(* Title: HOL/Library/RBT_Mapping.thy
Author: Florian Haftmann and Ondrej Kuncar
*)
section \<open>Implementation of mappings with Red-Black Trees\<close>
(*<*)
theory RBT_Mapping
imports RBT Mapping
begin
subsection \<open>Implementation of mappings\<close>
context includes rbt.lifting begin
lift_definition Mapping :: "('a::linorder, 'b) rbt \<Rightarrow> ('a, 'b) mapping" is RBT.lookup .
end
code_datatype Mapping
context includes rbt.lifting begin
lemma lookup_Mapping [simp, code]:
"Mapping.lookup (Mapping t) = RBT.lookup t"
by (transfer fixing: t) rule
lemma empty_Mapping [code]: "Mapping.empty = Mapping RBT.empty"
proof -
note RBT.empty.transfer[transfer_rule del]
show ?thesis by transfer simp
qed
lemma is_empty_Mapping [code]:
"Mapping.is_empty (Mapping t) \<longleftrightarrow> RBT.is_empty t"
unfolding is_empty_def by (transfer fixing: t) simp
lemma insert_Mapping [code]:
"Mapping.update k v (Mapping t) = Mapping (RBT.insert k v t)"
by (transfer fixing: t) simp
lemma delete_Mapping [code]:
"Mapping.delete k (Mapping t) = Mapping (RBT.delete k t)"
by (transfer fixing: t) simp
lemma map_entry_Mapping [code]:
"Mapping.map_entry k f (Mapping t) = Mapping (RBT.map_entry k f t)"
apply (transfer fixing: t)
apply (case_tac "RBT.lookup t k")
apply auto
done
lemma keys_Mapping [code]:
"Mapping.keys (Mapping t) = set (RBT.keys t)"
by (transfer fixing: t) (simp add: lookup_keys)
lemma ordered_keys_Mapping [code]:
"Mapping.ordered_keys (Mapping t) = RBT.keys t"
unfolding ordered_keys_def
by (transfer fixing: t) (auto simp add: lookup_keys intro: sorted_distinct_set_unique)
lemma Mapping_size_card_keys: (*FIXME*)
"Mapping.size m = card (Mapping.keys m)"
unfolding size_def by transfer simp
lemma size_Mapping [code]:
"Mapping.size (Mapping t) = length (RBT.keys t)"
unfolding size_def
by (transfer fixing: t) (simp add: lookup_keys distinct_card)
context
notes RBT.bulkload.transfer[transfer_rule del]
begin
lemma tabulate_Mapping [code]:
"Mapping.tabulate ks f = Mapping (RBT.bulkload (List.map (\<lambda>k. (k, f k)) ks))"
by transfer (simp add: map_of_map_restrict)
lemma bulkload_Mapping [code]:
"Mapping.bulkload vs = Mapping (RBT.bulkload (List.map (\<lambda>n. (n, vs ! n)) [0..<length vs]))"
by transfer (simp add: map_of_map_restrict fun_eq_iff)
end
lemma map_values_Mapping [code]:
"Mapping.map_values f (Mapping t) = Mapping (RBT.map f t)"
by (transfer fixing: t) (auto simp: fun_eq_iff)
lemma filter_Mapping [code]:
"Mapping.filter P (Mapping t) = Mapping (RBT.filter P t)"
by (transfer' fixing: P t) (simp add: RBT.lookup_filter fun_eq_iff)
lemma combine_with_key_Mapping [code]:
"Mapping.combine_with_key f (Mapping t1) (Mapping t2) =
Mapping (RBT.combine_with_key f t1 t2)"
by (transfer fixing: f t1 t2) (simp_all add: fun_eq_iff)
lemma combine_Mapping [code]:
"Mapping.combine f (Mapping t1) (Mapping t2) =
Mapping (RBT.combine f t1 t2)"
by (transfer fixing: f t1 t2) (simp_all add: fun_eq_iff)
lemma equal_Mapping [code]:
"HOL.equal (Mapping t1) (Mapping t2) \<longleftrightarrow> RBT.entries t1 = RBT.entries t2"
by (transfer fixing: t1 t2) (simp add: entries_lookup)
lemma [code nbe]:
"HOL.equal (x :: (_, _) mapping) x \<longleftrightarrow> True"
by (fact equal_refl)
end
(*>*)
text \<open>
This theory defines abstract red-black trees as an efficient
representation of finite maps, backed by the implementation
in @{theory RBT_Impl}.
