# HG changeset patch
# User schirmer
# Date 1142003134 -3600
# Node ID 054332e39e0a172c2c4c0100531e1a811a7410c2
# Parent  77ca20b0ed7782849ce2f9d26b5219c14005808b
Added Library/AssocList.thy

diff -r 77ca20b0ed77 -r 054332e39e0a src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Fri Mar 10 15:33:48 2006 +0100
+++ b/src/HOL/IsaMakefile	Fri Mar 10 16:05:34 2006 +0100
@@ -196,7 +196,7 @@
   Library/Library/document/root.bib Library/While_Combinator.thy \
   Library/Product_ord.thy Library/Char_ord.thy \
   Library/List_lexord.thy Library/Commutative_Ring.thy Library/comm_ring.ML \
-  Library/Coinductive_List.thy
+  Library/Coinductive_List.thy Library/AssocList.thy
 	@cd Library; $(ISATOOL) usedir $(OUT)/HOL Library
 
 
diff -r 77ca20b0ed77 -r 054332e39e0a src/HOL/Library/AssocList.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/AssocList.thy	Fri Mar 10 16:05:34 2006 +0100
@@ -0,0 +1,862 @@
+(*  Title:      HOL/Library/Library.thy
+    ID:         $Id$
+    Author:     Norbert Schirmer, Tobias Nipkow, Martin Wildmoser
+*)
+
+header {* Map operations implemented on association lists*}
+
+theory AssocList 
+imports Map
+
+begin
+
+text {* The operations preserve distinctness of keys and 
+        function @{term "clearjunk"} distributes over them.*}
+consts 
+  delete     :: "'key \<Rightarrow> ('key * 'val)list \<Rightarrow>  ('key * 'val)list"
+  update     :: "'key \<Rightarrow> 'val \<Rightarrow> ('key * 'val)list \<Rightarrow>  ('key * 'val)list"
+  updates    :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key * 'val)list \<Rightarrow>  ('key * 'val)list"
+  substitute :: "'val \<Rightarrow> 'val \<Rightarrow> ('key * 'val)list \<Rightarrow>  ('key * 'val)list"
+  map_at     :: "('val \<Rightarrow> 'val) \<Rightarrow> 'key \<Rightarrow> ('key * 'val)list \<Rightarrow>  ('key * 'val) list"
+  merge      :: "('key * 'val)list \<Rightarrow> ('key * 'val)list \<Rightarrow> ('key * 'val)list"
+  compose    :: "('key * 'a)list \<Rightarrow> ('a * 'b)list \<Rightarrow> ('key * 'b)list"
+  restrict   :: "('key set) \<Rightarrow> ('key * 'val)list \<Rightarrow> ('key * 'val)list"
+
+  clearjunk  :: "('key * 'val)list \<Rightarrow> ('key * 'val)list"
+
+defs
+delete_def: "delete k \<equiv> filter (\<lambda>p. fst p \<noteq> k)"
+
+primrec
+"update k v [] = [(k,v)]"
+"update k v (p#ps) = (if fst p = k then (k,v)#ps else p # update k v ps)"
+primrec
+"updates [] vs al = al"
+"updates (k#ks) vs al = (case vs of [] \<Rightarrow> al 
+                         | (v#vs') \<Rightarrow> updates ks vs' (update k v al))"
+primrec
+"substitute v v' [] = []"
+"substitute v v' (p#ps) = (if snd p = v then (fst p,v')#substitute v v' ps
+                          else p#substitute v v' ps)"
+primrec
+"map_at f k [] = []"
+"map_at f k (p#ps) = (if fst p = k then (k,f (snd p))#ps else p # map_at f k ps)"
+primrec
+"merge xs [] = xs"
+"merge xs (p#ps) = update (fst p) (snd p) (merge xs ps)"
+
+lemma length_delete_le: "length (delete k al) \<le> length al"
+proof (induct al)
+  case Nil thus ?case by (simp add: delete_def)
+next
+  case (Cons a al)
+  note length_filter_le [of "\<lambda>p. fst p \<noteq> fst a" al] 
+  also have "\<And>n. n \<le> Suc n"
+    by simp
+  finally have "length [p\<in>al . fst p \<noteq> fst a] \<le> Suc (length al)" .
