src/HOL/Library/Mapping.thy
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(*  Title:      HOL/Library/Mapping.thy
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    Author:     Florian Haftmann and Ondrej Kuncar
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*)
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header {* An abstract view on maps for code generation. *}
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theory Mapping
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imports Main
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begin
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subsection {* Parametricity transfer rules *}
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context
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begin
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interpretation lifting_syntax .
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lemma empty_transfer: "(A ===> option_rel B) Map.empty Map.empty" by transfer_prover
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lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)" by transfer_prover
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lemma update_transfer:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "(A ===> B ===> (A ===> option_rel B) ===> A ===> option_rel B) 
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          (\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
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by transfer_prover
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lemma delete_transfer:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "(A ===> (A ===> option_rel B) ===> A ===> option_rel B) 
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          (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
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by transfer_prover
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definition equal_None :: "'a option \<Rightarrow> bool" where "equal_None x \<equiv> x = None"
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lemma [transfer_rule]: "(option_rel A ===> op=) equal_None equal_None" 
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unfolding fun_rel_def option_rel_def equal_None_def by (auto split: option.split)
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lemma dom_transfer:
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  assumes [transfer_rule]: "bi_total A"
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  shows "((A ===> option_rel B) ===> set_rel A) dom dom" 
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unfolding dom_def[abs_def] equal_None_def[symmetric] 
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by transfer_prover
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lemma map_of_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique R1"
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  shows "(list_all2 (prod_rel R1 R2) ===> R1 ===> option_rel R2) map_of map_of"
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unfolding map_of_def by transfer_prover
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lemma tabulate_transfer: 
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  assumes [transfer_rule]: "bi_unique A"
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  shows "(list_all2 A ===> (A ===> B) ===> A ===> option_rel B) 
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    (\<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)))"
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by transfer_prover
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lemma bulkload_transfer: 
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  "(list_all2 A ===> op= ===> option_rel A) 
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    (\<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)"
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unfolding fun_rel_def 
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apply clarsimp 
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apply (erule list_all2_induct) 
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  apply simp 
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apply (case_tac xa) 
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  apply simp 
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by (auto dest: list_all2_lengthD list_all2_nthD)
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lemma map_transfer: 
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  "((A ===> B) ===> (C ===> D) ===> (B ===> option_rel C) ===> A ===> option_rel D) 
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    (\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
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by transfer_prover
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lemma map_entry_transfer:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "(A ===> (B ===> B) ===> (A ===> option_rel B) ===> A ===> option_rel B) 
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    (\<lambda>k f m. (case m k of None \<Rightarrow> m
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      | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
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      | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
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by transfer_prover
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end
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subsection {* Type definition and primitive operations *}
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typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
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  morphisms rep Mapping ..
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setup_lifting(no_code) type_definition_mapping
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lift_definition empty :: "('a, 'b) mapping" is Map.empty parametric empty_transfer .
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lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option" is "\<lambda>m k. m k" 
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  parametric lookup_transfer .
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lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k v m. m(k \<mapsto> v)" 
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  parametric update_transfer .
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lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k m. m(k := None)" 
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  parametric delete_transfer .
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lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" is dom parametric dom_transfer .
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lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" is
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  "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_transfer .
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lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" is
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  "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_transfer .
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lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping" is
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  "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_transfer .
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subsection {* Functorial structure *}
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enriched_type map: map
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  by (transfer, auto simp add: fun_eq_iff option.map_comp option.map_id)+
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subsection {* Derived operations *}
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definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where
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  "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
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definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where
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  "is_empty m \<longleftrightarrow> keys m = {}"
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definition size :: "('a, 'b) mapping \<Rightarrow> nat" where
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  "size m = (if finite (keys m) then card (keys m) else 0)"
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definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
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  "replace k v m = (if k \<in> keys m then update k v m else m)"
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definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
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  "default k v m = (if k \<in> keys m then m else update k v m)"
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lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
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  "\<lambda>k f m. (case m k of None \<Rightarrow> m
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    | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_transfer .
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lemma map_entry_code [code]: "map_entry k f m = (case lookup m k of None \<Rightarrow> m
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    | Some v \<Rightarrow> update k (f v) m)"
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  by transfer rule
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definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
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  "map_default k v f m = map_entry k f (default k v m)" 
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lift_definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
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is map_of parametric map_of_transfer .
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lemma of_alist_code [code]:
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  "of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
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by transfer(simp add: map_add_map_of_foldr[symmetric])
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instantiation mapping :: (type, type) equal
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begin
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definition
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  "HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. lookup m1 k = lookup m2 k)"
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instance proof
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qed (unfold equal_mapping_def, transfer, auto)
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end
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context
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begin
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interpretation lifting_syntax .
