header {* An abstract view on maps for code generation. *}
theory Mapping
imports Main Quotient_Option
begin
subsection {* Type definition and primitive operations *}
typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
morphisms rep Mapping ..
setup_lifting(no_code) type_definition_mapping
lift_definition empty :: "('a, 'b) mapping" is "(λ_. None)" .
lift_definition lookup :: "('a, 'b) mapping => 'a => 'b option" is "λm k. m k" .
lift_definition update :: "'a => 'b => ('a, 'b) mapping => ('a, 'b) mapping" is "λk v m. m(k \<mapsto> v)" .
lift_definition delete :: "'a => ('a, 'b) mapping => ('a, 'b) mapping" is "λk m. m(k := None)" .
lift_definition keys :: "('a, 'b) mapping => 'a set" is dom .
lift_definition tabulate :: "'a list => ('a => 'b) => ('a, 'b) mapping" is
"λks f. (map_of (List.map (λk. (k, f k)) ks))" .
lift_definition bulkload :: "'a list => (nat, 'a) mapping" is
"λxs k. if k < length xs then Some (xs ! k) else None" .
lift_definition map :: "('c => 'a) => ('b => 'd) => ('a, 'b) mapping => ('c, 'd) mapping" is
"λf g m. (Option.map g o m o f)" .
subsection {* Functorial structure *}
enriched_type map: map
by (transfer, auto simp add: fun_eq_iff Option.map.compositionality Option.map.id)+
subsection {* Derived operations *}
definition ordered_keys :: "('a::linorder, 'b) mapping => 'a list" where
"ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
definition is_empty :: "('a, 'b) mapping => bool" where
"is_empty m <-> keys m = {}"
definition size :: "('a, 'b) mapping => nat" where
"size m = (if finite (keys m) then card (keys m) else 0)"
definition replace :: "'a => 'b => ('a, 'b) mapping => ('a, 'b) mapping" where
"replace k v m = (if k ∈ keys m then update k v m else m)"
definition default :: "'a => 'b => ('a, 'b) mapping => ('a, 'b) mapping" where
"default k v m = (if k ∈ keys m then m else update k v m)"
lift_definition map_entry :: "'a => ('b => 'b) => ('a, 'b) mapping => ('a, 'b) mapping" is
"λk f m. (case m k of None => m
| Some v => m (k \<mapsto> (f v)))" .
lemma map_entry_code [code]: "map_entry k f m = (case lookup m k of None => m
| Some v => update k (f v) m)"
by transfer rule
definition map_default :: "'a => 'b => ('b => 'b) => ('a, 'b) mapping => ('a, 'b) mapping" where
"map_default k v f m = map_entry k f (default k v m)"
subsection {* Properties *}
lemma lookup_update: "lookup (update k v m) k = Some v"
by transfer simp
lemma lookup_update_neq: "k ≠ k' ==> lookup (update k v m) k' = lookup m k'"
by transfer simp
lemma lookup_empty: "lookup empty k = None"
by transfer simp
lemma keys_is_none_rep [code_unfold]:
"k ∈ keys m <-> ¬ (Option.is_none (lookup m k))"
by transfer (auto simp add: is_none_def)
lemma tabulate_alt_def:
"map_of (List.map (λk. (k, f k)) ks) = (Some o f) |` set ks"
by (induct ks) (auto simp add: tabulate_def restrict_map_def)
lemma update_update:
"update k v (update k w m) = update k v m"
"k ≠ l ==> update k v (update l w m) = update l w (update k v m)"
by (transfer, simp add: fun_upd_twist)+
lemma update_delete [simp]:
"update k v (delete k m) = update k v m"
by transfer simp
lemma delete_update:
"delete k (update k v m) = delete k m"
"k ≠ l ==> delete k (update l v m) = update l v (delete k m)"
by (transfer, simp add: fun_upd_twist)+
lemma delete_empty [simp]:
"delete k empty = empty"
by transfer simp
lemma replace_update:
"k ∉ keys m ==> replace k v m = m"
"k ∈ keys m ==> replace k v m = update k v m"
by (transfer, auto simp add: replace_def fun_upd_twist)+
lemma size_empty [simp]:
"size empty = 0"
unfolding size_def by transfer simp
lemma size_update:
"finite (keys m) ==> size (update k v m) =
(if k ∈ keys m then size m else Suc (size m))"
unfolding size_def by transfer (auto simp add: insert_dom)
lemma size_delete:
"size (delete k m) = (if k ∈ keys m then size m - 1 else size m)"
unfolding size_def by transfer simp
lemma size_tabulate [simp]:
"size (tabulate ks f) = length (remdups ks)"
unfolding size_def by transfer (auto simp add: tabulate_alt_def card_set comp_def)
lemma bulkload_tabulate:
"bulkload xs = tabulate [0..<length xs] (nth xs)"
