# HG changeset patch
# User haftmann
# Date 1266396532 -3600
# Node ID 73cd6f78c86d896c974c6c90258f9b537204db46
# Parent  37872c68a38552612d4bd903c7ba84b0bda16be5
more close integration with theory Map

diff -r 37872c68a385 -r 73cd6f78c86d src/HOL/Library/Mapping.thy
--- a/src/HOL/Library/Mapping.thy	Wed Feb 17 09:48:52 2010 +0100
+++ b/src/HOL/Library/Mapping.thy	Wed Feb 17 09:48:52 2010 +0100
@@ -3,50 +3,58 @@
 header {* An abstract view on maps for code generation. *}
 
 theory Mapping
-imports Map Main
+imports Main
 begin
 
 subsection {* Type definition and primitive operations *}
 
-datatype ('a, 'b) map = Map "'a \<rightharpoonup> 'b"
+datatype ('a, 'b) mapping = Mapping "'a \<rightharpoonup> 'b"
 
-definition empty :: "('a, 'b) map" where
-  "empty = Map (\<lambda>_. None)"
-
-primrec lookup :: "('a, 'b) map \<Rightarrow> 'a \<rightharpoonup> 'b" where
-  "lookup (Map f) = f"
+definition empty :: "('a, 'b) mapping" where
+  "empty = Mapping (\<lambda>_. None)"
 
-primrec update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) map \<Rightarrow> ('a, 'b) map" where
-  "update k v (Map f) = Map (f (k \<mapsto> v))"
+primrec lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<rightharpoonup> 'b" where
+  "lookup (Mapping f) = f"
 
-primrec delete :: "'a \<Rightarrow> ('a, 'b) map \<Rightarrow> ('a, 'b) map" where
-  "delete k (Map f) = Map (f (k := None))"
+primrec update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
+  "update k v (Mapping f) = Mapping (f (k \<mapsto> v))"
 
-primrec keys :: "('a, 'b) map \<Rightarrow> 'a set" where
-  "keys (Map f) = dom f"
+primrec delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
+  "delete k (Mapping f) = Mapping (f (k := None))"
 
 
 subsection {* Derived operations *}
 
-definition size :: "('a, 'b) map \<Rightarrow> nat" where
-  "size m = (if finite (keys m) then card (keys m) else 0)"
+definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" where
+  "keys m = dom (lookup m)"
 
-definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) map \<Rightarrow> ('a, 'b) map" where
+definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where
+  "is_empty m \<longleftrightarrow> dom (lookup m) = {}"
+
+definition size :: "('a, 'b) mapping \<Rightarrow> nat" where
+  "size m = (if finite (dom (lookup m)) then card (dom (lookup m)) else 0)"
+
+definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
   "replace k v m = (if lookup m k = None then m else update k v m)"
 
-definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) map" where
-  "tabulate ks f = Map (map_of (map (\<lambda>k. (k, f k)) ks))"
+definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" where
+  "tabulate ks f = Mapping (map_of (map (\<lambda>k. (k, f k)) ks))"
 
-definition bulkload :: "'a list \<Rightarrow> (nat, 'a) map" where
-  "bulkload xs = Map (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
+definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" where
+  "bulkload xs = Mapping (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
 
 
 subsection {* Properties *}
 
-lemma lookup_inject:
+lemma lookup_inject [simp]:
   "lookup m = lookup n \<longleftrightarrow> m = n"
   by (cases m, cases n) simp
 
+lemma mapping_eqI:
+  assumes "lookup m = lookup n"
+  shows "m = n"
+  using assms by simp
+
 lemma lookup_empty [simp]:
   "lookup empty = Map.empty"
   by (simp add: empty_def)
@@ -55,98 +63,82 @@
   "lookup (update k v m) = (lookup m) (k \<mapsto> v)"
   by (cases m) simp
 
-lemma lookup_delete:
-  "lookup (delete k m) k = None"
-  "k \<noteq> l \<Longrightarrow> lookup (delete k m) l = lookup m l"
-  by (cases m, simp)+
+lemma lookup_delete [simp]:
+  "lookup (delete k m) = (lookup m) (k := None)"
+  by (cases m) simp
 
-lemma lookup_tabulate:
+lemma lookup_tabulate [simp]:
   "lookup (tabulate ks f) = (Some o f) |` set ks"
   by (induct ks) (auto simp add: tabulate_def restrict_map_def expand_fun_eq)
 
