# Theory RBT

theory RBT
imports RBT_Impl
```(*  Title:      HOL/Library/RBT.thy
Author:     Lukas Bulwahn and Ondrej Kuncar
*)

section ‹Abstract type of RBT trees›

theory RBT
imports Main RBT_Impl
begin

subsection ‹Type definition›

typedef (overloaded) ('a, 'b) rbt = "{t :: ('a::linorder, 'b) RBT_Impl.rbt. is_rbt t}"
morphisms impl_of RBT
proof -
have "RBT_Impl.Empty ∈ ?rbt" by simp
then show ?thesis ..
qed

lemma rbt_eq_iff:
"t1 = t2 ⟷ impl_of t1 = impl_of t2"

lemma rbt_eqI:
"impl_of t1 = impl_of t2 ⟹ t1 = t2"

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"

subsection ‹Primitive operations›

setup_lifting type_definition_rbt

lift_definition lookup :: "('a::linorder, 'b) rbt ⇒ 'a ⇀ 'b" is "rbt_lookup" .

lift_definition empty :: "('a::linorder, 'b) rbt" is RBT_Impl.Empty

lift_definition insert :: "'a::linorder ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" is "rbt_insert"
by simp

lift_definition delete :: "'a::linorder ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" is "rbt_delete"
by simp

lift_definition entries :: "('a::linorder, 'b) rbt ⇒ ('a × 'b) list" is RBT_Impl.entries .

lift_definition keys :: "('a::linorder, 'b) rbt ⇒ 'a list" is RBT_Impl.keys .

lift_definition bulkload :: "('a::linorder × 'b) list ⇒ ('a, 'b) rbt" is "rbt_bulkload" ..

lift_definition map_entry :: "'a ⇒ ('b ⇒ 'b) ⇒ ('a::linorder, 'b) rbt ⇒ ('a, 'b) rbt" is rbt_map_entry
by simp

lift_definition map :: "('a ⇒ 'b ⇒ 'c) ⇒ ('a::linorder, 'b) rbt ⇒ ('a, 'c) rbt" is RBT_Impl.map
by simp

lift_definition fold :: "('a ⇒ 'b ⇒ 'c ⇒ 'c) ⇒ ('a::linorder, 'b) rbt ⇒ 'c ⇒ 'c"  is RBT_Impl.fold .

lift_definition union :: "('a::linorder, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" is "rbt_union"

lift_definition foldi :: "('c ⇒ bool) ⇒ ('a ⇒ 'b ⇒ 'c ⇒ 'c) ⇒ ('a :: linorder, 'b) rbt ⇒ 'c ⇒ 'c"
is RBT_Impl.foldi .

lift_definition combine_with_key :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a::linorder, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt"
is RBT_Impl.rbt_union_with_key by (rule is_rbt_rbt_unionwk)

lift_definition combine :: "('b ⇒ 'b ⇒ 'b) ⇒ ('a::linorder, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt"
is RBT_Impl.rbt_union_with by (rule rbt_unionw_is_rbt)

subsection ‹Derived operations›

definition is_empty :: "('a::linorder, 'b) rbt ⇒ bool" where
[code]: "is_empty t = (case impl_of t of RBT_Impl.Empty ⇒ True | _ ⇒ False)"

(* TODO: Is deleting more efficient than re-building the tree?
