src/HOL/Library/RBT.thy
 author haftmann Mon Jun 05 15:59:41 2017 +0200 (2017-06-05) changeset 66010 2f7d39285a1a parent 63219 a5697f7a3322 permissions -rw-r--r--
executable domain membership checks
1 (*  Title:      HOL/Library/RBT.thy
2     Author:     Lukas Bulwahn and Ondrej Kuncar
3 *)
5 section \<open>Abstract type of RBT trees\<close>
7 theory RBT
8 imports Main RBT_Impl
9 begin
11 subsection \<open>Type definition\<close>
13 typedef (overloaded) ('a, 'b) rbt = "{t :: ('a::linorder, 'b) RBT_Impl.rbt. is_rbt t}"
14   morphisms impl_of RBT
15 proof -
16   have "RBT_Impl.Empty \<in> ?rbt" by simp
17   then show ?thesis ..
18 qed
20 lemma rbt_eq_iff:
21   "t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"
24 lemma rbt_eqI:
25   "impl_of t1 = impl_of t2 \<Longrightarrow> t1 = t2"
28 lemma is_rbt_impl_of [simp, intro]:
29   "is_rbt (impl_of t)"
30   using impl_of [of t] by simp
32 lemma RBT_impl_of [simp, code abstype]:
33   "RBT (impl_of t) = t"
36 subsection \<open>Primitive operations\<close>
38 setup_lifting type_definition_rbt
40 lift_definition lookup :: "('a::linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" is "rbt_lookup" .
42 lift_definition empty :: "('a::linorder, 'b) rbt" is RBT_Impl.Empty
45 lift_definition insert :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_insert"
46 by simp
48 lift_definition delete :: "'a::linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_delete"
49 by simp
51 lift_definition entries :: "('a::linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" is RBT_Impl.entries .
53 lift_definition keys :: "('a::linorder, 'b) rbt \<Rightarrow> 'a list" is RBT_Impl.keys .
55 lift_definition bulkload :: "('a::linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" is "rbt_bulkload" ..
57 lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a::linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is rbt_map_entry
58 by simp
60 lift_definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a::linorder, 'b) rbt \<Rightarrow> ('a, 'c) rbt" is RBT_Impl.map
61 by simp
63 lift_definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a::linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"  is RBT_Impl.fold .
65 lift_definition union :: "('a::linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_union"
68 lift_definition foldi :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a :: linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"
69   is RBT_Impl.foldi .
71 lift_definition combine_with_key :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a::linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
72   is RBT_Impl.rbt_union_with_key by (rule is_rbt_rbt_unionwk)
74 lift_definition combine :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a::linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
75   is RBT_Impl.rbt_union_with by (rule rbt_unionw_is_rbt)
77 subsection \<open>Derived operations\<close>
79 definition is_empty :: "('a::linorder, 'b) rbt \<Rightarrow> bool" where
80   [code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
82 (* TODO: Is deleting more efficient than re-building the tree?
83    (Probably more difficult to prove though *)
84 definition filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a::linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
85   [code]: "filter P t = fold (\<lambda>k v t. if P k v then insert k v t else t) t empty"
87 subsection \<open>Abstract lookup properties\<close>
89 lemma lookup_RBT:
90   "is_rbt t \<Longrightarrow> lookup (RBT t) = rbt_lookup t"
91   by (simp add: lookup_def RBT_inverse)
93 lemma lookup_impl_of:
94   "rbt_lookup (impl_of t) = lookup t"
95   by transfer (rule refl)
97 lemma entries_impl_of:
98   "RBT_Impl.entries (impl_of t) = entries t"
99   by transfer (rule refl)
101 lemma keys_impl_of:
102   "RBT_Impl.keys (impl_of t) = keys t"
103   by transfer (rule refl)
105 lemma lookup_keys:
106   "dom (lookup t) = set (keys t)"
107   by transfer (simp add: rbt_lookup_keys)
109 lemma lookup_empty [simp]:
110   "lookup empty = Map.empty"
111   by (simp add: empty_def lookup_RBT fun_eq_iff)
113 lemma lookup_insert [simp]:
114   "lookup (insert k v t) = (lookup t)(k \<mapsto> v)"
115   by transfer (rule rbt_lookup_rbt_insert)
117 lemma lookup_delete [simp]:
118   "lookup (delete k t) = (lookup t)(k := None)"
