.234 subsection {* Quickcheck generators *}
1.235
2.1 --- a/src/HOL/Quotient_Examples/Lift_RBT.thy Tue Jul 31 13:55:39 2012 +0200
2.2 +++ b/src/HOL/Quotient_Examples/Lift_RBT.thy Tue Jul 31 13:55:39 2012 +0200
2.3 @@ -8,183 +8,6 @@
2.4 imports Main "~~/src/HOL/Library/RBT_Impl"
2.5 begin
2.6
2.7 -(* TODO: Replace the ancient Library/RBT theory by this example of the lifting and transfer mechanism. *)
2.8 -
2.9 -subsection {* Type definition *}
2.10 -
2.11 -typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}"
2.12 - morphisms impl_of RBT
2.13 -proof -
2.14 - have "RBT_Impl.Empty \<in> ?rbt" by simp
2.15 - then show ?thesis ..
2.16 -qed
2.17 -
2.18 -lemma rbt_eq_iff:
2.19 - "t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"
2.20 - by (simp add: impl_of_inject)
2.21 -
2.22 -lemma rbt_eqI:
2.23 - "impl_of t1 = impl_of t2 \<Longrightarrow> t1 = t2"
2.24 - by (simp add: rbt_eq_iff)
2.25 -
2.26 -lemma is_rbt_impl_of [simp, intro]:
2.27 - "is_rbt (impl_of t)"
2.28 - using impl_of [of t] by simp
2.29 -
2.30 -lemma RBT_impl_of [simp, code abstype]:
2.31 - "RBT (impl_of t) = t"
2.32 - by (simp add: impl_of_inverse)
2.33 -
2.34 -subsection {* Primitive operations *}
2.35 -
2.36 -setup_lifting type_definition_rbt
2.37 -
2.38 -lift_definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" is "rbt_lookup"
2.39 -by simp
2.40 -
2.41 -lift_definition empty :: "('a\<Colon>linorder, 'b) rbt" is RBT_Impl.Empty
2.42 -by (simp add: empty_def)
2.43 -
2.44 -lift_definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_insert"
2.45 -by simp
2.46 -
2.47 -lift_definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_delete"
2.48 -by simp
2.49 -
2.50 -lift_definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" is RBT_Impl.entries
2.51 -by simp
2.52 -
2.53 -lift_definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" is RBT_Impl.keys
2.54 -by simp
2.55 -
2.56 -lift_definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" is "rbt_bulkload"
2.57 -by simp
2.58 -
2.59 -lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is rbt_map_entry
2.60 -by simp
2.61 -
2.62 -lift_definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is RBT_Impl.map
2.63 -by simp
2.64 -
2.65 -lift_definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" is RBT_Impl.fold
2.66 -by simp
2.67 -
2.68 -lift_definition union :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_union"
2.69 -by (simp add: rbt_union_is_rbt)
2.70 -
2.71 -lift_definition foldi :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a :: linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"
2.72 - is RBT_Impl.foldi by simp
2.73 -
2.74 -export_code lookup empty insert delete entries keys bulkload map_entry map fold union foldi in SML
2.75 -
2.76 -subsection {* Derived operations *}
2.77 -
2.78 -definition is_empty :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where
2.79 - [code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
2.80 -
2.81 -
2.82 -subsection {* Abstract lookup properties *}
2.83 -
2.84 -lemma lookup_RBT:
2.85 - "is_rbt t \<Longrightarrow> lookup (RBT t) = rbt_lookup t"
2.86 - by (simp add: lookup_def RBT_inverse)
2.87 -
2.88 -lemma lookup_impl_of:
2.89 - "rbt_lookup (impl_of t) = lookup t"
2.90 - by transfer (rule refl)
2.91 -
2.92 -lemma entries_impl_of:
2.93 - "RBT_Impl.entries (impl_of t) = entries t"
2.94 - by transfer (rule refl)
2.95 -
2.96 -lemma keys_impl_of:
