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 (case_tac 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 (λk. (k, f k)) ks)"
by transfer (simp add: map_of_map_restrict)
lemma bulkload_Mapping [code]:
"Mapping.bulkload vs = Mapping (map (λn. (n, vs ! n)) [0..<length vs])"
by transfer (simp add: map_of_map_restrict fun_eq_iff)
lemma equal_Mapping [code]:
"HOL.equal (Mapping xs) (Mapping ys) ⟷
(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 aux: "⋀a b xs. (a, b) ∈ set xs ⟹ a ∈ fst ` set xs"
by (auto simp add: image_def intro!: bexI)
show ?thesis apply transfer
by (auto intro!: map_of_eqI) (auto dest!: map_of_eq_dom intro: aux)
qed
lemma [code nbe]:
"HOL.equal (x :: ('a, 'b) mapping) x ⟷ True"
by (fact equal_refl)
end