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lemma [transfer_rule]:
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  assumes [transfer_rule]: "bi_total A"
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  assumes [transfer_rule]: "bi_unique B"
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  shows  "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
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by (unfold equal) transfer_prover
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end
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subsection {* Properties *}
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lemma lookup_update: "lookup (update k v m) k = Some v" 
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  by transfer simp
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lemma lookup_update_neq: "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'" 
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  by transfer simp
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lemma lookup_empty: "lookup empty k = None" 
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  by transfer simp
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lemma keys_is_none_rep [code_unfold]:
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  "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
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  by transfer (auto simp add: is_none_def)
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lemma tabulate_alt_def:
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  "map_of (List.map (\<lambda>k. (k, f k)) ks) = (Some o f) |` set ks"
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  by (induct ks) (auto simp add: tabulate_def restrict_map_def)
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lemma update_update:
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  "update k v (update k w m) = update k v m"
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  "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
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  by (transfer, simp add: fun_upd_twist)+
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lemma update_delete [simp]:
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  "update k v (delete k m) = update k v m"
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  by transfer simp
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lemma delete_update:
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  "delete k (update k v m) = delete k m"
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  "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
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  by (transfer, simp add: fun_upd_twist)+
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lemma delete_empty [simp]:
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  "delete k empty = empty"
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  by transfer simp
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lemma replace_update:
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  "k \<notin> keys m \<Longrightarrow> replace k v m = m"
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  "k \<in> keys m \<Longrightarrow> replace k v m = update k v m"
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  by (transfer, auto simp add: replace_def fun_upd_twist)+
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lemma size_empty [simp]:
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  "size empty = 0"
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  unfolding size_def by transfer simp
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lemma size_update:
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  "finite (keys m) \<Longrightarrow> size (update k v m) =
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    (if k \<in> keys m then size m else Suc (size m))"
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  unfolding size_def by transfer (auto simp add: insert_dom)
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lemma size_delete:
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  "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
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  unfolding size_def by transfer simp
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lemma size_tabulate [simp]:
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  "size (tabulate ks f) = length (remdups ks)"
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  unfolding size_def by transfer (auto simp add: tabulate_alt_def card_set comp_def)
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lemma bulkload_tabulate:
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  "bulkload xs = tabulate [0..<length xs] (nth xs)"
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  by transfer (auto simp add: tabulate_alt_def)
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lemma is_empty_empty [simp]:
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  "is_empty empty"
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  unfolding is_empty_def by transfer simp 
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lemma is_empty_update [simp]:
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  "\<not> is_empty (update k v m)"
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  unfolding is_empty_def by transfer simp
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lemma is_empty_delete:
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  "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
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  unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)
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lemma is_empty_replace [simp]:
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  "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
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  unfolding is_empty_def replace_def by transfer auto
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lemma is_empty_default [simp]:
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  "\<not> is_empty (default k v m)"
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  unfolding is_empty_def default_def by transfer auto
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lemma is_empty_map_entry [simp]:
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  "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
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  unfolding is_empty_def 
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  apply transfer by (case_tac "m k") auto
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lemma is_empty_map_default [simp]:
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  "\<not> is_empty (map_default k v f m)"
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  by (simp add: map_default_def)
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lemma keys_empty [simp]:
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  "keys empty = {}"
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  by transfer simp
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lemma keys_update [simp]:
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  "keys (update k v m) = insert k (keys m)"
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  by transfer simp
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lemma keys_delete [simp]:
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  "keys (delete k m) = keys m - {k}"
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  by transfer simp
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lemma keys_replace [simp]:
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  "keys (replace k v m) = keys m"
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  unfolding replace_def by transfer (simp add: insert_absorb)
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lemma keys_default [simp]:
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  "keys (default k v m) = insert k (keys m)"
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  unfolding default_def by transfer (simp add: insert_absorb)
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lemma keys_map_entry [simp]:
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  "keys (map_entry k f m) = keys m"
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  apply transfer by (case_tac "m k") auto
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lemma keys_map_default [simp]:
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  "keys (map_default k v f m) = insert k (keys m)"
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  by (simp add: map_default_def)
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lemma keys_tabulate [simp]:
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  "keys (tabulate ks f) = set ks"
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  by transfer (simp add: map_of_map_restrict o_def)
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lemma keys_bulkload [simp]:
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  "keys (bulkload xs) = {0..<length xs}"
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  by (simp add: keys_tabulate bulkload_tabulate)
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lemma distinct_ordered_keys [simp]:
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  "distinct (ordered_keys m)"
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  by (simp add: ordered_keys_def)
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lemma ordered_keys_infinite [simp]:
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  "\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []"
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  by (simp add: ordered_keys_def)
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lemma ordered_keys_empty [simp]:
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  "ordered_keys empty = []"
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  by (simp add: ordered_keys_def)
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lemma ordered_keys_update [simp]:
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  "k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m"
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  "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)"
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  by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
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lemma ordered_keys_delete [simp]:
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  "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
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proof (cases "finite (keys m)")
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  case False then show ?thesis by simp
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next
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  case True note fin = True
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  show ?thesis
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  proof (cases "k \<in> keys m")
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    case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp
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    with False show ?thesis by (simp add: ordered_keys_def remove1_idem)
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  next
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    case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove)
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  qed
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qed
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lemma ordered_keys_replace [simp]:
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  "ordered_keys (replace k v m) = ordered_keys m"
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  by (simp add: replace_def)
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lemma ordered_keys_default [simp]:
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  "k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m"
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  "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)"
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  by (simp_all add: default_def)
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lemma ordered_keys_map_entry [simp]:
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  "ordered_keys (map_entry k f m) = ordered_keys m"
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  by (simp add: ordered_keys_def)
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lemma ordered_keys_map_default [simp]:
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  "k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m"
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  "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
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  by (simp_all add: map_default_def)
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lemma ordered_keys_tabulate [simp]:
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  "ordered_keys (tabulate ks f) = sort (remdups ks)"
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  by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)
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lemma ordered_keys_bulkload [simp]:
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  "ordered_keys (bulkload ks) = [0..<length ks]"
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  by (simp add: ordered_keys_def)
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subsection {* Code generator setup *}
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code_datatype empty update
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hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size
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  replace default map_entry map_default tabulate bulkload map of_alist
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end