2.97 - "RBT_Impl.keys (impl_of t) = keys t"
2.98 - by transfer (rule refl)
2.99 -
2.100 -lemma lookup_empty [simp]:
2.101 - "lookup empty = Map.empty"
2.102 - by (simp add: empty_def lookup_RBT fun_eq_iff)
2.103 -
2.104 -lemma lookup_insert [simp]:
2.105 - "lookup (insert k v t) = (lookup t)(k \<mapsto> v)"
2.106 - by transfer (rule rbt_lookup_rbt_insert)
2.107 -
2.108 -lemma lookup_delete [simp]:
2.109 - "lookup (delete k t) = (lookup t)(k := None)"
2.110 - by transfer (simp add: rbt_lookup_rbt_delete restrict_complement_singleton_eq)
2.111 -
2.112 -lemma map_of_entries [simp]:
2.113 - "map_of (entries t) = lookup t"
2.114 - by transfer (simp add: map_of_entries)
2.115 -
2.116 -lemma entries_lookup:
2.117 - "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
2.118 - by transfer (simp add: entries_rbt_lookup)
2.119 -
2.120 -lemma lookup_bulkload [simp]:
2.121 - "lookup (bulkload xs) = map_of xs"
2.122 - by transfer (rule rbt_lookup_rbt_bulkload)
2.123 -
2.124 -lemma lookup_map_entry [simp]:
2.125 - "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
2.126 - by transfer (rule rbt_lookup_rbt_map_entry)
2.127 -
2.128 -lemma lookup_map [simp]:
2.129 - "lookup (map f t) k = Option.map (f k) (lookup t k)"
2.130 - by transfer (rule rbt_lookup_map)
2.131 -
2.132 -lemma fold_fold:
2.133 - "fold f t = List.fold (prod_case f) (entries t)"
2.134 - by transfer (rule RBT_Impl.fold_def)
2.135 -
2.136 -lemma impl_of_empty:
2.137 - "impl_of empty = RBT_Impl.Empty"
2.138 - by transfer (rule refl)
2.139 -
2.140 -lemma is_empty_empty [simp]:
2.141 - "is_empty t \<longleftrightarrow> t = empty"
2.142 - unfolding is_empty_def by transfer (simp split: rbt.split)
2.143 -
2.144 -lemma RBT_lookup_empty [simp]: (*FIXME*)
2.145 - "rbt_lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
2.146 - by (cases t) (auto simp add: fun_eq_iff)
2.147 -
2.148 -lemma lookup_empty_empty [simp]:
2.149 - "lookup t = Map.empty \<longleftrightarrow> t = empty"
2.150 - by transfer (rule RBT_lookup_empty)
2.151 -
2.152 -lemma sorted_keys [iff]:
2.153 - "sorted (keys t)"
2.154 - by transfer (simp add: RBT_Impl.keys_def rbt_sorted_entries)
2.155 -
2.156 -lemma distinct_keys [iff]:
2.157 - "distinct (keys t)"
2.158 - by transfer (simp add: RBT_Impl.keys_def distinct_entries)
2.159 -
2.160 -lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
2.161 - by transfer simp
2.162 -
2.163 -lemma lookup_union: "lookup (union s t) = lookup s ++ lookup t"
2.164 - by transfer (simp add: rbt_lookup_rbt_union)
2.165 -
2.166 -lemma lookup_in_tree: "(lookup t k = Some v) = ((k, v) \<in> set (entries t))"
2.167 - by transfer (simp add: rbt_lookup_in_tree)
2.168 -
2.169 -lemma keys_entries: "(k \<in> set (keys t)) = (\<exists>v. (k, v) \<in> set (entries t))"
2.170 - by transfer (simp add: keys_entries)
2.171 -
2.172 -lemma fold_def_alt:
2.173 - "fold f t = List.fold (prod_case f) (entries t)"
2.174 - by transfer (auto simp: RBT_Impl.fold_def)
2.175 -
2.176 -lemma distinct_entries: "distinct (List.map fst (entries t))"
2.177 - by transfer (simp add: distinct_entries)
2.178 -
2.179 -lemma non_empty_keys: "t \<noteq> Lift_RBT.empty \<Longrightarrow> keys t \<noteq> []"
2.180 - by transfer (simp add: non_empty_rbt_keys)
2.181 -
2.182 -lemma keys_def_alt:
2.183 - "keys t = List.map fst (entries t)"
2.184 - by transfer (simp add: RBT_Impl.keys_def)
2.185 +(* Moved to ~~/HOL/Library/RBT" *)
2.186
2.187 end
2.188 \ No newline at end of file