author  wenzelm 
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parent 59485  792272e6ee6b 
child 60679  ade12ef2773c 
permissions  rwrr 
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(* Title: HOL/Library/Mapping.thy 
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Author: Florian Haftmann and Ondrej Kuncar 
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*) 
29708  4 

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section \<open>An abstract view on maps for code generation.\<close> 
29708  6 

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theory Mapping 

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imports Main 
29708  9 
begin 
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subsection \<open>Parametricity transfer rules\<close> 
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lemma map_of_foldr:  \<open>FIXME move\<close> 
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"map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty" 
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using map_add_map_of_foldr [of Map.empty] by auto 
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context 
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begin 
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interpretation lifting_syntax . 
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lemma empty_parametric: 
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"(A ===> rel_option B) Map.empty Map.empty" 
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by transfer_prover 

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lemma lookup_parametric: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)" 
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by transfer_prover 
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lemma update_parametric: 
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assumes [transfer_rule]: "bi_unique A" 
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shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option 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_parametric: 
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assumes [transfer_rule]: "bi_unique A" 
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shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option 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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lemma is_none_parametric [transfer_rule]: 
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"(rel_option A ===> HOL.eq) Option.is_none Option.is_none" 

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by (auto simp add: is_none_def rel_fun_def rel_option_iff split: option.split) 

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lemma dom_parametric: 
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assumes [transfer_rule]: "bi_total A" 
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shows "((A ===> rel_option B) ===> rel_set A) dom dom" 
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unfolding dom_def [abs_def] is_none_def [symmetric] by transfer_prover 
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lemma map_of_parametric [transfer_rule]: 
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assumes [transfer_rule]: "bi_unique R1" 
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shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of" 
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unfolding map_of_def by transfer_prover 
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lemma map_entry_parametric [transfer_rule]: 
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assumes [transfer_rule]: "bi_unique A" 
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shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option 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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lemma tabulate_parametric: 
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assumes [transfer_rule]: "bi_unique A" 
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shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B) 
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(\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))" 
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by transfer_prover 

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lemma bulkload_parametric: 
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"(list_all2 A ===> HOL.eq ===> rel_option 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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proof 
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fix xs ys 

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assume "list_all2 A xs ys" 

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then show "(HOL.eq ===> rel_option A) 

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(\<lambda>k. if k < length xs then Some (xs ! k) else None) 

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(\<lambda>k. if k < length ys then Some (ys ! k) else None)" 

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apply induct 

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apply auto 

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unfolding rel_fun_def 

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apply clarsimp 

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apply (case_tac xa) 

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apply (auto dest: list_all2_lengthD list_all2_nthD) 

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done 

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qed 

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lemma map_parametric: 
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"((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option 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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end 
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subsection \<open>Type definition and primitive operations\<close> 
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typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set" 
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morphisms rep Mapping 
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.. 

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setup_lifting type_definition_mapping 
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lift_definition empty :: "('a, 'b) mapping" 
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is Map.empty parametric empty_parametric . 
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lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option" 
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is "\<lambda>m k. m k" parametric lookup_parametric . 
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declare [[code drop: Mapping.lookup]] 
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setup \<open>Code.add_default_eqn @{thm Mapping.lookup.abs_eq}\<close>  \<open>FIXME lifting\<close> 

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lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" 
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is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_parametric . 
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lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" 
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is "\<lambda>k m. m(k := None)" parametric delete_parametric . 
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lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" 
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is dom parametric dom_parametric . 
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lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" 
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is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_parametric . 
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lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" 
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is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_parametric . 
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lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping" 
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is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric . 
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declare [[code drop: map]] 
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subsection \<open>Functorial structure\<close> 
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functor 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 \<open>Derived operations\<close> 
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definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" 
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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" 
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where 

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"is_empty m \<longleftrightarrow> keys m = {}" 
35157  148 

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definition size :: "('a, 'b) mapping \<Rightarrow> nat" 
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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" 
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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" 
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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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text \<open>Manual derivation of transfer rule is nontrivial\<close> 
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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_parametric . 
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lemma map_entry_code [code]: 
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"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" 
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where 

