author  blanchet 
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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. *} 

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theory Mapping 

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imports Main 
29708  9 
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 ===> rel_option B) Map.empty Map.empty" by transfer_prover 
51379  18 

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lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)" by transfer_prover 

20 

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lemma update_transfer: 

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assumes [transfer_rule]: "bi_unique A" 

55525  23 
shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B) 
51379  24 
(\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))" 
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by transfer_prover 

26 

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lemma delete_transfer: 

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assumes [transfer_rule]: "bi_unique A" 

55525  29 
shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B) 
51379  30 
(\<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" 

34 

55525  35 
lemma [transfer_rule]: "(rel_option A ===> op=) equal_None equal_None" 
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unfolding fun_rel_def rel_option_iff equal_None_def by (auto split: option.split) 

51379  37 

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lemma dom_transfer: 

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assumes [transfer_rule]: "bi_total A" 

55938  40 
shows "((A ===> rel_option B) ===> rel_set A) dom dom" 
51379  41 
unfolding dom_def[abs_def] equal_None_def[symmetric] 
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by transfer_prover 

43 

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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 ===> rel_option R2) map_of map_of" 
51379  47 
unfolding map_of_def by transfer_prover 
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lemma tabulate_transfer: 

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assumes [transfer_rule]: "bi_unique A" 

55525  51 
shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B) 
51379  52 
(\<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= ===> 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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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) 

65 

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lemma map_transfer: 

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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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lemma map_entry_transfer: 

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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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end 
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29708  81 
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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51379  90 
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 . 
29708  100 

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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 . 
29708  103 

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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 . 
29708  106 

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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 . 
29708  109 

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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)+ 
40605  115 

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subsection {* Derived operations *} 
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definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where 
37052  120 
"ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])" 
35194  121 

35157  122 
definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where 
37052  123 
"is_empty m \<longleftrightarrow> keys m = {}" 
35157  124 

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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)" 
35157  127 

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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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54853  145 
lift_definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping" 
51379  146 
is map_of parametric map_of_transfer . 
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54853  148 
lemma of_alist_code [code]: 
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"of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty" 

51379  150 
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" 
51379  171 
by (unfold equal) transfer_prover 
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end 
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29708  175 
subsection {* Properties *} 
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49973  177 
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) 
29708  189 

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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) 
29826  193 

29708  194 
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  198 

35157  199 
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)+ 
29708  207 

35157  208 
lemma delete_empty [simp]: 
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"delete k empty = empty" 

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by transfer simp 
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35157  212 
lemma replace_update: 
37052  213 
"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)+ 
29708  216 

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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 
29708  220 

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lemma size_update: 

37052  222 
"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) 
29708  225 

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lemma size_delete: 

37052  227 
"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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37052  230 
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) 
29708  233 

29831  234 
lemma bulkload_tabulate: 
29826  235 
"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 
37052  253 

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lemma is_empty_default [simp]: 

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

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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) 

294 

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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)" 

305 
by (simp add: ordered_keys_def) 

306 

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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) 

310 

311 
lemma ordered_keys_empty [simp]: 

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"ordered_keys empty = []" 

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by (simp add: ordered_keys_def) 

314 

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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) 

319 

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lemma ordered_keys_delete [simp]: 

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

322 
proof (cases "finite (keys m)") 

323 
case False then show ?thesis by simp 

324 
next 

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case True note fin = True 

326 
show ?thesis 

327 
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 

329 
with False show ?thesis by (simp add: ordered_keys_def remove1_idem) 

330 
next 

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

332 
qed 

333 
qed 

334 

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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) 

338 

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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) 

343 

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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) 

347 

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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) 

352 

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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) 

356 

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lemma ordered_keys_bulkload [simp]: 

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

359 
by (simp add: ordered_keys_def) 

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subsection {* Code generator setup *} 
31459  363 

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code_datatype empty update 
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365 

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hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size 
54853  367 
replace default map_entry map_default tabulate bulkload map of_alist 
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end 