Theory Mapping

theory Mapping
imports Main
(*  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 ===> option_rel B) Map.empty Map.empty" by transfer_prover

lemma lookup_transfer: "((A ===> B) ===> A ===> B) (λm k. m k) (λm k. m k)" by transfer_prover

lemma update_transfer:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> B ===> (A ===> option_rel B) ===> A ===> option_rel B)
(λk v m. m(k \<mapsto> v)) (λk v m. m(k \<mapsto> v))"

by transfer_prover

lemma delete_transfer:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> (A ===> option_rel B) ===> A ===> option_rel B)
(λk m. m(k := None)) (λk m. m(k := None))"

by transfer_prover

definition equal_None :: "'a option => bool" where "equal_None x ≡ x = None"

lemma [transfer_rule]: "(option_rel A ===> op=) equal_None equal_None"
unfolding fun_rel_def option_rel_def equal_None_def by (auto split: option.split)

lemma dom_transfer:
assumes [transfer_rule]: "bi_total A"
shows "((A ===> option_rel B) ===> set_rel 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 (prod_rel R1 R2) ===> R1 ===> option_rel 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 ===> option_rel B)
(λks f. (map_of (List.map (λk. (k, f k)) ks))) (λks f. (map_of (List.map (λk. (k, f k)) ks)))"

by transfer_prover

lemma bulkload_transfer:
"(list_all2 A ===> op= ===> option_rel A)
(λxs k. if k < length xs then Some (xs ! k) else None) (λxs k. if k < length xs then Some (xs ! k) else None)"

unfolding fun_rel_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 ===> option_rel C) ===> A ===> option_rel D)
(λf g m. (Option.map g o m o f)) (λf g m. (Option.map g o m o f))"

by transfer_prover

lemma map_entry_transfer:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> (B ===> B) ===> (A ===> option_rel B) ===> A ===> option_rel B)
(λk f m. (case m k of None => m
| Some v => m (k \<mapsto> (f v)))) (λk f m. (case m k of None => m
| Some v => 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 => 'a => 'b option" is "λm k. m k"
parametric lookup_transfer .

lift_definition update :: "'a => 'b => ('a, 'b) mapping => ('a, 'b) mapping" is "λk v m. m(k \<mapsto> v)"
parametric update_transfer .

lift_definition delete :: "'a => ('a, 'b) mapping => ('a, 'b) mapping" is "λk m. m(k := None)"
parametric delete_transfer .

lift_definition keys :: "('a, 'b) mapping => 'a set" is dom parametric dom_transfer .

lift_definition tabulate :: "'a list => ('a => 'b) => ('a, 'b) mapping" is
"λks f. (map_of (List.map (λk. (k, f k)) ks))" parametric tabulate_transfer .

lift_definition bulkload :: "'a list => (nat, 'a) mapping" is
"λxs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_transfer .

lift_definition map :: "('c => 'a) => ('b => 'd) => ('a, 'b) mapping => ('c, 'd) mapping" is
"λf g m. (Option.map g o m o f)" parametric map_transfer .


subsection {* Functorial structure *}

enriched_type map: map
by (transfer, auto simp add: fun_eq_iff Option.map.compositionality Option.map.id)+


subsection {* Derived operations *}

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)"

lift_definition map_entry :: "'a => ('b => 'b) => ('a, 'b) mapping => ('a, 'b) mapping" is
"λk f m. (case m k of None => m
| Some v => 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 => m
| Some v => update k (f v) m)"

by transfer rule

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)"

lift_definition assoc_list_to_mapping :: "('k × 'v) list => ('k, 'v) mapping"
is map_of parametric map_of_transfer .

lemma assoc_list_to_mapping_code [code]:
"assoc_list_to_mapping xs = foldr (λ(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 <-> (∀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 ≠ k' ==> 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 ∈ keys m <-> ¬ (Option.is_none (lookup m k))"
by transfer (auto simp add: is_none_def)

lemma tabulate_alt_def:
"map_of (List.map (λk. (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 ≠ l ==> 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 ≠ l ==> 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 ∉ keys m ==> replace k v m = m"
"k ∈ keys m ==> 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) ==> size (update k v m) =
(if k ∈ 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 ∈ 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]:
"¬ is_empty (update k v m)"
unfolding is_empty_def by transfer simp

lemma is_empty_delete:
"is_empty (delete k m) <-> is_empty m ∨ 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) <-> is_empty m"
unfolding is_empty_def replace_def by transfer auto

lemma is_empty_default [simp]:
"¬ 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) <-> is_empty m"
unfolding is_empty_def
apply transfer by (case_tac "m k") auto

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 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]:
"¬ 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 {* 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

end