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"map_default k v f m = map_entry k f (default k v m)" 
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definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping" 
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where 
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"of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty" 
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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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lemma of_alist_transfer [transfer_rule]: 
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assumes [transfer_rule]: "bi_unique R1" 
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shows "(list_all2 (rel_prod R1 R2) ===> pcr_mapping R1 R2) map_of of_alist" 
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unfolding of_alist_def [abs_def] map_of_foldr [abs_def] by transfer_prover 
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end 
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subsection \<open>Properties\<close> 
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lemma lookup_update: 
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"lookup (update k v m) k = Some v" 

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by transfer simp 
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lemma lookup_update_neq: 
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"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: 
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"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 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)+ 
29708  232 

35157  233 
lemma update_delete [simp]: 
234 
"update k v (delete k m) = update k v m" 

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by transfer simp 
29708  236 

237 
lemma delete_update: 

238 
"delete k (update k v m) = delete k m" 

239 
"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)+ 
29708  241 

35157  242 
lemma delete_empty [simp]: 
243 
"delete k empty = empty" 

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by transfer simp 
29708  245 

35157  246 
lemma replace_update: 
37052  247 
"k \<notin> keys m \<Longrightarrow> replace k v m = m" 
248 
"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)+ 
29708  250 

251 
lemma size_empty [simp]: 

252 
"size empty = 0" 

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253 
unfolding size_def by transfer simp 
29708  254 

255 
lemma size_update: 

37052  256 
"finite (keys m) \<Longrightarrow> size (update k v m) = 
257 
(if k \<in> keys m then size m else Suc (size m))" 

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258 
unfolding size_def by transfer (auto simp add: insert_dom) 
29708  259 

260 
lemma size_delete: 

37052  261 
"size (delete k m) = (if k \<in> keys m then size m  1 else size m)" 
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262 
unfolding size_def by transfer simp 
29708  263 

37052  264 
lemma size_tabulate [simp]: 
29708  265 
"size (tabulate ks f) = length (remdups ks)" 
56528  266 
unfolding size_def by transfer (auto simp add: map_of_map_restrict card_set comp_def) 
29708  267 

29831  268 
lemma bulkload_tabulate: 
29826  269 
"bulkload xs = tabulate [0..<length xs] (nth xs)" 
56528  270 
by transfer (auto simp add: map_of_map_restrict) 
29826  271 

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lemma is_empty_empty [simp]: 
37052  273 
"is_empty empty" 
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unfolding is_empty_def by transfer simp 
37052  275 

276 
lemma is_empty_update [simp]: 

277 
"\<not> is_empty (update k v m)" 

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278 
unfolding is_empty_def by transfer simp 
37052  279 

280 
lemma is_empty_delete: 

281 
"is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}" 

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282 
unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv) 
37052  283 

284 
lemma is_empty_replace [simp]: 

285 
"is_empty (replace k v m) \<longleftrightarrow> is_empty m" 

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286 
unfolding is_empty_def replace_def by transfer auto 
37052  287 

288 
lemma is_empty_default [simp]: 

289 
"\<not> is_empty (default k v m)" 

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unfolding is_empty_def default_def by transfer auto 
37052  291 

292 
lemma is_empty_map_entry [simp]: 

293 
"is_empty (map_entry k f m) \<longleftrightarrow> is_empty m" 

56528  294 
unfolding is_empty_def by transfer (auto split: option.split) 
37052  295 

296 
lemma is_empty_map_default [simp]: 

297 
"\<not> is_empty (map_default k v f m)" 

298 
by (simp add: map_default_def) 

299 

56545  300 
lemma keys_dom_lookup: 
301 
"keys m = dom (Mapping.lookup m)" 

302 
by transfer rule 

303 

37052  304 
lemma keys_empty [simp]: 
305 
"keys empty = {}" 

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by transfer simp 
37052  307 

308 
lemma keys_update [simp]: 