119   by transfer (simp add: rbt_lookup_rbt_delete restrict_complement_singleton_eq)
121 lemma map_of_entries [simp]:
122   "map_of (entries t) = lookup t"
123   by transfer (simp add: map_of_entries)
125 lemma entries_lookup:
126   "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
127   by transfer (simp add: entries_rbt_lookup)
130   "lookup (bulkload xs) = map_of xs"
133 lemma lookup_map_entry [simp]:
134   "lookup (map_entry k f t) = (lookup t)(k := map_option f (lookup t k))"
135   by transfer (rule rbt_lookup_rbt_map_entry)
137 lemma lookup_map [simp]:
138   "lookup (map f t) k = map_option (f k) (lookup t k)"
139   by transfer (rule rbt_lookup_map)
141 lemma lookup_combine_with_key [simp]:
142   "lookup (combine_with_key f t1 t2) k = combine_options (f k) (lookup t1 k) (lookup t2 k)"
143   by transfer (simp_all add: combine_options_def rbt_lookup_rbt_unionwk)
145 lemma combine_altdef: "combine f t1 t2 = combine_with_key (\<lambda>_. f) t1 t2"
146   by transfer (simp add: rbt_union_with_def)
148 lemma lookup_combine [simp]:
149   "lookup (combine f t1 t2) k = combine_options f (lookup t1 k) (lookup t2 k)"
152 lemma fold_fold:
153   "fold f t = List.fold (case_prod f) (entries t)"
154   by transfer (rule RBT_Impl.fold_def)
156 lemma impl_of_empty:
157   "impl_of empty = RBT_Impl.Empty"
158   by transfer (rule refl)
160 lemma is_empty_empty [simp]:
161   "is_empty t \<longleftrightarrow> t = empty"
162   unfolding is_empty_def by transfer (simp split: rbt.split)
164 lemma RBT_lookup_empty [simp]: (*FIXME*)
165   "rbt_lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
166   by (cases t) (auto simp add: fun_eq_iff)
168 lemma lookup_empty_empty [simp]:
169   "lookup t = Map.empty \<longleftrightarrow> t = empty"
170   by transfer (rule RBT_lookup_empty)
172 lemma sorted_keys [iff]:
173   "sorted (keys t)"
174   by transfer (simp add: RBT_Impl.keys_def rbt_sorted_entries)
176 lemma distinct_keys [iff]:
177   "distinct (keys t)"
178   by transfer (simp add: RBT_Impl.keys_def distinct_entries)
180 lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
181   by transfer simp
183 lemma lookup_union: "lookup (union s t) = lookup s ++ lookup t"
184   by transfer (simp add: rbt_lookup_rbt_union)
186 lemma lookup_in_tree: "(lookup t k = Some v) = ((k, v) \<in> set (entries t))"
187   by transfer (simp add: rbt_lookup_in_tree)
189 lemma keys_entries: "(k \<in> set (keys t)) = (\<exists>v. (k, v) \<in> set (entries t))"
190   by transfer (simp add: keys_entries)
192 lemma fold_def_alt:
193   "fold f t = List.fold (case_prod f) (entries t)"
194   by transfer (auto simp: RBT_Impl.fold_def)
196 lemma distinct_entries: "distinct (List.map fst (entries t))"
197   by transfer (simp add: distinct_entries)
199 lemma non_empty_keys: "t \<noteq> empty \<Longrightarrow> keys t \<noteq> []"
200   by transfer (simp add: non_empty_rbt_keys)
202 lemma keys_def_alt:
203   "keys t = List.map fst (entries t)"
204   by transfer (simp add: RBT_Impl.keys_def)
206 context
207 begin
209 private lemma lookup_filter_aux:
210   assumes "distinct (List.map fst xs)"
211   shows   "lookup (List.fold (\<lambda>(k, v) t. if P k v then insert k v t else t) xs t) k =
212              (case map_of xs k of
213                 None \<Rightarrow> lookup t k
214               | Some v \<Rightarrow> if P k v then Some v else lookup t k)"
215   using assms by (induction xs arbitrary: t) (force split: option.splits)+
217 lemma lookup_filter:
218   "lookup (filter P t) k =
219      (case lookup t k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None)"
220   unfolding filter_def using lookup_filter_aux[of "entries t" P empty k]
221   by (simp add: fold_fold distinct_entries split: option.splits)
223 end
226 subsection \<open>Quickcheck generators\<close>
228 quickcheck_generator rbt predicate: is_rbt constructors: empty, insert
230 subsection \<open>Hide implementation details\<close>
232 lifting_update rbt.lifting
233 lifting_forget rbt.lifting
235 hide_const (open) impl_of empty lookup keys entries bulkload delete map fold union insert map_entry foldi
236   is_empty filter
237 hide_fact (open) empty_def lookup_def keys_def entries_def bulkload_def delete_def map_def fold_def
238   union_def insert_def map_entry_def foldi_def is_empty_def filter_def
240 end