Theory RBT_Mapping

theory RBT_Mapping
imports RBT Mapping
(*  Title:      HOL/Library/RBT_Mapping.thy
Author: Florian Haftmann and Ondrej Kuncar
*)


header {* Implementation of mappings with Red-Black Trees *}

(*<*)
theory RBT_Mapping
imports RBT Mapping
begin

subsection {* Implementation of mappings *}

lift_definition Mapping :: "('a::linorder, 'b) rbt => ('a, 'b) mapping" is lookup .

code_datatype Mapping

lemma lookup_Mapping [simp, code]:
"Mapping.lookup (Mapping t) = lookup t"
by (transfer fixing: t) rule

lemma empty_Mapping [code]: "Mapping.empty = Mapping 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) <-> is_empty t"
unfolding is_empty_def by (transfer fixing: t) simp

lemma insert_Mapping [code]:
"Mapping.update k v (Mapping t) = Mapping (insert k v t)"
by (transfer fixing: t) simp

lemma delete_Mapping [code]:
"Mapping.delete k (Mapping t) = Mapping (delete k t)"
by (transfer fixing: t) simp

lemma map_entry_Mapping [code]:
"Mapping.map_entry k f (Mapping t) = Mapping (map_entry k f t)"
apply (transfer fixing: t) by (case_tac "lookup t k") auto

lemma keys_Mapping [code]:
"Mapping.keys (Mapping t) = set (keys t)"
by (transfer fixing: t) (simp add: lookup_keys)

lemma ordered_keys_Mapping [code]:
"Mapping.ordered_keys (Mapping t) = 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 (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 (bulkload (List.map (λk. (k, f k)) ks))"
by transfer (simp add: map_of_map_restrict)

lemma bulkload_Mapping [code]:
"Mapping.bulkload vs = Mapping (bulkload (List.map (λn. (n, vs ! n)) [0..<length vs]))"
by transfer (simp add: map_of_map_restrict fun_eq_iff)
end

lemma equal_Mapping [code]:
"HOL.equal (Mapping t1) (Mapping t2) <-> entries t1 = entries t2"
by (transfer fixing: t1 t2) (simp add: entries_lookup)

lemma [code nbe]:
"HOL.equal (x :: (_, _) mapping) x <-> True"
by (fact equal_refl)


hide_const (open) impl_of lookup empty insert delete
entries keys bulkload map_entry map fold
(*>*)

text {*
This theory defines abstract red-black trees as an efficient
representation of finite maps, backed by the implementation
in @{theory 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 @{text "linorder"}
class.

A value @{term t} of this type is a valid red-black tree if it
satisfies the invariant @{text "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(@{text "Empty_is_rbt"})

\noindent
@{thm rbt_insert_is_rbt}\hfill(@{text "rbt_insert_is_rbt"})

\noindent
@{thm rbt_delete_is_rbt}\hfill(@{text "delete_is_rbt"})

\noindent
@{thm rbt_bulkload_is_rbt}\hfill(@{text "bulkload_is_rbt"})

\noindent
@{thm rbt_map_entry_is_rbt}\hfill(@{text "map_entry_is_rbt"})

\noindent
@{thm map_is_rbt}\hfill(@{text "map_is_rbt"})

\noindent
@{thm rbt_union_is_rbt}\hfill(@{text "union_is_rbt"})
*}



subsection {* Map Semantics *}

text {*
\noindent
\underline{@{text "lookup_empty"}}
@{thm [display] lookup_empty}
\vspace{1ex}

\noindent
\underline{@{text "lookup_insert"}}
@{thm [display] lookup_insert}
\vspace{1ex}

\noindent
\underline{@{text "lookup_delete"}}
@{thm [display] lookup_delete}
\vspace{1ex}

\noindent
\underline{@{text "lookup_bulkload"}}
@{thm [display] lookup_bulkload}
\vspace{1ex}

\noindent
\underline{@{text "lookup_map"}}
@{thm [display] lookup_map}
\vspace{1ex}
*}


end