author | haftmann |
Fri, 11 Jun 2010 17:14:02 +0200 | |
changeset 37407 | 61dd8c145da7 |
parent 37299 | 6315d1bd8388 |
child 37700 | bd90378b8171 |
permissions | -rw-r--r-- |
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(* Author: Florian Haftmann, TU Muenchen *) |
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header {* An abstract view on maps for code generation. *} |
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theory Mapping |
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imports Main |
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begin |
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subsection {* Type definition and primitive operations *} |
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datatype ('a, 'b) mapping = Mapping "'a \<rightharpoonup> 'b" |
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definition empty :: "('a, 'b) mapping" where |
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"empty = Mapping (\<lambda>_. None)" |
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primrec lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<rightharpoonup> 'b" where |
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"lookup (Mapping f) = f" |
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primrec update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where |
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"update k v (Mapping f) = Mapping (f (k \<mapsto> v))" |
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primrec delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where |
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"delete k (Mapping f) = Mapping (f (k := None))" |
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subsection {* Derived operations *} |
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definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" where |
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"keys m = dom (lookup m)" |
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definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where |
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"ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])" |
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definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where |
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"is_empty m \<longleftrightarrow> keys m = {}" |
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definition size :: "('a, 'b) mapping \<Rightarrow> nat" where |
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"size m = (if finite (keys m) then card (keys m) else 0)" |
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definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where |
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"replace k v m = (if k \<in> keys m then update k v m else m)" |
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definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where |
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"default k v m = (if k \<in> keys m then m else update k v m)" |
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definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where |
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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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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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definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" where |
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"tabulate ks f = Mapping (map_of (map (\<lambda>k. (k, f k)) ks))" |
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definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" where |
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"bulkload xs = Mapping (\<lambda>k. if k < length xs then Some (xs ! k) else None)" |
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subsection {* Properties *} |
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lemma lookup_inject [simp]: |
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"lookup m = lookup n \<longleftrightarrow> m = n" |
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by (cases m, cases n) simp |
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lemma mapping_eqI: |
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assumes "lookup m = lookup n" |
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shows "m = n" |
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using assms by simp |
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lemma keys_is_none_lookup [code_inline]: |
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"k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))" |
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by (auto simp add: keys_def is_none_def) |
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lemma lookup_empty [simp]: |
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"lookup empty = Map.empty" |
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by (simp add: empty_def) |
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lemma lookup_update [simp]: |
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"lookup (update k v m) = (lookup m) (k \<mapsto> v)" |
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by (cases m) simp |
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lemma lookup_delete [simp]: |
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"lookup (delete k m) = (lookup m) (k := None)" |
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by (cases m) simp |
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lemma lookup_map_entry [simp]: |
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"lookup (map_entry k f m) = (lookup m) (k := Option.map f (lookup m k))" |
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by (cases "lookup m k") (simp_all add: map_entry_def expand_fun_eq) |
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lemma lookup_tabulate [simp]: |
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"lookup (tabulate ks f) = (Some o f) |` set ks" |
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by (induct ks) (auto simp add: tabulate_def restrict_map_def expand_fun_eq) |
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lemma lookup_bulkload [simp]: |
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"lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)" |
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by (simp add: bulkload_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 (rule mapping_eqI, simp add: fun_upd_twist)+ |
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lemma update_delete [simp]: |
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"update k v (delete k m) = update k v m" |
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by (rule mapping_eqI) 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 (rule mapping_eqI, simp add: fun_upd_twist)+ |
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lemma delete_empty [simp]: |
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"delete k empty = empty" |
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by (rule mapping_eqI) simp |
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lemma replace_update: |
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"k \<notin> keys m \<Longrightarrow> replace k v m = m" |
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"k \<in> keys m \<Longrightarrow> replace k v m = update k v m" |
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by (rule mapping_eqI) (auto simp add: replace_def fun_upd_twist)+ |
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lemma size_empty [simp]: |
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"size empty = 0" |
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by (simp add: size_def keys_def) |
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lemma size_update: |
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"finite (keys m) \<Longrightarrow> size (update k v m) = |
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(if k \<in> keys m then size m else Suc (size m))" |
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by (auto simp add: size_def insert_dom keys_def) |
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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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by (simp add: size_def keys_def) |
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lemma size_tabulate [simp]: |
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"size (tabulate ks f) = length (remdups ks)" |
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by (simp add: size_def distinct_card [of "remdups ks", symmetric] comp_def keys_def) |
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lemma bulkload_tabulate: |
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"bulkload xs = tabulate [0..<length xs] (nth xs)" |
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by (rule mapping_eqI) (simp add: expand_fun_eq) |
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lemma is_empty_empty: (*FIXME*) |
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"is_empty m \<longleftrightarrow> m = Mapping Map.empty" |
