(* Title: HOL/Library/Mapping.thy
Author: Florian Haftmann, TU Muenchen
*)
header {* An abstract view on maps for code generation. *}
theory Mapping
imports Map Main
begin
subsection {* Type definition and primitive operations *}
datatype ('a, 'b) map = Map "'a \<rightharpoonup> 'b"
definition empty :: "('a, 'b) map" where
"empty = Map (\<lambda>_. None)"
primrec lookup :: "('a, 'b) map \<Rightarrow> 'a \<rightharpoonup> 'b" where
"lookup (Map f) = f"
primrec update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) map \<Rightarrow> ('a, 'b) map" where
"update k v (Map f) = Map (f (k \<mapsto> v))"
primrec delete :: "'a \<Rightarrow> ('a, 'b) map \<Rightarrow> ('a, 'b) map" where
"delete k (Map f) = Map (f (k := None))"
primrec keys :: "('a, 'b) map \<Rightarrow> 'a set" where
"keys (Map f) = dom f"
subsection {* Derived operations *}
definition size :: "('a, 'b) map \<Rightarrow> nat" where
"size m = (if finite (keys m) then card (keys m) else 0)"
definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) map \<Rightarrow> ('a, 'b) map" where
"replace k v m = (if lookup m k = None then m else update k v m)"
definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) map" where
"tabulate ks f = Map (map_of (map (\<lambda>k. (k, f k)) ks))"
definition bulkload :: "'a list \<Rightarrow> (nat, 'a) map" where
"bulkload xs = Map (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
subsection {* Properties *}
lemma lookup_inject:
"lookup m = lookup n \<longleftrightarrow> m = n"
by (cases m, cases n) 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:
"lookup (delete k m) k = None"
"k \<noteq> l \<Longrightarrow> lookup (delete k m) l = lookup m l"
by (cases m, simp)+
lemma lookup_tabulate:
"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:
"lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
unfolding bulkload_def by simp
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 (cases m, simp add: expand_fun_eq)+
lemma replace_update:
"lookup m k = None \<Longrightarrow> replace k v m = m"
"lookup m k \<noteq> None \<Longrightarrow> replace k v m = update k v m"
by (auto simp add: replace_def)
lemma delete_empty [simp]:
"delete k empty = empty"
by (simp add: empty_def)
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 (cases m, simp add: expand_fun_eq)+
lemma update_delete [simp]:
"update k v (delete k m) = update k v m"
by (cases m) simp
lemma keys_empty [simp]:
"keys empty = {}"
unfolding empty_def by simp
lemma keys_update [simp]:
"keys (update k v m) = insert k (keys m)"
by (cases m) simp
lemma keys_delete [simp]:
"keys (delete k m) = keys m - {k}"
by (cases m) simp
lemma keys_tabulate [simp]:
"keys (tabulate ks f) = set ks"
by (auto simp add: tabulate_def dest: map_of_SomeD intro!: weak_map_of_SomeI)
lemma size_empty [simp]:
"size empty = 0"
by (simp add: size_def keys_empty)
lemma size_update:
"finite (keys m) \<Longrightarrow> size (update k v m) =
(if k \<in> keys m then size m else Suc (size m))"
by (simp add: size_def keys_update)
(auto simp only: card_insert card_Suc_Diff1)
lemma size_delete:
"size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
by (simp add: size_def keys_delete)
lemma size_tabulate:
"size (tabulate ks f) = length (remdups ks)"
by (simp add: size_def keys_tabulate distinct_card [of "remdups ks", symmetric])
lemma bulkload_tabulate:
"bulkload xs = tabulate [0..<length xs] (nth xs)"
by (rule sym)
(auto simp add: bulkload_def tabulate_def expand_fun_eq map_of_eq_None_iff map_compose [symmetric] comp_def)
end