Theory RBT

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

header {* Abstract type of RBT trees *}

theory RBT
imports Main RBT_Impl

subsection {* Type definition *}

typedef ('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 ..

lemma rbt_eq_iff:
"t1 = t2 <-> impl_of t1 = impl_of t2"
by (simp add: impl_of_inject)

lemma rbt_eqI:
"impl_of t1 = impl_of t2 ==> 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 *}

setup_lifting type_definition_rbt

lift_definition lookup :: "('a::linorder, 'b) rbt => 'a \<rightharpoonup> 'b" is "rbt_lookup"
by simp

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

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
by simp

lift_definition keys :: "('a::linorder, 'b) rbt => 'a list" is RBT_Impl.keys
by simp

lift_definition bulkload :: "('a::linorder × 'b) list => ('a, 'b) rbt" is "rbt_bulkload"
by simp

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
by simp

lift_definition fold :: "('a => 'b => 'c => 'c) => ('a::linorder, 'b) rbt => 'c => 'c" is RBT_Impl.fold
by simp

lift_definition union :: "('a::linorder, 'b) rbt => ('a, 'b) rbt => ('a, 'b) rbt" is "rbt_union"
by (simp add: rbt_union_is_rbt)

lift_definition foldi :: "('c => bool) => ('a => 'b => 'c => 'c) => ('a :: linorder, 'b) rbt => 'c => 'c"
is RBT_Impl.foldi by simp

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)"

subsection {* Abstract lookup properties *}

lemma lookup_RBT:
"is_rbt t ==> lookup (RBT t) = rbt_lookup t"
by (simp add: lookup_def RBT_inverse)

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)"
by transfer (simp add: rbt_lookup_keys)

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 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"
by transfer (simp add: map_of_entries)

lemma entries_lookup:
"entries t1 = entries t2 <-> lookup t1 = lookup t2"
by transfer (simp add: entries_rbt_lookup)

lemma lookup_bulkload [simp]:
"lookup (bulkload xs) = map_of xs"
by transfer (rule rbt_lookup_rbt_bulkload)

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

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

lemma fold_fold:
"fold f t = List.fold (prod_case 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"
by transfer (simp add: rbt_lookup_rbt_union)

lemma lookup_in_tree: "(lookup t k = Some v) = ((k, v) ∈ set (entries t))"
by transfer (simp add: rbt_lookup_in_tree)

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

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

lemma distinct_entries: "distinct ( fst (entries t))"
by transfer (simp add: distinct_entries)

lemma non_empty_keys: "t ≠ empty ==> keys t ≠ []"
by transfer (simp add: non_empty_rbt_keys)

lemma keys_def_alt:
"keys t = fst (entries t)"
by transfer (simp add: RBT_Impl.keys_def)

subsection {* Quickcheck generators *}

quickcheck_generator rbt predicate: is_rbt constructors: empty, insert