bot comes before top, inf before sup etc.
(* Author: Florian Haftmann, TU Muenchen *)
header {* Abstract type of Red-Black Trees *}
(*<*)
theory RBT
imports Main RBT_Impl Mapping
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)
subsection {* Implementation of mappings *}
definition Mapping :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) mapping" where
"Mapping t = Mapping.Mapping (lookup t)"
code_datatype Mapping
lemma lookup_Mapping [simp, code]:
"Mapping.lookup (Mapping t) = lookup t"
by (simp add: Mapping_def)
lemma empty_Mapping [code]:
"Mapping.empty = Mapping empty"
by (rule mapping_eqI) simp
lemma is_empty_Mapping [code]:
"Mapping.is_empty (Mapping t) \<longleftrightarrow> is_empty t"
by (simp add: rbt_eq_iff Mapping.is_empty_empty Mapping_def)
lemma insert_Mapping [code]:
"Mapping.update k v (Mapping t) = Mapping (insert k v t)"
by (rule mapping_eqI) simp
lemma delete_Mapping [code]:
"Mapping.delete k (Mapping t) = Mapping (delete k t)"
by (rule mapping_eqI) simp
lemma map_entry_Mapping [code]:
"Mapping.map_entry k f (Mapping t) = Mapping (map_entry k f t)"
by (rule mapping_eqI) simp
lemma keys_Mapping [code]:
"Mapping.keys (Mapping t) = set (keys t)"
by (simp add: keys_def Mapping_def Mapping.keys_def lookup_def lookup_keys)
lemma ordered_keys_Mapping [code]:
"Mapping.ordered_keys (Mapping t) = keys t"
by (rule sorted_distinct_set_unique) (simp_all add: ordered_keys_def keys_Mapping)
lemma Mapping_size_card_keys: (*FIXME*)
"Mapping.size m = card (Mapping.keys m)"
by (simp add: Mapping.size_def Mapping.keys_def)
lemma size_Mapping [code]:
"Mapping.size (Mapping t) = length (keys t)"
by (simp add: Mapping_size_card_keys keys_Mapping distinct_card)
lemma tabulate_Mapping [code]:
"Mapping.tabulate ks f = Mapping (bulkload (List.map (\<lambda>k. (k, f k)) ks))"
by (rule mapping_eqI) (simp add: map_of_map_restrict)
lemma bulkload_Mapping [code]:
"Mapping.bulkload vs = Mapping (bulkload (List.map (\<lambda>n. (n, vs ! n)) [0..<length vs]))"
by (rule mapping_eqI) (simp add: map_of_map_restrict fun_eq_iff)
lemma equal_Mapping [code]:
"HOL.equal (Mapping t1) (Mapping t2) \<longleftrightarrow> entries t1 = entries t2"
by (simp add: equal Mapping_def entries_lookup)
lemma [code nbe]:
"HOL.equal (x :: (_, _) mapping) x \<longleftrightarrow> 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 insert_is_rbt}\hfill(@{text "insert_is_rbt"})
\noindent
@{thm delete_is_rbt}\hfill(@{text "delete_is_rbt"})
\noindent
@{thm bulkload_is_rbt}\hfill(@{text "bulkload_is_rbt"})
\noindent
@{thm map_entry_is_rbt}\hfill(@{text "map_entry_is_rbt"})
\noindent
@{thm map_is_rbt}\hfill(@{text "map_is_rbt"})
\noindent
@{thm 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