+  with Cons show ?case
+    by (auto simp add: delete_def)
+qed
+
+lemma compose_hint: "length (delete k al) < Suc (length al)"
+proof -
+  note length_delete_le
+  also have "\<And>n. n < Suc n"
+    by simp
+  finally show ?thesis .
+qed
+
+recdef compose "measure size"
+"compose [] = (\<lambda>ys. [])"
+"compose (x#xs) = (\<lambda>ys. (case (map_of ys (snd x)) of
+                          None \<Rightarrow> compose (delete (fst x) xs) ys
+                         | Some v \<Rightarrow> (fst x,v)#compose xs ys))"
+(hints intro: compose_hint)
+
+defs  
+restrict_def: "restrict A \<equiv> filter (\<lambda>(k,v). k \<in> A)"
+
+recdef clearjunk "measure size"
+"clearjunk [] = []"
+"clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
+(hints intro: compose_hint)
+
+
+(* ******************************************************************************** *)
+subsection {* Lookup *}
+(* ******************************************************************************** *)
+
+lemma lookup_simps: 
+  "map_of [] k = None"
+  "map_of (p#ps) k = (if fst p = k then Some (snd p) else map_of ps k)"
+  by simp_all
+
+(* ******************************************************************************** *)
+subsection {* @{const delete} *}
+(* ******************************************************************************** *)
+
+lemma delete_simps [simp]:
+"delete k [] = []"
+"delete k (p#ps) = (if fst p = k then delete k ps else p # delete k ps)"
+  by (simp_all add: delete_def)
+
+lemma delete_id[simp]: "k \<notin> fst ` set al \<Longrightarrow> delete k al = al"
+by(induct al, auto)
+
+lemma delete_conv: "map_of (delete k al) k' = ((map_of al)(k := None)) k'"
+  by (induct al) auto
+
+lemma delete_conv': "map_of (delete k al) = ((map_of al)(k := None))"
+  by (rule ext) (rule delete_conv)
+
+lemma delete_idem: "delete k (delete k al) = delete k al"
+  by (induct al) auto
+
+lemma map_of_delete[simp]:
+ "k' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'"
+by(induct al, auto)
+
+lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)"
+  by (induct al) auto
+
+lemma dom_delete_subset: "fst ` set (delete k al) \<subseteq> fst ` set al"
+  by (induct al) auto
+
+lemma distinct_delete:
+  assumes "distinct (map fst al)" 
+  shows "distinct (map fst (delete k al))"
+using prems
+proof (induct al)
+  case Nil thus ?case by simp
+next
+  case (Cons a al)
+  from Cons.prems obtain 
+    a_notin_al: "fst a \<notin> fst ` set al" and
+    dist_al: "distinct (map fst al)"
+    by auto
+  show ?case
+  proof (cases "fst a = k")
+    case True
+    from True dist_al show ?thesis by simp
+  next
+    case False
+    from dist_al
+    have "distinct (map fst (delete k al))"
+      by (rule Cons.hyps)
+    moreover from a_notin_al dom_delete_subset [of k al] 
+    have "fst a \<notin> fst ` set (delete k al)"
+      by blast
+    ultimately show ?thesis using False by simp
+  qed
+qed
+
+lemma delete_twist: "delete x (delete y al) = delete y (delete x al)"
+  by (induct al) auto
+
+lemma clearjunk_delete: "clearjunk (delete x al) = delete x (clearjunk al)"
+  by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist)
+
+(* ******************************************************************************** *)
+subsection {* @{const clearjunk} *}
+(* ******************************************************************************** *)
+
+lemma insert_fst_filter: 
+  "insert a(fst ` {x \<in> set ps. fst x \<noteq> a}) = insert a (fst ` set ps)"
+  by (induct ps) auto
+
+lemma dom_clearjunk: "fst ` set (clearjunk al) = fst ` set al"
+  by (induct al rule: clearjunk.induct) (simp_all add: insert_fst_filter delete_def)
