(* 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 *}
context
begin
interpretation lifting_syntax .
lemma empty_transfer: "(A ===> rel_option B) Map.empty Map.empty" by transfer_prover
lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)" by transfer_prover
lemma update_transfer:
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_transfer:
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
definition equal_None :: "'a option \<Rightarrow> bool" where "equal_None x \<equiv> x = None"
lemma [transfer_rule]: "(rel_option A ===> op=) equal_None equal_None"
unfolding rel_fun_def rel_option_iff equal_None_def by (auto split: option.split)
lemma dom_transfer:
assumes [transfer_rule]: "bi_total A"
shows "((A ===> rel_option B) ===> rel_set A) dom dom"
unfolding dom_def[abs_def] equal_None_def[symmetric]
by transfer_prover
lemma map_of_transfer [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 tabulate_transfer:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
(\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks)))"
by transfer_prover
lemma bulkload_transfer:
"(list_all2 A ===> op= ===> 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)"
unfolding rel_fun_def
apply clarsimp
apply (erule list_all2_induct)
apply simp
apply (case_tac xa)
apply simp
by (auto dest: list_all2_lengthD list_all2_nthD)
lemma map_transfer:
"((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
lemma map_entry_transfer:
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
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_transfer .
lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option" is "\<lambda>m k. m k"
parametric lookup_transfer .
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_transfer .
lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k m. m(k := None)"
parametric delete_transfer .
lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" is dom parametric dom_transfer .
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_transfer .
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_transfer .
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_transfer .
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)"
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_transfer .
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)"
lift_definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
is map_of parametric map_of_transfer .
lemma of_alist_code [code]:
"of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
by transfer(simp add: map_add_map_of_foldr[symmetric])
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
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 tabulate_alt_def:
"map_of (List.map (\<lambda>k. (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 \<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: 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]:
"\<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
apply transfer by (case_tac "m k") auto
lemma is_empty_map_default [simp]:
"\<not> 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]:
"\<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)
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