(* Author: Florian Haftmann, TU Muenchen *)
header {* Abstract type of Red-Black Trees *}
(*<*)
theory RBT
imports Main RBT_Impl
begin
subsection {* Type definition *}
typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}"
morphisms impl_of RBT
proof -
have "RBT_Impl.Empty \<in> ?rbt" by simp
then show ?thesis ..
qed
lemma rbt_eq_iff:
"t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"
by (simp add: impl_of_inject)
lemma rbt_eqI:
"impl_of t1 = impl_of t2 \<Longrightarrow> t1 = t2"
by (simp add: rbt_eq_iff)
lemma is_rbt_impl_of [simp, intro]:
"is_rbt (impl_of t)"
using impl_of [of t] by simp
lemma RBT_impl_of [simp, code abstype]:
"RBT (impl_of t) = t"
by (simp add: impl_of_inverse)
subsection {* Primitive operations *}
definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" where
[code]: "lookup t = RBT_Impl.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))"
lemma impl_of_insert [code abstract]:
"impl_of (insert k v t) = RBT_Impl.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))"
lemma impl_of_delete [code abstract]:
"impl_of (delete k t) = RBT_Impl.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)"
lemma impl_of_bulkload [code abstract]:
"impl_of (bulkload xs) = RBT_Impl.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))"
lemma impl_of_map_entry [code abstract]:
"impl_of (map_entry k f t) = RBT_Impl.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))"
lemma impl_of_map [code abstract]:
"impl_of (map f t) = RBT_Impl.map f (impl_of t)"
by (simp add: map_def RBT_inverse)
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
[code]: "fold f t = RBT_Impl.fold f (impl_of t)"
subsection {* Derived operations *}
definition is_empty :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where
[code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
subsection {* Abstract lookup properties *}
lemma lookup_RBT:
"is_rbt t \<Longrightarrow> lookup (RBT t) = RBT_Impl.lookup t"
by (simp add: lookup_def RBT_inverse)
lemma lookup_impl_of:
"RBT_Impl.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)
lemma keys_impl_of:
"RBT_Impl.keys (impl_of t) = keys t"
by (simp add: keys_def)
lemma lookup_empty [simp]:
"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)
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)
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)
lemma lookup_bulkload [simp]:
"lookup (bulkload xs) = map_of xs"
by (simp add: bulkload_def lookup_RBT RBT_Impl.lookup_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)
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)
lemma fold_fold:
"fold f t = More_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"
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)
lemma distinct_keys [iff]:
"distinct (keys t)"
by (simp add: keys_def RBT_Impl.keys_def distinct_entries)
end