-lemma lookup_bulkload:
+lemma lookup_bulkload [simp]:
   "lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
-  unfolding bulkload_def by simp
+  by (simp add: bulkload_def)
 
 lemma update_update:
   "update k v (update k w m) = update k v m"
   "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
-  by (cases m, simp add: expand_fun_eq)+
+  by (rule mapping_eqI, simp add: fun_upd_twist)+
 
-lemma replace_update:
-  "lookup m k = None \<Longrightarrow> replace k v m = m"
-  "lookup m k \<noteq> None \<Longrightarrow> replace k v m = update k v m"
-  by (auto simp add: replace_def)
-
-lemma delete_empty [simp]:
-  "delete k empty = empty"
-  by (simp add: empty_def)
+lemma update_delete [simp]:
+  "update k v (delete k m) = update k v m"
+  by (rule mapping_eqI) simp
 
 lemma delete_update:
   "delete k (update k v m) = delete k m"
   "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
-  by (cases m, simp add: expand_fun_eq)+
-
-lemma update_delete [simp]:
-  "update k v (delete k m) = update k v m"
-  by (cases m) simp
-
-lemma keys_empty [simp]:
-  "keys empty = {}"
-  unfolding empty_def by simp
+  by (rule mapping_eqI, simp add: fun_upd_twist)+
 
-lemma keys_update [simp]:
-  "keys (update k v m) = insert k (keys m)"
-  by (cases m) simp
+lemma delete_empty [simp]:
+  "delete k empty = empty"
+  by (rule mapping_eqI) simp
 
-lemma keys_delete [simp]:
-  "keys (delete k m) = keys m - {k}"
-  by (cases m) simp
-
-lemma keys_tabulate [simp]:
-  "keys (tabulate ks f) = set ks"
-  by (auto simp add: tabulate_def dest: map_of_SomeD intro!: weak_map_of_SomeI)
+lemma replace_update:
+  "k \<notin> dom (lookup m) \<Longrightarrow> replace k v m = m"
+  "k \<in> dom (lookup m) \<Longrightarrow> replace k v m = update k v m"
+  by (rule mapping_eqI, auto simp add: replace_def fun_upd_twist)+
 
 lemma size_empty [simp]:
   "size empty = 0"
-  by (simp add: size_def keys_empty)
+  by (simp add: size_def)
 
 lemma size_update:
-  "finite (keys m) \<Longrightarrow> size (update k v m) =
-    (if k \<in> keys m then size m else Suc (size m))"
-  by (simp add: size_def keys_update)
-    (auto simp only: card_insert card_Suc_Diff1)
+  "finite (dom (lookup m)) \<Longrightarrow> size (update k v m) =
+    (if k \<in> dom (lookup m) then size m else Suc (size m))"
+  by (auto simp add: size_def insert_dom)
 
 lemma size_delete:
-  "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
-  by (simp add: size_def keys_delete)
+  "size (delete k m) = (if k \<in> dom (lookup m) then size m - 1 else size m)"
+  by (simp add: size_def)
 
 lemma size_tabulate:
   "size (tabulate ks f) = length (remdups ks)"
-  by (simp add: size_def keys_tabulate distinct_card [of "remdups ks", symmetric])
+  by (simp add: size_def distinct_card [of "remdups ks", symmetric] comp_def)
 
 lemma bulkload_tabulate:
   "bulkload xs = tabulate [0..<length xs] (nth xs)"
-  by (rule sym)
-    (auto simp add: bulkload_def tabulate_def expand_fun_eq map_of_eq_None_iff comp_def)
+  by (rule mapping_eqI) (simp add: expand_fun_eq)
 
 
 subsection {* Some technical code lemmas *}
 
 lemma [code]:
-  "map_case f m = f (Mapping.lookup m)"
+  "mapping_case f m = f (Mapping.lookup m)"
   by (cases m) simp
 
 lemma [code]:
-  "map_rec f m = f (Mapping.lookup m)"
+  "mapping_rec f m = f (Mapping.lookup m)"
   by (cases m) simp
 
 lemma [code]:
-  "Nat.size (m :: (_, _) map) = 0"
+  "Nat.size (m :: (_, _) mapping) = 0"
   by (cases m) simp
 
 lemma [code]:
-  "map_size f g m = 0"
+  "mapping_size f g m = 0"
   by (cases m) simp
 
+
+hide (open) const empty is_empty lookup update delete keys size replace tabulate bulkload
+
 end
\ No newline at end of file