by transfer (auto simp add: tabulate_alt_def)
lemma is_empty_empty [simp]:
"is_empty empty"
unfolding is_empty_def by transfer simp
lemma is_empty_update [simp]:
"¬ is_empty (update k v m)"
unfolding is_empty_def by transfer simp
lemma is_empty_delete:
"is_empty (delete k m) <-> is_empty m ∨ keys m = {k}"
unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)
lemma is_empty_replace [simp]:
"is_empty (replace k v m) <-> is_empty m"
unfolding is_empty_def replace_def by transfer auto
lemma is_empty_default [simp]:
"¬ is_empty (default k v m)"
unfolding is_empty_def default_def by transfer auto
lemma is_empty_map_entry [simp]:
"is_empty (map_entry k f m) <-> is_empty m"
unfolding is_empty_def
apply transfer by (case_tac "m k") auto
lemma is_empty_map_default [simp]:
"¬ is_empty (map_default k v f m)"
by (simp add: map_default_def)
lemma keys_empty [simp]:
"keys empty = {}"
by transfer simp
lemma keys_update [simp]:
"keys (update k v m) = insert k (keys m)"
by transfer simp
lemma keys_delete [simp]:
"keys (delete k m) = keys m - {k}"
by transfer simp
lemma keys_replace [simp]:
"keys (replace k v m) = keys m"
unfolding replace_def by transfer (simp add: insert_absorb)
lemma keys_default [simp]:
"keys (default k v m) = insert k (keys m)"
unfolding default_def by transfer (simp add: insert_absorb)
lemma keys_map_entry [simp]:
"keys (map_entry k f m) = keys m"
apply transfer by (case_tac "m k") auto
lemma keys_map_default [simp]:
"keys (map_default k v f m) = insert k (keys m)"
by (simp add: map_default_def)
lemma keys_tabulate [simp]:
"keys (tabulate ks f) = set ks"
by transfer (simp add: map_of_map_restrict o_def)
lemma keys_bulkload [simp]:
"keys (bulkload xs) = {0..<length xs}"
by (simp add: keys_tabulate bulkload_tabulate)
lemma distinct_ordered_keys [simp]:
"distinct (ordered_keys m)"
by (simp add: ordered_keys_def)
lemma ordered_keys_infinite [simp]:
"¬ finite (keys m) ==> ordered_keys m = []"
by (simp add: ordered_keys_def)
lemma ordered_keys_empty [simp]:
"ordered_keys empty = []"
by (simp add: ordered_keys_def)
lemma ordered_keys_update [simp]:
"k ∈ keys m ==> ordered_keys (update k v m) = ordered_keys m"
"finite (keys m) ==> k ∉ keys m ==> ordered_keys (update k v m) = insort k (ordered_keys m)"
by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
lemma ordered_keys_delete [simp]:
"ordered_keys (delete k m) = remove1 k (ordered_keys m)"
proof (cases "finite (keys m)")
case False then show ?thesis by simp
next
case True note fin = True
show ?thesis
proof (cases "k ∈ keys m")
case False with fin have "k ∉ set (sorted_list_of_set (keys m))" by simp
with False show ?thesis by (simp add: ordered_keys_def remove1_idem)
next
case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove)
qed
qed
lemma ordered_keys_replace [simp]:
"ordered_keys (replace k v m) = ordered_keys m"
by (simp add: replace_def)
lemma ordered_keys_default [simp]:
"k ∈ keys m ==> ordered_keys (default k v m) = ordered_keys m"
"finite (keys m) ==> k ∉ keys m ==> ordered_keys (default k v m) = insort k (ordered_keys m)"
by (simp_all add: default_def)
lemma ordered_keys_map_entry [simp]:
"ordered_keys (map_entry k f m) = ordered_keys m"
by (simp add: ordered_keys_def)
lemma ordered_keys_map_default [simp]:
"k ∈ keys m ==> ordered_keys (map_default k v f m) = ordered_keys m"
"finite (keys m) ==> k ∉ keys m ==> ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
by (simp_all add: map_default_def)
lemma ordered_keys_tabulate [simp]:
"ordered_keys (tabulate ks f) = sort (remdups ks)"
by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)
lemma ordered_keys_bulkload [simp]:
"ordered_keys (bulkload ks) = [0..<length ks]"
by (simp add: ordered_keys_def)
subsection {* Code generator setup *}
code_datatype empty update
instantiation mapping :: (type, type) equal
begin
lift_definition equal_mapping :: "('a, 'b) mapping => ('a, 'b) mapping => bool" is "op=" .
instance proof
qed(transfer, rule)
end
hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size
replace default map_entry map_default tabulate bulkload map
end