+
+lemma notin_filter_fst: "a \<notin> fst ` {x \<in> set ps. fst x \<noteq> a}"
+  by (induct ps) auto
+
+lemma distinct_clearjunk [simp]: "distinct (map fst (clearjunk al))"
+  by (induct al rule: clearjunk.induct) 
+     (simp_all add: dom_clearjunk notin_filter_fst delete_def)
+
+lemma map_of_filter: "k \<noteq> a \<Longrightarrow> map_of [q\<in>ps . fst q \<noteq> a] k = map_of ps k"
+  by (induct ps) auto
+
+lemma map_of_clearjunk: "map_of (clearjunk al) = map_of al"
+  apply (rule ext)
+  apply (induct al rule: clearjunk.induct)
+  apply  simp
+  apply (simp add: map_of_filter)
+  done
+
+lemma length_clearjunk: "length (clearjunk al) \<le> length al"
+proof (induct al rule: clearjunk.induct [case_names Nil Cons])
+  case Nil thus ?case by simp
+next
+  case (Cons k v ps)
+  from Cons have "length (clearjunk [q\<in>ps . fst q \<noteq> k]) \<le> length [q\<in>ps . fst q \<noteq> k]" 
+    by (simp add: delete_def)
+  also have "\<dots> \<le> length ps"
+    by simp
+  finally show ?case
+    by (simp add: delete_def)
+qed
+
+lemma notin_fst_filter: "a \<notin> fst ` set ps \<Longrightarrow> [q\<in>ps . fst q \<noteq> a] = ps"
+  by (induct ps) auto
+            
+lemma distinct_clearjunk_id [simp]: "distinct (map fst al) \<Longrightarrow> clearjunk al = al"
+  by (induct al rule: clearjunk.induct) (auto simp add: notin_fst_filter)
+
+lemma clearjunk_idem: "clearjunk (clearjunk al) = clearjunk al"
+  by simp
+
+(* ******************************************************************************** *)
+subsection {* @{const dom} and @{term "ran"} *}
+(* ******************************************************************************** *)
+
+lemma dom_map_of': "fst ` set al = dom (map_of al)"
+  by (induct al) auto
+
+lemmas dom_map_of = dom_map_of' [symmetric]
+
+lemma ran_clearjunk: "ran (map_of (clearjunk al)) = ran (map_of al)"
+  by (simp add: map_of_clearjunk)
+
+lemma ran_distinct: 
+  assumes dist: "distinct (map fst al)" 
+  shows "ran (map_of al) = snd ` set al"
+using dist
+proof (induct al) 
+  case Nil
+  thus ?case by simp
+next
+  case (Cons a al)
+  hence hyp: "snd ` set al = ran (map_of al)"
+    by simp
+
+  have "ran (map_of (a # al)) = {snd a} \<union> ran (map_of al)"
+  proof 
+    show "ran (map_of (a # al)) \<subseteq> {snd a} \<union> ran (map_of al)"
+    proof   
+      fix v
+      assume "v \<in> ran (map_of (a#al))"
+      then obtain x where "map_of (a#al) x = Some v"
+	by (auto simp add: ran_def)
+      then show "v \<in> {snd a} \<union> ran (map_of al)"
+	by (auto split: split_if_asm simp add: ran_def)
+    qed
+  next
+    show "{snd a} \<union> ran (map_of al) \<subseteq> ran (map_of (a # al))"
+    proof 
+      fix v
+      assume v_in: "v \<in> {snd a} \<union> ran (map_of al)"
+      show "v \<in> ran (map_of (a#al))"
+      proof (cases "v=snd a")
+	case True
+	with v_in show ?thesis
+	  by (auto simp add: ran_def)
+      next
+	case False
+	with v_in have "v \<in> ran (map_of al)" by auto
+	then obtain x where al_x: "map_of al x = Some v"
+	  by (auto simp add: ran_def)
+	from map_of_SomeD [OF this]
+	have "x \<in> fst ` set al"
+	  by (force simp add: image_def)
+	with Cons.prems have "x\<noteq>fst a"
+	  by - (rule ccontr,simp)
+	with al_x
+	show ?thesis
+	  by (auto simp add: ran_def)
+      qed
+    qed
+  qed
+  with hyp show ?case
+    by (simp only:) auto
+qed
+
+lemma ran_map_of: "ran (map_of al) = snd ` set (clearjunk al)"
+proof -
+  have "ran (map_of al) = ran (map_of (clearjunk al))"
+    by (simp add: ran_clearjunk)
+  also have "\<dots> = snd ` set (clearjunk al)"
+    by (simp add: ran_distinct)
+  finally show ?thesis .
