Theory Mapping

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theory Mapping
imports Main

(* 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 *}

datatype ('a, 'b) mapping = Mapping "'a \<rightharpoonup> 'b"

definition empty :: "('a, 'b) mapping" where
"empty = Mapping (λ_. None)"


primrec lookup :: "('a, 'b) mapping => 'a \<rightharpoonup> 'b" where
"lookup (Mapping f) = f"


primrec update :: "'a => 'b => ('a, 'b) mapping => ('a, 'b) mapping" where
"update k v (Mapping f) = Mapping (f (k \<mapsto> v))"


primrec delete :: "'a => ('a, 'b) mapping => ('a, 'b) mapping" where
"delete k (Mapping f) = Mapping (f (k := None))"



subsection {* Derived operations *}

definition keys :: "('a, 'b) mapping => 'a set" where
"keys m = dom (lookup m)"


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)"


definition map_entry :: "'a => ('b => 'b) => ('a, 'b) mapping => ('a, 'b) mapping" where
"map_entry k f m = (case lookup m k of None => m
| Some v => update k (f v) m)"


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)"


definition tabulate :: "'a list => ('a => 'b) => ('a, 'b) mapping" where
"tabulate ks f = Mapping (map_of (map (λk. (k, f k)) ks))"


definition bulkload :: "'a list => (nat, 'a) mapping" where
"bulkload xs = Mapping (λk. if k < length xs then Some (xs ! k) else None)"



subsection {* Properties *}

lemma lookup_inject [simp]:
"lookup m = lookup n <-> m = n"

by (cases m, cases n) simp

lemma mapping_eqI:
assumes "lookup m = lookup n"
shows "m = n"

using assms by simp

lemma keys_is_none_lookup [code_inline]:
"k ∈ keys m <-> ¬ (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 (cases m) simp

lemma lookup_delete [simp]:
"lookup (delete k m) = (lookup m) (k := None)"

by (cases m) simp

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 expand_fun_eq)

lemma lookup_tabulate [simp]:
"lookup (tabulate ks f) = (Some o f) |` set ks"

by (induct ks) (auto simp add: tabulate_def restrict_map_def expand_fun_eq)

lemma lookup_bulkload [simp]:
"lookup (bulkload xs) = (λk. 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 ≠ l ==> 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 ≠ l ==> 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 ∉ keys m ==> replace k v m = m"
"k ∈ keys m ==> 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) ==> size (update k v m) =
(if k ∈ 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 ∈ 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: expand_fun_eq)

lemma is_empty_empty: (*FIXME*)
"is_empty m <-> 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]:
"¬ is_empty (update k v m)"

by (cases m) (simp add: is_empty_empty)

lemma is_empty_delete:
"is_empty (delete k m) <-> is_empty m ∨ keys m = {k}"

by (cases m) (auto simp add: is_empty_empty keys_def dom_eq_empty_conv [symmetric] simp del: dom_eq_empty_conv)

lemma is_empty_replace [simp]:
"is_empty (replace k v m) <-> is_empty m"

by (auto simp add: replace_def) (simp add: is_empty_def)

lemma is_empty_default [simp]:
"¬ 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) <-> 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]:
"¬ 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]:
"¬ 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 {* Some technical code lemmas *}

lemma [code]:
"mapping_case f m = f (Mapping.lookup m)"

by (cases m) simp

lemma [code]:
"mapping_rec f m = f (Mapping.lookup m)"

by (cases m) simp

lemma [code]:
"Nat.size (m :: (_, _) mapping) = 0"

by (cases m) simp

lemma [code]:
"mapping_size f g m = 0"

by (cases m) simp


hide_const (open) empty is_empty lookup update delete ordered_keys keys size
replace default map_entry map_default tabulate bulkload


end