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by (cases m) (simp add: is_empty_def keys_def) |
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lemma is_empty_empty' [simp]: |
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"is_empty empty" |
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by (simp add: is_empty_empty empty_def) |
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lemma is_empty_update [simp]: |
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"\<not> is_empty (update k v m)" |
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by (cases m) (simp add: is_empty_empty) |
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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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by (cases m) (auto simp add: is_empty_empty keys_def dom_eq_empty_conv [symmetric] 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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by (auto simp add: replace_def) (simp add: is_empty_def) |
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lemma is_empty_default [simp]: |
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"\<not> is_empty (default k v m)" |
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by (auto simp add: default_def) (simp add: is_empty_def) |
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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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by (cases "lookup m k") |
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(auto simp add: map_entry_def, simp add: is_empty_empty) |
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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 (simp add: keys_def) |
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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 (simp add: keys_def) |
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lemma keys_delete [simp]: |
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"keys (delete k m) = keys m - {k}" |
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by (simp add: keys_def) |
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lemma keys_replace [simp]: |
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"keys (replace k v m) = keys m" |
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by (auto simp add: keys_def replace_def) |
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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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by (auto simp add: keys_def default_def) |
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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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by (auto simp add: keys_def) |
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lemma keys_map_default [simp]: |
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"keys (map_default k v f m) = insert k (keys m)" |
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by (simp add: map_default_def) |
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lemma keys_tabulate [simp]: |
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"keys (tabulate ks f) = set ks" |
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by (simp add: tabulate_def keys_def map_of_map_restrict o_def) |
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lemma keys_bulkload [simp]: |
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"keys (bulkload xs) = {0..<length xs}" |
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by (simp add: keys_tabulate bulkload_tabulate) |
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lemma distinct_ordered_keys [simp]: |
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"distinct (ordered_keys m)" |
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by (simp add: ordered_keys_def) |
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lemma ordered_keys_infinite [simp]: |
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"\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []" |
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by (simp add: ordered_keys_def) |
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lemma ordered_keys_empty [simp]: |
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"ordered_keys empty = []" |
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by (simp add: ordered_keys_def) |
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lemma ordered_keys_update [simp]: |
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"k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m" |
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"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)" |
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by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb) |
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lemma ordered_keys_delete [simp]: |
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"ordered_keys (delete k m) = remove1 k (ordered_keys m)" |
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proof (cases "finite (keys m)") |
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case False then show ?thesis by simp |
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next |
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case True note fin = True |
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show ?thesis |
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proof (cases "k \<in> keys m") |
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case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp |
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with False show ?thesis by (simp add: ordered_keys_def remove1_idem) |
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next |
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case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove) |
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qed |
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qed |
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lemma ordered_keys_replace [simp]: |
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"ordered_keys (replace k v m) = ordered_keys m" |
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by (simp add: replace_def) |
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lemma ordered_keys_default [simp]: |
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"k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m" |
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"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)" |
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by (simp_all add: default_def) |
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lemma ordered_keys_map_entry [simp]: |
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"ordered_keys (map_entry k f m) = ordered_keys m" |
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by (simp add: ordered_keys_def) |
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lemma ordered_keys_map_default [simp]: |
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"k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m" |
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"finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)" |
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by (simp_all add: map_default_def) |
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lemma ordered_keys_tabulate [simp]: |
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"ordered_keys (tabulate ks f) = sort (remdups ks)" |
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by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups) |
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lemma ordered_keys_bulkload [simp]: |
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"ordered_keys (bulkload ks) = [0..<length ks]" |
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by (simp add: ordered_keys_def) |
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subsection {* Some technical code lemmas *} |
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lemma [code]: |
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"mapping_case f m = f (Mapping.lookup m)" |
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by (cases m) simp |
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lemma [code]: |
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"mapping_rec f m = f (Mapping.lookup m)" |
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by (cases m) simp |
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lemma [code]: |
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"Nat.size (m :: (_, _) mapping) = 0" |
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by (cases m) simp |
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lemma [code]: |
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"mapping_size f g m = 0" |
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by (cases m) simp |
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hide_const (open) empty is_empty lookup update delete ordered_keys keys size |
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replace default map_entry map_default tabulate bulkload |
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end |