+qed
+   
+(* ******************************************************************************** *)
+subsection {* @{const update} *}
+(* ******************************************************************************** *)
+
+lemma update_conv: "map_of (update k v al) k' = ((map_of al)(k\<mapsto>v)) k'"
+  by (induct al) auto
+
+lemma update_conv': "map_of (update k v al)  = ((map_of al)(k\<mapsto>v))"
+  by (rule ext) (rule update_conv)
+
+lemma dom_update: "fst ` set (update k v al) = {k} \<union> fst ` set al"
+  by (induct al) auto
+
+lemma distinct_update:
+  assumes "distinct (map fst al)" 
+  shows "distinct (map fst (update k v al))"
+using prems
+proof (induct al)
+  case Nil thus ?case by simp
+next
+  case (Cons a al)
+  from Cons.prems obtain 
+    a_notin_al: "fst a \<notin> fst ` set al" and
+    dist_al: "distinct (map fst al)"
+    by auto
+  show ?case
+  proof (cases "fst a = k")
+    case True
+    from True dist_al a_notin_al show ?thesis by simp
+  next
+    case False
+    from dist_al
+    have "distinct (map fst (update k v al))"
+      by (rule Cons.hyps)
+    with False a_notin_al show ?thesis by (simp add: dom_update)
+  qed
+qed
+
+lemma update_filter: 
+  "a\<noteq>k \<Longrightarrow> update k v [q\<in>ps . fst q \<noteq> a] = [q\<in>update k v ps . fst q \<noteq> a]"
+  by (induct ps) auto
+
+lemma clearjunk_update: "clearjunk (update k v al) = update k v (clearjunk al)"
+  by (induct al rule: clearjunk.induct) (auto simp add: update_filter delete_def)
+
+lemma update_triv: "map_of al k = Some v \<Longrightarrow> update k v al = al"
+  by (induct al) auto
+
+lemma update_nonempty [simp]: "update k v al \<noteq> []"
+  by (induct al) auto
+
+lemma update_eqD: "update k v al = update k v' al' \<Longrightarrow> v=v'"
+proof (induct al fixing: al') 
+  case Nil thus ?case 
+    by (cases al') (auto split: split_if_asm)
+next
+  case Cons thus ?case
+    by (cases al') (auto split: split_if_asm)
+qed
+
+lemma update_last [simp]: "update k v (update k v' al) = update k v al"
+  by (induct al) auto
+
+text {* Note that the lists are not necessarily the same:
+        @{term "update k v (update k' v' []) = [(k',v'),(k,v)]"} and 
+        @{term "update k' v' (update k v []) = [(k,v),(k',v')]"}.*}
+lemma update_swap: "k\<noteq>k' 
+  \<Longrightarrow> map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"
+  by (auto simp add: update_conv' intro: ext)
+
+lemma update_Some_unfold: 
+  "(map_of (update k v al) x = Some y) = 
+     (x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y)"
+  by (simp add: update_conv' map_upd_Some_unfold)
+
+lemma image_update[simp]: "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A"
+  by (simp add: update_conv' image_map_upd)
+
+
+(* ******************************************************************************** *)
+subsection {* @{const updates} *}
+(* ******************************************************************************** *)
+
+lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\<mapsto>]vs)) k"
+proof (induct ks fixing: vs al)
+  case Nil
+  thus ?case by simp
+next
+  case (Cons k ks)
+  show ?case
+  proof (cases vs)
+    case Nil
+    with Cons show ?thesis by simp
+  next
+    case (Cons k ks')
+    with Cons.hyps show ?thesis
+      by (simp add: update_conv fun_upd_def)
+  qed
+qed
+
+lemma updates_conv': "map_of (updates ks vs al) = ((map_of al)(ks[\<mapsto>]vs))"
+  by (rule ext) (rule updates_conv)
+
+lemma distinct_updates:
+  assumes "distinct (map fst al)"
