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