(Probably more difficult to prove though *)
definition filter :: "('a ⇒ 'b ⇒ bool) ⇒ ('a::linorder, 'b) rbt ⇒ ('a, 'b) rbt" where
[code]: "filter P t = fold (λk v t. if P k v then insert k v t else t) t empty"

subsection ‹Abstract lookup properties›

lemma lookup_RBT:
"is_rbt t ⟹ lookup (RBT t) = rbt_lookup t"

lemma lookup_impl_of:
"rbt_lookup (impl_of t) = lookup t"
by transfer (rule refl)

lemma entries_impl_of:
"RBT_Impl.entries (impl_of t) = entries t"
by transfer (rule refl)

lemma keys_impl_of:
"RBT_Impl.keys (impl_of t) = keys t"
by transfer (rule refl)

lemma lookup_keys:
"dom (lookup t) = set (keys t)"

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 ↦ v)"
by transfer (rule rbt_lookup_rbt_insert)

lemma lookup_delete [simp]:
"lookup (delete k t) = (lookup t)(k := None)"
by transfer (simp add: rbt_lookup_rbt_delete restrict_complement_singleton_eq)

lemma map_of_entries [simp]:
"map_of (entries t) = lookup t"

lemma entries_lookup:
"entries t1 = entries t2 ⟷ lookup t1 = lookup t2"

"lookup (bulkload xs) = map_of xs"

lemma lookup_map_entry [simp]:
"lookup (map_entry k f t) = (lookup t)(k := map_option f (lookup t k))"
by transfer (rule rbt_lookup_rbt_map_entry)

lemma lookup_map [simp]:
"lookup (map f t) k = map_option (f k) (lookup t k)"
by transfer (rule rbt_lookup_map)

lemma lookup_combine_with_key [simp]:
"lookup (combine_with_key f t1 t2) k = combine_options (f k) (lookup t1 k) (lookup t2 k)"
by transfer (simp_all add: combine_options_def rbt_lookup_rbt_unionwk)

lemma combine_altdef: "combine f t1 t2 = combine_with_key (λ_. f) t1 t2"

lemma lookup_combine [simp]:
"lookup (combine f t1 t2) k = combine_options f (lookup t1 k) (lookup t2 k)"

lemma fold_fold:
"fold f t = List.fold (case_prod f) (entries t)"
by transfer (rule RBT_Impl.fold_def)

lemma impl_of_empty:
"impl_of empty = RBT_Impl.Empty"
by transfer (rule refl)

lemma is_empty_empty [simp]:
"is_empty t ⟷ t = empty"
unfolding is_empty_def by transfer (simp split: rbt.split)

lemma RBT_lookup_empty [simp]: (*FIXME*)
"rbt_lookup t = Map.empty ⟷ t = RBT_Impl.Empty"
by (cases t) (auto simp add: fun_eq_iff)

lemma lookup_empty_empty [simp]:
"lookup t = Map.empty ⟷ t = empty"
by transfer (rule RBT_lookup_empty)

lemma sorted_keys [iff]:
"sorted (keys t)"
by transfer (simp add: RBT_Impl.keys_def rbt_sorted_entries)

lemma distinct_keys [iff]:
"distinct (keys t)"
by transfer (simp add: RBT_Impl.keys_def distinct_entries)

lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
by transfer simp

lemma lookup_union: "lookup (union s t) = lookup s ++ lookup t"

lemma lookup_in_tree: "(lookup t k = Some v) = ((k, v) ∈ set (entries t))"

lemma keys_entries: "(k ∈ set (keys t)) = (∃v. (k, v) ∈ set (entries t))"

lemma fold_def_alt:
"fold f t = List.fold (case_prod f) (entries t)"
by transfer (auto simp: RBT_Impl.fold_def)

lemma distinct_entries: "distinct (List.map fst (entries t))"

lemma non_empty_keys: "t ≠ empty ⟹ keys t ≠ []"

lemma keys_def_alt:
"keys t = List.map fst (entries t)"

context
begin

private lemma lookup_filter_aux:
assumes "distinct (List.map fst xs)"
shows   "lookup (List.fold (λ(k, v) t. if P k v then insert k v t else t) xs t) k =
(case map_of xs k of
None ⇒ lookup t k
| Some v ⇒ if P k v then Some v else lookup t k)"
using assms by (induction xs arbitrary: t) (force split: option.splits)+

lemma lookup_filter:
"lookup (filter P t) k =
(case lookup t k of None ⇒ None | Some v ⇒ if P k v then Some v else None)"
unfolding filter_def using lookup_filter_aux[of "entries t" P empty k]
by (simp add: fold_fold distinct_entries split: option.splits)

end

subsection ‹Quickcheck generators›

quickcheck_generator rbt predicate: is_rbt constructors: empty, insert

subsection ‹Hide implementation details›

lifting_update rbt.lifting
lifting_forget rbt.lifting

hide_const (open) impl_of empty lookup keys entries bulkload delete map fold union insert map_entry foldi
is_empty filter
hide_fact (open) empty_def lookup_def keys_def entries_def bulkload_def delete_def map_def fold_def
union_def insert_def map_entry_def foldi_def is_empty_def filter_def

end
```