+  shows "distinct (map fst (updates ks vs al))"
+  using prems
+by (induct ks fixing: vs al) (auto simp add: distinct_update split: list.splits)
+
+lemma clearjunk_updates:
+ "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
+  by (induct ks fixing: vs al) (auto simp add: clearjunk_update split: list.splits)
+
+lemma updates_empty[simp]: "updates vs [] al = al"
+  by (induct vs) auto 
+
+lemma updates_Cons: "updates (k#ks) (v#vs) al = updates ks vs (update k v al)"
+  by simp
+
+lemma updates_append1[simp]: "size ks < size vs \<Longrightarrow>
+  updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)"
+  by (induct ks fixing: vs al) (auto split: list.splits)
+
+lemma updates_list_update_drop[simp]:
+ "\<lbrakk>size ks \<le> i; i < size vs\<rbrakk>
+   \<Longrightarrow> updates ks (vs[i:=v]) al = updates ks vs al"
+  by (induct ks fixing: al vs i) (auto split:list.splits nat.splits)
+
+lemma update_updates_conv_if: "
+ map_of (updates xs ys (update x y al)) =
+ map_of (if x \<in>  set(take (length ys) xs) then updates xs ys al
+                                  else (update x y (updates xs ys al)))"
+  by (simp add: updates_conv' update_conv' map_upd_upds_conv_if)
+
+lemma updates_twist [simp]:
+ "k \<notin> set ks \<Longrightarrow> 
+  map_of (updates ks vs (update k v al)) = map_of (update k v (updates ks vs al))"
+  by (simp add: updates_conv' update_conv' map_upds_twist)
+
+lemma updates_apply_notin[simp]:
+ "k \<notin> set ks ==> map_of (updates ks vs al) k = map_of al k"
+  by (simp add: updates_conv)
+
+lemma updates_append_drop[simp]:
+  "size xs = size ys \<Longrightarrow> updates (xs@zs) ys al = updates xs ys al"
+  by (induct xs fixing: ys al) (auto split: list.splits)
+
+lemma updates_append2_drop[simp]:
+  "size xs = size ys \<Longrightarrow> updates xs (ys@zs) al = updates xs ys al"
+  by (induct xs fixing: ys al) (auto split: list.splits)
+
+
+(* ******************************************************************************** *)
+subsection {* @{const substitute} *}
+(* ******************************************************************************** *)
+
+lemma substitute_conv: "map_of (substitute v v' al) k = ((map_of al)(v ~> v')) k"
+  by (induct al) auto
+
+lemma substitute_conv': "map_of (substitute v v' al) = ((map_of al)(v ~> v'))"
+  by (rule ext) (rule substitute_conv)
+
+lemma dom_substitute: "fst ` set (substitute v v' al) = fst ` set al"
+  by (induct al) auto
+
+lemma distinct_substitute: 
+  "distinct (map fst al) \<Longrightarrow> distinct (map fst (substitute v v' al))"
+  by (induct al) (auto simp add: dom_substitute)
+
+lemma substitute_filter: 
+  "(substitute v v' [q\<in>ps . fst q \<noteq> a]) = [q\<in>substitute v v' ps . fst q \<noteq> a]"
+  by (induct ps) auto
+
+lemma clearjunk_substitute:
+ "clearjunk (substitute v v' al) = substitute v v' (clearjunk al)"
+  by (induct al rule: clearjunk.induct) (auto simp add: substitute_filter delete_def)
+
+(* ******************************************************************************** *)
+subsection {* @{const map_at} *}
+(* ******************************************************************************** *)
+  
+lemma map_at_conv: "map_of (map_at f k al) k' = (chg_map f k (map_of al)) k'"
+  by (induct al) (auto simp add: chg_map_def split: option.splits)
+
+lemma map_at_conv': "map_of (map_at f k al) = (chg_map f k (map_of al))"
+  by (rule ext) (rule map_at_conv)
+
+lemma dom_map_at: "fst ` set (map_at f k al) = fst ` set al"
