(* Title: HOL/Library/Mapping.thy
Author: Florian Haftmann and Ondrej Kuncar
*)
header {* An abstract view on maps for code generation. *}
theory Mapping
imports Main
begin
subsection {* Parametricity transfer rules *}
lemma map_of_foldr: -- {* FIXME move *}
"map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"
using map_add_map_of_foldr [of Map.empty] by auto
context
begin
interpretation lifting_syntax .
lemma empty_parametric:
"(A ===> rel_option B) Map.empty Map.empty"
by transfer_prover
lemma lookup_parametric: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
by transfer_prover
lemma update_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
(\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
by transfer_prover
lemma delete_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B)
(\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
by transfer_prover
lemma is_none_parametric [transfer_rule]:
"(rel_option A ===> HOL.eq) Option.is_none Option.is_none"
by (auto simp add: is_none_def rel_fun_def rel_option_iff split: option.split)
lemma dom_parametric:
assumes [transfer_rule]: "bi_total A"
shows "((A ===> rel_option B) ===> rel_set A) dom dom"
unfolding dom_def [abs_def] is_none_def [symmetric] by transfer_prover
lemma map_of_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique R1"
shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of"
unfolding map_of_def by transfer_prover
lemma map_entry_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B)
(\<lambda>k f m. (case m k of None \<Rightarrow> m
| Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
| Some v \<Rightarrow> m (k \<mapsto> (f v))))"
by transfer_prover
lemma tabulate_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
(\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))"
by transfer_prover
lemma bulkload_parametric:
"(list_all2 A ===> HOL.eq ===> rel_option A)
(\<lambda>xs k. if k < length xs then Some (xs ! k) else None) (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)"
proof
fix xs ys
assume "list_all2 A xs ys"
then show "(HOL.eq ===> rel_option A)
(\<lambda>k. if k < length xs then Some (xs ! k) else None)
(\<lambda>k. if k < length ys then Some (ys ! k) else None)"
apply induct
apply auto
unfolding rel_fun_def
apply clarsimp
apply (case_tac xa)
apply (auto dest: list_all2_lengthD list_all2_nthD)
done
qed
lemma map_parametric:
"((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D)
(\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
by transfer_prover
end
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 Map.empty parametric empty_parametric .
lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option"
is "\<lambda>m k. m k" parametric lookup_parametric .
lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_parametric .
lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
is "\<lambda>k m. m(k := None)" parametric delete_parametric .
lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set"
is dom parametric dom_parametric .
lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping"
is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_parametric .
lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping"
is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_parametric .
lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping"
is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric .
subsection {* Functorial structure *}
functor map: map
by (transfer, auto simp add: fun_eq_iff option.map_comp option.map_id)+
subsection {* Derived operations *}
definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list"
where
"ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool"
where
"is_empty m \<longleftrightarrow> keys m = {}"
definition size :: "('a, 'b) mapping \<Rightarrow> nat"
where
"size m = (if finite (keys m) then card (keys m) else 0)"
definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
where
"replace k v m = (if k \<in> keys m then update k v m else m)"
definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
where
"default k v m = (if k \<in> keys m then m else update k v m)"
text {* Manual derivation of transfer rule is non-trivial *}
lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
"\<lambda>k f m. (case m k of None \<Rightarrow> m
| Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_parametric .
lemma map_entry_code [code]:
"map_entry k f m = (case lookup m k of None \<Rightarrow> m
| Some v \<Rightarrow> update k (f v) m)"
by transfer rule
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 of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
where
"of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
instantiation mapping :: (type, type) equal
begin
definition
"HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. lookup m1 k = lookup m2 k)"
instance proof
qed (unfold equal_mapping_def, transfer, auto)
end
context
begin
interpretation lifting_syntax .
lemma [transfer_rule]:
assumes [transfer_rule]: "bi_total A"
assumes [transfer_rule]: "bi_unique B"
shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
by (unfold equal) transfer_prover
lemma of_alist_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique R1"
shows "(list_all2 (rel_prod R1 R2) ===> pcr_mapping R1 R2) map_of of_alist"
unfolding of_alist_def [abs_def] map_of_foldr [abs_def] by transfer_prover
end
subsection {* Properties *}
lemma lookup_update:
"lookup (update k v m) k = Some v"
by transfer simp
lemma lookup_update_neq:
"k \<noteq> k' \<Longrightarrow> 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 \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
by transfer (auto simp add: is_none_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 (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 \<noteq> l \<Longrightarrow> 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 \<notin> keys m \<Longrightarrow> replace k v m = m"
"k \<in> keys m \<Longrightarrow> 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) \<Longrightarrow> size (update k v m) =
(if k \<in> 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 \<in> 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: map_of_map_restrict card_set comp_def)
lemma bulkload_tabulate:
"bulkload xs = tabulate [0..<length xs] (nth xs)"
by transfer (auto simp add: map_of_map_restrict)
lemma is_empty_empty [simp]:
"is_empty empty"
unfolding is_empty_def by transfer simp
lemma is_empty_update [simp]:
"\<not> is_empty (update k v m)"
unfolding is_empty_def by transfer simp
lemma is_empty_delete:
"is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> 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) \<longleftrightarrow> is_empty m"
unfolding is_empty_def replace_def by transfer auto
lemma is_empty_default [simp]:
"\<not> 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) \<longleftrightarrow> is_empty m"
unfolding is_empty_def by transfer (auto split: option.split)
lemma is_empty_map_default [simp]:
"\<not> is_empty (map_default k v f m)"
by (simp add: map_default_def)
lemma keys_dom_lookup:
"keys m = dom (Mapping.lookup m)"
by transfer rule
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"
by transfer (auto split: option.split)
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: bulkload_tabulate)
lemma distinct_ordered_keys [simp]:
"distinct (ordered_keys m)"
by (simp add: ordered_keys_def)
lemma ordered_keys_infinite [simp]:
"\<not> finite (keys m) \<Longrightarrow> 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 \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m"
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> 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 \<in> keys m")
case False with fin have "k \<notin> 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 \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m"
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> 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 \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m"
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> 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)
lemma tabulate_fold:
"tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
proof transfer
fix f :: "'a \<Rightarrow> 'b" and xs
have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
by (simp add: foldr_map comp_def map_of_foldr)
also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs"
by (rule foldr_fold) (simp add: fun_eq_iff)
ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
by simp
qed
subsection {* Code generator setup *}
code_datatype empty update
hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size
replace default map_entry map_default tabulate bulkload map of_alist
end