author | haftmann |
Sat, 19 Dec 2015 17:03:17 +0100 | |
changeset 61891 | 76189756ff65 |
parent 61585 | a9599d3d7610 |
child 63194 | 0b7bdb75f451 |
permissions | -rw-r--r-- |
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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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section \<open>An abstract view on maps for code generation.\<close> |
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theory Mapping |
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imports Main |
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begin |
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subsection \<open>Parametricity transfer rules\<close> |
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lemma map_of_foldr: \<comment> \<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: Option.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] Option.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> \<comment> \<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::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 \<open>Manual derivation of transfer rule is non-trivial\<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 |
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by standard (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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|
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subsection \<open>Properties\<close> |
29708 | 211 |
|
56528 | 212 |
lemma lookup_update: |
213 |
"lookup (update k v m) k = Some v" |
|
49973 | 214 |
by transfer simp |
215 |
||
56528 | 216 |
lemma lookup_update_neq: |
217 |
"k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'" |
|
49973 | 218 |
by transfer simp |
219 |
||
56528 | 220 |
lemma lookup_empty: |
221 |
"lookup empty k = None" |
|
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by transfer simp |
223 |
||
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lemma keys_is_none_rep [code_unfold]: |
37052 | 225 |
"k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))" |
61068 | 226 |
by transfer (auto simp add: Option.is_none_def) |
29708 | 227 |
|
228 |
lemma update_update: |
|
229 |
"update k v (update k w m) = update k v m" |
|
230 |
"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 |
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|
237 |
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 | 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: |
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"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)+ |
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|
251 |
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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|
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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unfolding size_def by transfer (auto simp add: insert_dom) |
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|
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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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|
37052 | 264 |
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: 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)" |
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by transfer (auto simp add: map_of_map_restrict) |
29826 | 271 |
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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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|
276 |
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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|
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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) |
37052 | 283 |
|
284 |
lemma is_empty_replace [simp]: |
|
285 |
"is_empty (replace k v m) \<longleftrightarrow> is_empty m" |
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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 |
||
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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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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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|
333 |
"keys (tabulate ks f) = set ks" |
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|
334 |
by transfer (simp add: map_of_map_restrict o_def) |
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|
335 |
|
37052 | 336 |
lemma keys_bulkload [simp]: |
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|
337 |
"keys (bulkload xs) = {0..<length xs}" |
56528 | 338 |
by (simp add: bulkload_tabulate) |
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|
339 |
|
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 |
|
56529
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haftmann
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56528
diff
changeset
|
402 |
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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changeset
|
413 |
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 |
|
49975
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huffman
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changeset
|
416 |
end |
59485 | 417 |