explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;
(* Author: Florian Haftmann, TU Muenchen *)
header {* An abstract view on maps for code generation. *}
theory Mapping
imports Main
begin
subsection {* Type definition and primitive operations *}
typedef (open) ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
morphisms lookup Mapping ..
lemma lookup_Mapping [simp]:
"lookup (Mapping f) = f"
by (rule Mapping_inverse) rule
lemma Mapping_lookup [simp]:
"Mapping (lookup m) = m"
by (fact lookup_inverse)
lemma Mapping_inject [simp]:
"Mapping f = Mapping g \<longleftrightarrow> f = g"
by (simp add: Mapping_inject)
lemma mapping_eq_iff:
"m = n \<longleftrightarrow> lookup m = lookup n"
by (simp add: lookup_inject)
lemma mapping_eqI:
"lookup m = lookup n \<Longrightarrow> m = n"
by (simp add: mapping_eq_iff)
definition empty :: "('a, 'b) mapping" where
"empty = Mapping (\<lambda>_. None)"
definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"update k v m = Mapping ((lookup m)(k \<mapsto> v))"
definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"delete k m = Mapping ((lookup m)(k := None))"
subsection {* Functorial structure *}
definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping" where
"map f g m = Mapping (Option.map g \<circ> lookup m \<circ> f)"
lemma lookup_map [simp]:
"lookup (map f g m) = Option.map g \<circ> lookup m \<circ> f"
by (simp add: map_def)
enriched_type map: map
by (simp_all add: mapping_eq_iff fun_eq_iff Option.map.compositionality Option.map.id)
subsection {* Derived operations *}
definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" where
"keys m = dom (lookup m)"
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)"
definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
"map_entry k f m = (case lookup m k of None \<Rightarrow> m
| Some v \<Rightarrow> update k (f v) m)"
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 tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" where
"tabulate ks f = Mapping (map_of (List.map (\<lambda>k. (k, f k)) ks))"
definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" where
"bulkload xs = Mapping (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
subsection {* Properties *}
lemma keys_is_none_lookup [code_unfold]:
"k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
by (auto simp add: keys_def is_none_def)
lemma lookup_empty [simp]:
"lookup empty = Map.empty"
by (simp add: empty_def)
lemma lookup_update [simp]:
"lookup (update k v m) = (lookup m) (k \<mapsto> v)"
by (simp add: update_def)
lemma lookup_delete [simp]:
"lookup (delete k m) = (lookup m) (k := None)"
by (simp add: delete_def)
lemma lookup_map_entry [simp]:
"lookup (map_entry k f m) = (lookup m) (k := Option.map f (lookup m k))"
by (cases "lookup m k") (simp_all add: map_entry_def fun_eq_iff)
lemma lookup_tabulate [simp]:
"lookup (tabulate ks f) = (Some o f) |` set ks"
by (induct ks) (auto simp add: tabulate_def restrict_map_def fun_eq_iff)
lemma lookup_bulkload [simp]:
"lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
by (simp add: bulkload_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 (rule mapping_eqI, simp add: fun_upd_twist)+
lemma update_delete [simp]:
"update k v (delete k m) = update k v m"
by (rule mapping_eqI) 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 (rule mapping_eqI, simp add: fun_upd_twist)+
lemma delete_empty [simp]:
"delete k empty = empty"
by (rule mapping_eqI) 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 (rule mapping_eqI) (auto simp add: replace_def fun_upd_twist)+
lemma size_empty [simp]:
"size empty = 0"
by (simp add: size_def keys_def)
lemma size_update:
"finite (keys m) \<Longrightarrow> size (update k v m) =
(if k \<in> keys m then size m else Suc (size m))"
by (auto simp add: size_def insert_dom keys_def)
lemma size_delete:
"size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
by (simp add: size_def keys_def)
lemma size_tabulate [simp]:
"size (tabulate ks f) = length (remdups ks)"
by (simp add: size_def distinct_card [of "remdups ks", symmetric] comp_def keys_def)
lemma bulkload_tabulate:
"bulkload xs = tabulate [0..<length xs] (nth xs)"
by (rule mapping_eqI) (simp add: fun_eq_iff)
lemma is_empty_empty: (*FIXME*)
"is_empty m \<longleftrightarrow> m = Mapping Map.empty"
by (cases m) (simp add: is_empty_def keys_def)
lemma is_empty_empty' [simp]:
"is_empty empty"
by (simp add: is_empty_empty empty_def)
lemma is_empty_update [simp]:
"\<not> is_empty (update k v m)"
by (simp add: is_empty_empty update_def)
lemma is_empty_delete:
"is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
by (auto simp add: delete_def is_empty_def keys_def simp del: dom_eq_empty_conv)
lemma is_empty_replace [simp]:
"is_empty (replace k v m) \<longleftrightarrow> is_empty m"
by (auto simp add: replace_def) (simp add: is_empty_def)
lemma is_empty_default [simp]:
"\<not> is_empty (default k v m)"
by (auto simp add: default_def) (simp add: is_empty_def)
lemma is_empty_map_entry [simp]:
"is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
by (cases "lookup m k")
(auto simp add: map_entry_def, simp add: is_empty_empty)
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 (simp add: keys_def)
lemma keys_update [simp]:
"keys (update k v m) = insert k (keys m)"
by (simp add: keys_def)
lemma keys_delete [simp]:
"keys (delete k m) = keys m - {k}"
by (simp add: keys_def)
lemma keys_replace [simp]:
"keys (replace k v m) = keys m"
by (auto simp add: keys_def replace_def)
lemma keys_default [simp]:
"keys (default k v m) = insert k (keys m)"
by (auto simp add: keys_def default_def)
lemma keys_map_entry [simp]:
"keys (map_entry k f m) = keys m"
by (auto simp add: keys_def)
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 (simp add: tabulate_def keys_def 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
instantiation mapping :: (type, type) equal
begin
definition [code del]:
"HOL.equal m n \<longleftrightarrow> lookup m = lookup n"
instance proof
qed (simp add: equal_mapping_def mapping_eq_iff)
end
hide_const (open) empty is_empty lookup update delete ordered_keys keys size
replace default map_entry map_default tabulate bulkload map
end