author  haftmann 
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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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header {* An abstract view on maps for code generation. *} 

6 

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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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lemma map_of_foldr:  {* FIXME move *} 
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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 {* 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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.. 

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setup_lifting (no_code) 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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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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subsection {* Functorial structure *} 
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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 {* Derived operations *} 
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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 = {}" 
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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 {* Manual derivation of transfer rule is nontrivial *} 
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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" 

56528  194 
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 {* Properties *} 
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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  227 

35157  228 
lemma update_delete [simp]: 
229 
"update k v (delete k m) = update k v m" 

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by transfer simp 
29708  231 

232 
lemma delete_update: 

233 
"delete k (update k v m) = delete k m" 

234 
"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  236 

35157  237 
lemma delete_empty [simp]: 
238 
"delete k empty = empty" 

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by transfer simp 
29708  240 

35157  241 
lemma replace_update: 
37052  242 
"k \<notin> keys m \<Longrightarrow> replace k v m = m" 
243 
"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  245 

246 
lemma size_empty [simp]: 

247 
"size empty = 0" 

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248 
unfolding size_def by transfer simp 
29708  249 

250 
lemma size_update: 

37052  251 
"finite (keys m) \<Longrightarrow> size (update k v m) = 
252 
(if k \<in> keys m then size m else Suc (size m))" 

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253 
unfolding size_def by transfer (auto simp add: insert_dom) 
29708  254 

255 
lemma size_delete: 

37052  256 
"size (delete k m) = (if k \<in> keys m then size m  1 else size m)" 
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257 
unfolding size_def by transfer simp 
29708  258 

37052  259 
lemma size_tabulate [simp]: 
29708  260 
"size (tabulate ks f) = length (remdups ks)" 
56528  261 
unfolding size_def by transfer (auto simp add: map_of_map_restrict card_set comp_def) 
29708  262 

29831  263 
lemma bulkload_tabulate: 
29826  264 
"bulkload xs = tabulate [0..<length xs] (nth xs)" 
56528  265 
by transfer (auto simp add: map_of_map_restrict) 
29826  266 

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lemma is_empty_empty [simp]: 
37052  268 
"is_empty empty" 
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unfolding is_empty_def by transfer simp 
37052  270 

271 
lemma is_empty_update [simp]: 

272 
"\<not> is_empty (update k v m)" 

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273 
unfolding is_empty_def by transfer simp 
37052  274 

275 
lemma is_empty_delete: 

276 
"is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}" 

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277 
unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv) 
37052  278 

279 
lemma is_empty_replace [simp]: 

280 
"is_empty (replace k v m) \<longleftrightarrow> is_empty m" 

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281 
unfolding is_empty_def replace_def by transfer auto 
37052  282 

283 
lemma is_empty_default [simp]: 

284 
"\<not> is_empty (default k v m)" 

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285 
unfolding is_empty_def default_def by transfer auto 
37052  286 

287 
lemma is_empty_map_entry [simp]: 

288 
"is_empty (map_entry k f m) \<longleftrightarrow> is_empty m" 

56528  289 
unfolding is_empty_def by transfer (auto split: option.split) 
37052  290 

291 
lemma is_empty_map_default [simp]: 

292 
"\<not> is_empty (map_default k v f m)" 

293 
by (simp add: map_default_def) 

294 

56545  295 
lemma keys_dom_lookup: 
296 
"keys m = dom (Mapping.lookup m)" 

297 
by transfer rule 

298 

37052  299 
lemma keys_empty [simp]: 
300 
"keys empty = {}" 

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by transfer simp 
37052  302 

303 
lemma keys_update [simp]: 

304 
"keys (update k v m) = insert k (keys m)" 

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by transfer simp 
37052  306 

307 
lemma keys_delete [simp]: 

308 
"keys (delete k m) = keys m  {k}" 

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by transfer simp 
37052  310 

311 
lemma keys_replace [simp]: 

312 
"keys (replace k v m) = keys m" 

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313 
unfolding replace_def by transfer (simp add: insert_absorb) 
37052  314 

315 
lemma keys_default [simp]: 

316 
"keys (default k v m) = insert k (keys m)" 

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unfolding default_def by transfer (simp add: insert_absorb) 
37052  318 

319 
lemma keys_map_entry [simp]: 

320 
"keys (map_entry k f m) = keys m" 

56528  321 
by transfer (auto split: option.split) 
37052  322 

323 
lemma keys_map_default [simp]: 

324 
"keys (map_default k v f m) = insert k (keys m)" 

325 
by (simp add: map_default_def) 

326 

327 
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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330 

37052  331 
lemma keys_bulkload [simp]: 
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"keys (bulkload xs) = {0..<length xs}" 
56528  333 
by (simp add: bulkload_tabulate) 
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334 

37052  335 
lemma distinct_ordered_keys [simp]: 
336 
"distinct (ordered_keys m)" 

337 
by (simp add: ordered_keys_def) 

338 

339 
lemma ordered_keys_infinite [simp]: 

340 
"\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []" 

341 
by (simp add: ordered_keys_def) 

342 

343 
lemma ordered_keys_empty [simp]: 

344 
"ordered_keys empty = []" 

345 
by (simp add: ordered_keys_def) 

346 

347 
lemma ordered_keys_update [simp]: 

348 
"k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m" 

349 
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)" 

350 
by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb) 

351 

352 
lemma ordered_keys_delete [simp]: 

353 
"ordered_keys (delete k m) = remove1 k (ordered_keys m)" 

354 
proof (cases "finite (keys m)") 

355 
case False then show ?thesis by simp 

356 
next 

357 
case True note fin = True 

358 
show ?thesis 

359 
proof (cases "k \<in> keys m") 

360 
case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp 

361 
with False show ?thesis by (simp add: ordered_keys_def remove1_idem) 

362 
next 

363 
case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove) 

364 
qed 

365 
qed 

366 

367 
lemma ordered_keys_replace [simp]: 

368 
"ordered_keys (replace k v m) = ordered_keys m" 

369 
by (simp add: replace_def) 

370 

371 
lemma ordered_keys_default [simp]: 

372 
"k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m" 

373 
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)" 

374 
by (simp_all add: default_def) 

375 

376 
lemma ordered_keys_map_entry [simp]: 

377 
"ordered_keys (map_entry k f m) = ordered_keys m" 

378 
by (simp add: ordered_keys_def) 

379 

380 
lemma ordered_keys_map_default [simp]: 

381 
"k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m" 

382 
"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)" 

383 
by (simp_all add: map_default_def) 

384 

385 
lemma ordered_keys_tabulate [simp]: 

386 
"ordered_keys (tabulate ks f) = sort (remdups ks)" 

387 
by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups) 

388 

389 
lemma ordered_keys_bulkload [simp]: 

390 
"ordered_keys (bulkload ks) = [0..<length ks]" 

391 
by (simp add: ordered_keys_def) 

36110  392 

56528  393 
lemma tabulate_fold: 
394 
"tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty" 

395 
proof transfer 

396 
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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398 
by (simp add: foldr_map comp_def map_of_foldr) 
56528  399 
also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs" 
400 
by (rule foldr_fold) (simp add: fun_eq_iff) 

401 
ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty" 

402 
by simp 

403 
qed 

404 

31459  405 

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406 
subsection {* Code generator setup *} 
31459  407 

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408 
code_datatype empty update 
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409 

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hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size 
54853  411 
replace default map_entry map_default tabulate bulkload map of_alist 
35157  412 

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413 
end 