\<close>
subsection \<open>Data type and invariant\<close>
text \<open>
The type @{typ "('k, 'v) RBT_Impl.rbt"} denotes red-black trees with
keys of type @{typ "'k"} and values of type @{typ "'v"}. To function
properly, the key type musorted belong to the \<open>linorder\<close>
class.
A value @{term t} of this type is a valid red-black tree if it
satisfies the invariant \<open>is_rbt t\<close>. The abstract type @{typ
"('k, 'v) rbt"} always obeys this invariant, and for this reason you
should only use this in our application. Going back to @{typ "('k,
'v) RBT_Impl.rbt"} may be necessary in proofs if not yet proven
properties about the operations must be established.
The interpretation function @{const "RBT.lookup"} returns the partial
map represented by a red-black tree:
@{term_type[display] "RBT.lookup"}
This function should be used for reasoning about the semantics of the RBT
operations. Furthermore, it implements the lookup functionality for
the data structure: It is executable and the lookup is performed in
$O(\log n)$.
\<close>
subsection \<open>Operations\<close>
text \<open>
Currently, the following operations are supported:
@{term_type [display] "RBT.empty"}
Returns the empty tree. $O(1)$
@{term_type [display] "RBT.insert"}
Updates the map at a given position. $O(\log n)$
@{term_type [display] "RBT.delete"}
Deletes a map entry at a given position. $O(\log n)$
@{term_type [display] "RBT.entries"}
Return a corresponding key-value list for a tree.
@{term_type [display] "RBT.bulkload"}
Builds a tree from a key-value list.
@{term_type [display] "RBT.map_entry"}
Maps a single entry in a tree.
@{term_type [display] "RBT.map"}
Maps all values in a tree. $O(n)$
@{term_type [display] "RBT.fold"}
Folds over all entries in a tree. $O(n)$
\<close>
subsection \<open>Invariant preservation\<close>
text \<open>
\noindent
@{thm Empty_is_rbt}\hfill(\<open>Empty_is_rbt\<close>)
\noindent
@{thm rbt_insert_is_rbt}\hfill(\<open>rbt_insert_is_rbt\<close>)
\noindent
@{thm rbt_delete_is_rbt}\hfill(\<open>delete_is_rbt\<close>)
\noindent
@{thm rbt_bulkload_is_rbt}\hfill(\<open>bulkload_is_rbt\<close>)
\noindent
@{thm rbt_map_entry_is_rbt}\hfill(\<open>map_entry_is_rbt\<close>)
\noindent
@{thm map_is_rbt}\hfill(\<open>map_is_rbt\<close>)
\noindent
@{thm rbt_union_is_rbt}\hfill(\<open>union_is_rbt\<close>)
\<close>
subsection \<open>Map Semantics\<close>
text \<open>
\noindent
\underline{\<open>lookup_empty\<close>}
@{thm [display] lookup_empty}
\vspace{1ex}
\noindent
\underline{\<open>lookup_insert\<close>}
@{thm [display] lookup_insert}
\vspace{1ex}
\noindent
\underline{\<open>lookup_delete\<close>}
@{thm [display] lookup_delete}
\vspace{1ex}
\noindent
\underline{\<open>lookup_bulkload\<close>}
@{thm [display] lookup_bulkload}
\vspace{1ex}
\noindent
\underline{\<open>lookup_map\<close>}
@{thm [display] lookup_map}
\vspace{1ex}
\<close>
end