(*  Title:      HOL/Library/Mapping.thy

    Author:     Florian Haftmann and Ondrej Kuncar

*)

section \<open>An abstract view on maps for code generation.\<close>

theory Mapping

imports Main

begin

subsection \<open>Parametricity transfer rules\<close>

lemma map_of_foldr: -- \<open>FIXME move\<close>

  "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 \<open>Type definition and primitive operations\<close>

typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"

  morphisms rep Mapping

  ..

setup_lifting 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 .

declare [[code drop: Mapping.lookup]]

setup \<open>Code.add_default_eqn @{thm Mapping.lookup.abs_eq}\<close> -- \<open>FIXME lifting\<close>

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 .

declare [[code drop: map]]

subsection \<open>Functorial structure\<close>

functor map: map

  by (transfer, auto simp add: fun_eq_iff option.map_comp option.map_id)+

subsection \<open>Derived operations\<close>

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 \<open>Manual derivation of transfer rule is non-trivial\<close>

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

  by standard (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 \<open>Properties\<close>

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 \<open>Code generator setup\<close>

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