(* Author: Florian Haftmann, TU Muenchen *)
header {* An abstract view on maps for code generation. *}
theory Mapping
imports Main
begin
subsection {* Type definition and primitive operations *}
datatype ('a, 'b) mapping = Mapping "'a \<rightharpoonup> 'b"
definition empty :: "('a, 'b) mapping" where
"empty = Mapping (\<lambda>_. None)"
primrec lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<rightharpoonup> 'b" where
"lookup (Mapping f) = f"
primrec update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"update k v (Mapping f) = Mapping (f (k \<mapsto> v))"
primrec delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"delete k (Mapping f) = Mapping (f (k := None))"
subsection {* Derived operations *}
definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" where
"keys m = dom (lookup m)"
definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where
"ordered_keys m = sorted_list_of_set (keys m)"
definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where
"is_empty m \<longleftrightarrow> dom (lookup m) = {}"
definition size :: "('a, 'b) mapping \<Rightarrow> nat" where
"size m = (if finite (dom (lookup m)) then card (dom (lookup m)) else 0)"
definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"replace k v m = (if lookup m k = None then m else update k v m)"
definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"default k v m = (if lookup m k = None then update k v m else m)"
definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"map_entry k f m = (case lookup m k of None \<Rightarrow> m
| Some v \<Rightarrow> update k (f v) m)"
definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"map_default k v f m = map_entry k f (default k v m)"
definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" where
"tabulate ks f = Mapping (map_of (map (\<lambda>k. (k, f k)) ks))"
definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" where
"bulkload xs = Mapping (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
subsection {* Properties *}
lemma lookup_inject [simp]:
"lookup m = lookup n \<longleftrightarrow> m = n"
by (cases m, cases n) simp
lemma mapping_eqI:
assumes "lookup m = lookup n"
shows "m = n"
using assms by simp
lemma lookup_empty [simp]:
"lookup empty = Map.empty"
by (simp add: empty_def)
lemma lookup_update [simp]:
"lookup (update k v m) = (lookup m) (k \<mapsto> v)"
by (cases m) simp
lemma lookup_delete [simp]:
"lookup (delete k m) = (lookup m) (k := None)"
by (cases m) simp
lemma lookup_map_entry [simp]:
"lookup (map_entry k f m) = (lookup m) (k := Option.map f (lookup m k))"
by (cases "lookup m k") (simp_all add: map_entry_def expand_fun_eq)
lemma lookup_tabulate [simp]:
"lookup (tabulate ks f) = (Some o f) |` set ks"
by (induct ks) (auto simp add: tabulate_def restrict_map_def expand_fun_eq)
lemma lookup_bulkload [simp]:
"lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
by (simp add: bulkload_def)
lemma update_update:
"update k v (update k w m) = update k v m"
"k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
by (rule mapping_eqI, simp add: fun_upd_twist)+
lemma update_delete [simp]:
"update k v (delete k m) = update k v m"
by (rule mapping_eqI) simp
lemma delete_update:
"delete k (update k v m) = delete k m"
"k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
by (rule mapping_eqI, simp add: fun_upd_twist)+
lemma delete_empty [simp]:
"delete k empty = empty"
by (rule mapping_eqI) simp
lemma replace_update:
"k \<notin> dom (lookup m) \<Longrightarrow> replace k v m = m"
"k \<in> dom (lookup m) \<Longrightarrow> replace k v m = update k v m"
by (rule mapping_eqI, auto simp add: replace_def fun_upd_twist)+
lemma size_empty [simp]:
"size empty = 0"
by (simp add: size_def)
lemma size_update:
"finite (dom (lookup m)) \<Longrightarrow> size (update k v m) =
(if k \<in> dom (lookup m) then size m else Suc (size m))"
by (auto simp add: size_def insert_dom)
lemma size_delete:
"size (delete k m) = (if k \<in> dom (lookup m) then size m - 1 else size m)"
by (simp add: size_def)
lemma size_tabulate:
"size (tabulate ks f) = length (remdups ks)"
by (simp add: size_def distinct_card [of "remdups ks", symmetric] comp_def)
lemma bulkload_tabulate:
"bulkload xs = tabulate [0..<length xs] (nth xs)"
by (rule mapping_eqI) (simp add: expand_fun_eq)
lemma keys_tabulate:
"keys (tabulate ks f) = set ks"
by (simp add: tabulate_def keys_def map_of_map_restrict o_def)
lemma keys_bulkload:
"keys (bulkload xs) = {0..<length xs}"
by (simp add: keys_tabulate bulkload_tabulate)
lemma is_empty_empty:
"is_empty m \<longleftrightarrow> m = Mapping Map.empty"
by (cases m) (simp add: is_empty_def)
subsection {* Some technical code lemmas *}
lemma [code]:
"mapping_case f m = f (Mapping.lookup m)"
by (cases m) simp
lemma [code]:
"mapping_rec f m = f (Mapping.lookup m)"
by (cases m) simp
lemma [code]:
"Nat.size (m :: (_, _) mapping) = 0"
by (cases m) simp
lemma [code]:
"mapping_size f g m = 0"
by (cases m) simp
hide_const (open) empty is_empty lookup update delete ordered_keys keys size replace tabulate bulkload
end