309 
"keys (update k v m) = insert k (keys m)" 

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by transfer simp 
37052  311 

312 
lemma keys_delete [simp]: 

313 
"keys (delete k m) = keys m  {k}" 

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by transfer simp 
37052  315 

316 
lemma keys_replace [simp]: 

317 
"keys (replace k v m) = keys m" 

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318 
unfolding replace_def by transfer (simp add: insert_absorb) 
37052  319 

320 
lemma keys_default [simp]: 

321 
"keys (default k v m) = insert k (keys m)" 

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322 
unfolding default_def by transfer (simp add: insert_absorb) 
37052  323 

324 
lemma keys_map_entry [simp]: 

325 
"keys (map_entry k f m) = keys m" 

56528  326 
by transfer (auto split: option.split) 
37052  327 

328 
lemma keys_map_default [simp]: 

329 
"keys (map_default k v f m) = insert k (keys m)" 

330 
by (simp add: map_default_def) 

331 

332 
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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37052  336 
lemma keys_bulkload [simp]: 
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"keys (bulkload xs) = {0..<length xs}" 
56528  338 
by (simp add: bulkload_tabulate) 
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37052  340 
lemma distinct_ordered_keys [simp]: 
341 
"distinct (ordered_keys m)" 

342 
by (simp add: ordered_keys_def) 

343 

344 
lemma ordered_keys_infinite [simp]: 

345 
"\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []" 

346 
by (simp add: ordered_keys_def) 

347 

348 
lemma ordered_keys_empty [simp]: 

349 
"ordered_keys empty = []" 

350 
by (simp add: ordered_keys_def) 

351 

352 
lemma ordered_keys_update [simp]: 

353 
"k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m" 

354 
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)" 

355 
by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb) 

356 

357 
lemma ordered_keys_delete [simp]: 

358 
"ordered_keys (delete k m) = remove1 k (ordered_keys m)" 

359 
proof (cases "finite (keys m)") 

360 
case False then show ?thesis by simp 

361 
next 

362 
case True note fin = True 

363 
show ?thesis 

364 
proof (cases "k \<in> keys m") 

365 
case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp 

366 
with False show ?thesis by (simp add: ordered_keys_def remove1_idem) 

367 
next 

368 
case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove) 

369 
qed 

370 
qed 

371 

372 
lemma ordered_keys_replace [simp]: 

373 
"ordered_keys (replace k v m) = ordered_keys m" 

374 
by (simp add: replace_def) 

375 

376 
lemma ordered_keys_default [simp]: 

377 
"k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m" 

378 
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)" 

379 
by (simp_all add: default_def) 

380 

381 
lemma ordered_keys_map_entry [simp]: 

382 
"ordered_keys (map_entry k f m) = ordered_keys m" 

383 
by (simp add: ordered_keys_def) 

384 

385 
lemma ordered_keys_map_default [simp]: 

386 
"k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m" 

387 
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)" 

388 
by (simp_all add: map_default_def) 

389 

390 
lemma ordered_keys_tabulate [simp]: 

391 
"ordered_keys (tabulate ks f) = sort (remdups ks)" 

392 
by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups) 

393 

394 
lemma ordered_keys_bulkload [simp]: 

395 
"ordered_keys (bulkload ks) = [0..<length ks]" 

396 
by (simp add: ordered_keys_def) 

36110  397 

56528  398 
lemma tabulate_fold: 
399 
"tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty" 

400 
proof transfer 

401 
fix f :: "'a \<Rightarrow> 'b" and xs 

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have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty" 
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403 
by (simp add: foldr_map comp_def map_of_foldr) 
56528  404 
also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs" 
405 
by (rule foldr_fold) (simp add: fun_eq_iff) 

406 
ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty" 

407 
by simp 

408 
qed 

409 

31459  410 

60500  411 
subsection \<open>Code generator setup\<close> 
31459  412 

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hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size 
54853  414 
replace default map_entry map_default tabulate bulkload map of_alist 
35157  415 

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end 
59485  417 