Theory HOL-Library.RBT_Mapping
section ‹Implementation of mappings with Red-Black Trees›
theory RBT_Mapping
imports RBT Mapping
begin
subsection ‹Implementation of mappings›
context includes rbt.lifting begin
lift_definition Mapping :: "('a::linorder, 'b) rbt ⇒ ('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) ⟷ 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 Map_graph_lookup: "Map.graph (RBT.lookup t) = set (RBT.entries t)"
by (metis RBT.distinct_entries RBT.map_of_entries graph_map_of_if_distinct_dom)
lemma entries_Mapping [code]: "Mapping.entries (Mapping t) = set (RBT.entries t)"
by (transfer fixing: t) (fact Map_graph_lookup)
lemma ordered_entries_Mapping [code]: "Mapping.ordered_entries (Mapping t) = RBT.entries t"
proof -
note folding_Map_graph.idem_if_sorted_distinct[
where ?m="RBT.lookup t", OF _ _ folding_Map_graph.distinct_if_distinct_map]
then show ?thesis
unfolding ordered_entries_def
by (transfer fixing: t) (auto simp: Map_graph_lookup distinct_entries sorted_entries)
qed
lemma fold_Mapping [code]: "Mapping.fold f (Mapping t) a = RBT.fold f t a"
by (simp add: Mapping.fold_def fold_fold ordered_entries_Mapping)
lemma Mapping_size_card_keys:
"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 (λk. (k, f k)) ks))"
by transfer (simp add: map_of_map_restrict)
lemma bulkload_Mapping [code]:
"Mapping.bulkload vs = Mapping (RBT.bulkload (List.map (λn. (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) ⟷ RBT.entries t1 = RBT.entries t2"
by (transfer fixing: t1 t2) (simp add: RBT.entries_lookup)
lemma [code nbe]:
"HOL.equal (x :: (_, _) mapping) x ⟷ True"
by (fact equal_refl)
end
text ‹
This theory defines abstract red-black trees as an efficient
representation of finite maps, backed by the implementation
in \<^theory>‹HOL-Library.RBT_Impl›.
›
subsection ‹Data type and invariant›
text ‹
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 ‹linorder›
class.
A value \<^term>‹t› of this type is a valid red-black tree if it
satisfies the invariant ‹is_rbt t›. 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
$O(\log n)$.
›
subsection ‹Operations›
text ‹
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. $O(n)$
›
subsection ‹Invariant preservation›
text ‹
\noindent
@{thm Empty_is_rbt}\hfill(‹Empty_is_rbt›)
\noindent
@{thm rbt_insert_is_rbt}\hfill(‹rbt_insert_is_rbt›)
\noindent
@{thm rbt_delete_is_rbt}\hfill(‹delete_is_rbt›)
\noindent
@{thm rbt_bulkload_is_rbt}\hfill(‹bulkload_is_rbt›)
\noindent
@{thm rbt_map_entry_is_rbt}\hfill(‹map_entry_is_rbt›)
\noindent
@{thm map_is_rbt}\hfill(‹map_is_rbt›)
\noindent
@{thm rbt_union_is_rbt}\hfill(‹union_is_rbt›)
›
subsection ‹Map Semantics›
text ‹
\noindent
\underline{‹lookup_empty›}
@{thm [display] lookup_empty}
\vspace{1ex}
\noindent
\underline{‹lookup_insert›}
@{thm [display] lookup_insert}
\vspace{1ex}
\noindent
\underline{‹lookup_delete›}
@{thm [display] lookup_delete}
\vspace{1ex}
\noindent
\underline{‹lookup_bulkload›}
@{thm [display] lookup_bulkload}
\vspace{1ex}
\noindent
\underline{‹lookup_map›}
@{thm [display] lookup_map}
\vspace{1ex}
›
end