src/HOL/Library/RBT.thy

 
 
 subsection {* Primitive operations *}
 
 definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" where
-  [code]: "lookup t = RBT_Impl.lookup (impl_of t)"
+  [code]: "lookup t = rbt_lookup (impl_of t)"
 
 definition empty :: "('a\<Colon>linorder, 'b) rbt" where
   "empty = RBT RBT_Impl.Empty"
 
 lemma impl_of_empty [code abstract]:
   "impl_of empty = RBT_Impl.Empty"
   by (simp add: empty_def RBT_inverse)
 
 definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "insert k v t = RBT (RBT_Impl.insert k v (impl_of t))"
+  "insert k v t = RBT (rbt_insert k v (impl_of t))"
 
 lemma impl_of_insert [code abstract]:
-  "impl_of (insert k v t) = RBT_Impl.insert k v (impl_of t)"
+  "impl_of (insert k v t) = rbt_insert k v (impl_of t)"
   by (simp add: insert_def RBT_inverse)
 
 definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "delete k t = RBT (RBT_Impl.delete k (impl_of t))"
+  "delete k t = RBT (rbt_delete k (impl_of t))"
 
 lemma impl_of_delete [code abstract]:
-  "impl_of (delete k t) = RBT_Impl.delete k (impl_of t)"
+  "impl_of (delete k t) = rbt_delete k (impl_of t)"
   by (simp add: delete_def RBT_inverse)
 
 definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" where
   [code]: "entries t = RBT_Impl.entries (impl_of t)"
 
 definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" where
   [code]: "keys t = RBT_Impl.keys (impl_of t)"
 
 definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where
-  "bulkload xs = RBT (RBT_Impl.bulkload xs)"
+  "bulkload xs = RBT (rbt_bulkload xs)"
 
 lemma impl_of_bulkload [code abstract]:
-  "impl_of (bulkload xs) = RBT_Impl.bulkload xs"
+  "impl_of (bulkload xs) = rbt_bulkload xs"
   by (simp add: bulkload_def RBT_inverse)
 
 definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "map_entry k f t = RBT (RBT_Impl.map_entry k f (impl_of t))"
+  "map_entry k f t = RBT (rbt_map_entry k f (impl_of t))"
 
 lemma impl_of_map_entry [code abstract]:
-  "impl_of (map_entry k f t) = RBT_Impl.map_entry k f (impl_of t)"
+  "impl_of (map_entry k f t) = rbt_map_entry k f (impl_of t)"
   by (simp add: map_entry_def RBT_inverse)
 
 definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "map f t = RBT (RBT_Impl.map f (impl_of t))"
 

 
 
 subsection {* Abstract lookup properties *}
 
 lemma lookup_RBT:
-  "is_rbt t \<Longrightarrow> lookup (RBT t) = RBT_Impl.lookup t"
+  "is_rbt t \<Longrightarrow> lookup (RBT t) = rbt_lookup t"
   by (simp add: lookup_def RBT_inverse)
 
 lemma lookup_impl_of:
-  "RBT_Impl.lookup (impl_of t) = lookup t"
+  "rbt_lookup (impl_of t) = lookup t"
   by (simp add: lookup_def)
 
 lemma entries_impl_of:
   "RBT_Impl.entries (impl_of t) = entries t"
   by (simp add: entries_def)

   "lookup empty = Map.empty"
   by (simp add: empty_def lookup_RBT fun_eq_iff)
 
 lemma lookup_insert [simp]:
   "lookup (insert k v t) = (lookup t)(k \<mapsto> v)"
-  by (simp add: insert_def lookup_RBT lookup_insert lookup_impl_of)
+  by (simp add: insert_def lookup_RBT rbt_lookup_rbt_insert lookup_impl_of)
 
 lemma lookup_delete [simp]:
   "lookup (delete k t) = (lookup t)(k := None)"
-  by (simp add: delete_def lookup_RBT RBT_Impl.lookup_delete lookup_impl_of restrict_complement_singleton_eq)
+  by (simp add: delete_def lookup_RBT rbt_lookup_rbt_delete lookup_impl_of restrict_complement_singleton_eq)
 
 lemma map_of_entries [simp]:
   "map_of (entries t) = lookup t"
   by (simp add: entries_def map_of_entries lookup_impl_of)
 
 lemma entries_lookup:
   "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
-  by (simp add: entries_def lookup_def entries_lookup)
+  by (simp add: entries_def lookup_def entries_rbt_lookup)
 
 lemma lookup_bulkload [simp]:
   "lookup (bulkload xs) = map_of xs"
-  by (simp add: bulkload_def lookup_RBT RBT_Impl.lookup_bulkload)
+  by (simp add: bulkload_def lookup_RBT rbt_lookup_rbt_bulkload)
 
 lemma lookup_map_entry [simp]:
   "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
-  by (simp add: map_entry_def lookup_RBT RBT_Impl.lookup_map_entry lookup_impl_of)
+  by (simp add: map_entry_def lookup_RBT rbt_lookup_rbt_map_entry lookup_impl_of)
 
 lemma lookup_map [simp]:
   "lookup (map f t) k = Option.map (f k) (lookup t k)"
-  by (simp add: map_def lookup_RBT RBT_Impl.lookup_map lookup_impl_of)
+  by (simp add: map_def lookup_RBT rbt_lookup_map lookup_impl_of)
 
 lemma fold_fold:
   "fold f t = List.fold (prod_case f) (entries t)"
   by (simp add: fold_def fun_eq_iff RBT_Impl.fold_def entries_impl_of)
 
 lemma is_empty_empty [simp]:
   "is_empty t \<longleftrightarrow> t = empty"
   by (simp add: rbt_eq_iff is_empty_def impl_of_empty split: rbt.split)
 
 lemma RBT_lookup_empty [simp]: (*FIXME*)
-  "RBT_Impl.lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
+  "rbt_lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
   by (cases t) (auto simp add: fun_eq_iff)
 
 lemma lookup_empty_empty [simp]:
   "lookup t = Map.empty \<longleftrightarrow> t = empty"
   by (cases t) (simp add: empty_def lookup_def RBT_inject RBT_inverse)
 
 lemma sorted_keys [iff]:
   "sorted (keys t)"
-  by (simp add: keys_def RBT_Impl.keys_def sorted_entries)
+  by (simp add: keys_def RBT_Impl.keys_def rbt_sorted_entries)
 
 lemma distinct_keys [iff]:
   "distinct (keys t)"
   by (simp add: keys_def RBT_Impl.keys_def distinct_entries)