(* Title: HOL/Library/AList_Mapping.thy
Author: Florian Haftmann, TU Muenchen
*)
section \Implementation of mappings with Association Lists\
theory AList_Mapping
imports AList Mapping
begin
lift_definition Mapping :: "('a \ 'b) list \ ('a, 'b) mapping" is map_of .
code_datatype Mapping
lemma lookup_Mapping [simp, code]: "Mapping.lookup (Mapping xs) = map_of xs"
by transfer rule
lemma keys_Mapping [simp, code]: "Mapping.keys (Mapping xs) = set (map fst xs)"
by transfer (simp add: dom_map_of_conv_image_fst)
lemma empty_Mapping [code]: "Mapping.empty = Mapping []"
by transfer simp
lemma is_empty_Mapping [code]: "Mapping.is_empty (Mapping xs) \ List.null xs"
by (cases xs) (simp_all add: is_empty_def null_def)
lemma update_Mapping [code]: "Mapping.update k v (Mapping xs) = Mapping (AList.update k v xs)"
by transfer (simp add: update_conv')
lemma delete_Mapping [code]: "Mapping.delete k (Mapping xs) = Mapping (AList.delete k xs)"
by transfer (simp add: delete_conv')
lemma ordered_keys_Mapping [code]:
"Mapping.ordered_keys (Mapping xs) = sort (remdups (map fst xs))"
by (simp only: ordered_keys_def keys_Mapping sorted_list_of_set_sort_remdups) simp
lemma size_Mapping [code]: "Mapping.size (Mapping xs) = length (remdups (map fst xs))"
by (simp add: size_def length_remdups_card_conv dom_map_of_conv_image_fst)
lemma tabulate_Mapping [code]: "Mapping.tabulate ks f = Mapping (map (\k. (k, f k)) ks)"
by transfer (simp add: map_of_map_restrict)
lemma bulkload_Mapping [code]:
"Mapping.bulkload vs = Mapping (map (\n. (n, vs ! n)) [0..
(let ks = map fst xs; ls = map fst ys
in (\l\set ls. l \ set ks) \ (\k\set ks. k \ set ls \ map_of xs k = map_of ys k))"
proof -
have *: "(a, b) \ set xs \ a \ fst ` set xs" for a b xs
by (auto simp add: image_def intro!: bexI)
show ?thesis
apply transfer
apply (auto intro!: map_of_eqI)
apply (auto dest!: map_of_eq_dom intro: *)
done
qed
lemma map_values_Mapping [code]:
"Mapping.map_values f (Mapping xs) = Mapping (map (\(x,y). (x, f x y)) xs)"
for f :: "'c \ 'a \ 'b" and xs :: "('c \ 'a) list"
apply transfer
apply (rule ext)
subgoal for f xs x by (induct xs) auto
done
lemma combine_with_key_code [code]:
"Mapping.combine_with_key f (Mapping xs) (Mapping ys) =
Mapping.tabulate (remdups (map fst xs @ map fst ys))
(\x. the (combine_options (f x) (map_of xs x) (map_of ys x)))"
apply transfer
apply (rule ext)
apply (rule sym)
subgoal for f xs ys x
apply (cases "map_of xs x"; cases "map_of ys x"; simp)
apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
dest: map_of_SomeD split: option.splits)+
done
done
lemma combine_code [code]:
"Mapping.combine f (Mapping xs) (Mapping ys) =
Mapping.tabulate (remdups (map fst xs @ map fst ys))
(\x. the (combine_options f (map_of xs x) (map_of ys x)))"
apply transfer
apply (rule ext)
apply (rule sym)
subgoal for f xs ys x
apply (cases "map_of xs x"; cases "map_of ys x"; simp)
apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
dest: map_of_SomeD split: option.splits)+
done
done
lemma map_of_filter_distinct: (* TODO: move? *)
assumes "distinct (map fst xs)"
shows "map_of (filter P xs) x =
(case map_of xs x of
None \ None
| Some y \ if P (x,y) then Some y else None)"
using assms
by (auto simp: map_of_eq_None_iff filter_map distinct_map_filter dest: map_of_SomeD
simp del: map_of_eq_Some_iff intro!: map_of_is_SomeI split: option.splits)
lemma filter_Mapping [code]:
"Mapping.filter P (Mapping xs) = Mapping (filter (\(k,v). P k v) (AList.clearjunk xs))"
apply transfer
apply (rule ext)
apply (subst map_of_filter_distinct)
apply (simp_all add: map_of_clearjunk split: option.split)
done
lemma [code nbe]: "HOL.equal (x :: ('a, 'b) mapping) x \ True"
by (fact equal_refl)
end