1 (* Author: Florian Haftmann, TU Muenchen *)
3 header {* An abstract view on maps for code generation. *}
9 subsection {* Type definition and primitive operations *}
11 datatype ('a, 'b) mapping = Mapping "'a \<rightharpoonup> 'b"
13 definition empty :: "('a, 'b) mapping" where
14 "empty = Mapping (\<lambda>_. None)"
16 primrec lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<rightharpoonup> 'b" where
17 "lookup (Mapping f) = f"
19 primrec update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
20 "update k v (Mapping f) = Mapping (f (k \<mapsto> v))"
22 primrec delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
23 "delete k (Mapping f) = Mapping (f (k := None))"
26 subsection {* Derived operations *}
28 definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" where
29 "keys m = dom (lookup m)"
31 definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where
32 "ordered_keys m = sorted_list_of_set (keys m)"
34 definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where
35 "is_empty m \<longleftrightarrow> dom (lookup m) = {}"
37 definition size :: "('a, 'b) mapping \<Rightarrow> nat" where
38 "size m = (if finite (dom (lookup m)) then card (dom (lookup m)) else 0)"
40 definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
41 "replace k v m = (if lookup m k = None then m else update k v m)"
43 definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" where
44 "tabulate ks f = Mapping (map_of (map (\<lambda>k. (k, f k)) ks))"
46 definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" where
47 "bulkload xs = Mapping (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
50 subsection {* Properties *}
52 lemma lookup_inject [simp]:
53 "lookup m = lookup n \<longleftrightarrow> m = n"
54 by (cases m, cases n) simp
57 assumes "lookup m = lookup n"
61 lemma lookup_empty [simp]:
62 "lookup empty = Map.empty"
63 by (simp add: empty_def)
65 lemma lookup_update [simp]:
66 "lookup (update k v m) = (lookup m) (k \<mapsto> v)"
69 lemma lookup_delete [simp]:
70 "lookup (delete k m) = (lookup m) (k := None)"
73 lemma lookup_tabulate [simp]:
74 "lookup (tabulate ks f) = (Some o f) |` set ks"
75 by (induct ks) (auto simp add: tabulate_def restrict_map_def expand_fun_eq)
77 lemma lookup_bulkload [simp]:
78 "lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
79 by (simp add: bulkload_def)
82 "update k v (update k w m) = update k v m"
83 "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
84 by (rule mapping_eqI, simp add: fun_upd_twist)+
86 lemma update_delete [simp]:
87 "update k v (delete k m) = update k v m"
88 by (rule mapping_eqI) simp
91 "delete k (update k v m) = delete k m"
92 "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
93 by (rule mapping_eqI, simp add: fun_upd_twist)+
95 lemma delete_empty [simp]:
96 "delete k empty = empty"
97 by (rule mapping_eqI) simp
100 "k \<notin> dom (lookup m) \<Longrightarrow> replace k v m = m"
101 "k \<in> dom (lookup m) \<Longrightarrow> replace k v m = update k v m"
102 by (rule mapping_eqI, auto simp add: replace_def fun_upd_twist)+
104 lemma size_empty [simp]:
106 by (simp add: size_def)
109 "finite (dom (lookup m)) \<Longrightarrow> size (update k v m) =
110 (if k \<in> dom (lookup m) then size m else Suc (size m))"
111 by (auto simp add: size_def insert_dom)
114 "size (delete k m) = (if k \<in> dom (lookup m) then size m - 1 else size m)"
115 by (simp add: size_def)
118 "size (tabulate ks f) = length (remdups ks)"
119 by (simp add: size_def distinct_card [of "remdups ks", symmetric] comp_def)
121 lemma bulkload_tabulate:
122 "bulkload xs = tabulate [0..<length xs] (nth xs)"
123 by (rule mapping_eqI) (simp add: expand_fun_eq)
126 subsection {* Some technical code lemmas *}
129 "mapping_case f m = f (Mapping.lookup m)"
133 "mapping_rec f m = f (Mapping.lookup m)"
137 "Nat.size (m :: (_, _) mapping) = 0"
141 "mapping_size f g m = 0"
145 hide (open) const empty is_empty lookup update delete ordered_keys keys size replace tabulate bulkload