+  by (induct al) auto
+
+lemma distinct_map_at: 
+  assumes "distinct (map fst al)"
+  shows "distinct (map fst (map_at f k al))"
+using prems by (induct al) (auto simp add: dom_map_at)
+
+lemma map_at_notin_filter: 
+  "a \<noteq> k \<Longrightarrow> (map_at f k [q\<in>ps . fst q \<noteq> a]) = [q\<in>map_at f k ps . fst q \<noteq> a]"
+  by (induct ps) auto
+
+lemma clearjunk_map_at:
+ "clearjunk (map_at f k al) = map_at f k (clearjunk al)"
+  by (induct al rule: clearjunk.induct) (auto simp add: map_at_notin_filter delete_def)
+
+lemma map_at_new[simp]: "map_of al k = None \<Longrightarrow> map_at f k al = al"
+  by (induct al) auto
+
+lemma map_at_update: "map_of al k = Some v \<Longrightarrow> map_at f k al = update k (f v) al"
+  by (induct al) auto
+
+lemma map_at_other [simp]: "a \<noteq> b \<Longrightarrow> map_of (map_at f a al) b = map_of al b"
+  by (simp add: map_at_conv')
+
+(* ******************************************************************************** *)
+subsection {* @{const merge} *}
+(* ******************************************************************************** *)
+
+lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys"
+  by (induct ys fixing: xs) (auto simp add: dom_update)
+
+lemma distinct_merge:
+  assumes "distinct (map fst xs)"
+  shows "distinct (map fst (merge xs ys))"
+  using prems
+by (induct ys fixing: xs) (auto simp add: dom_merge distinct_update)
+
+lemma clearjunk_merge:
+ "clearjunk (merge xs ys) = merge (clearjunk xs) ys"
+  by (induct ys) (auto simp add: clearjunk_update)
+
+lemma merge_conv: "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"
+proof (induct ys)
+  case Nil thus ?case by simp 
+next
+  case (Cons y ys)
+  show ?case
+  proof (cases "k = fst y")
+    case True
+    from True show ?thesis
+      by (simp add: update_conv)
+  next
+    case False
+    from False show ?thesis
+      by (auto simp add: update_conv Cons.hyps map_add_def)
+  qed
+qed
+
+lemma merge_conv': "map_of (merge xs ys) = (map_of xs ++ map_of ys)"
+  by (rule ext) (rule merge_conv)
+
+lemma merge_emty: "map_of (merge [] ys) = map_of ys"
+  by (simp add: merge_conv')
+
+lemma merge_assoc[simp]: "map_of (merge m1 (merge m2 m3)) = 
+                           map_of (merge (merge m1 m2) m3)"
+  by (simp add: merge_conv')
+
+lemma merge_Some_iff: 
+ "(map_of (merge m n) k = Some x) = 
+  (map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x)"
+  by (simp add: merge_conv' map_add_Some_iff)
+
+lemmas merge_SomeD = merge_Some_iff [THEN iffD1, standard]
+declare merge_SomeD [dest!]
+
+lemma merge_find_right[simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v"
+  by (simp add: merge_conv')
+
+lemma merge_None [iff]: 
+  "(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)"
+  by (simp add: merge_conv')
+
+lemma merge_upd[simp]: 
+  "map_of (merge m (update k v n)) = map_of (update k v (merge m n))"
+  by (simp add: update_conv' merge_conv')
+
+lemma merge_updatess[simp]: 
+  "map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))"
+  by (simp add: updates_conv' merge_conv')
+
+lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)"
+  by (simp add: merge_conv')
+
+(* ******************************************************************************** *)
+subsection {* @{const compose} *}
+(* ******************************************************************************** *)
+
+lemma compose_induct [case_names Nil Cons]: 
+  assumes Nil: "P [] ys"
+  assumes Cons: "\<And>x xs.
+     \<lbrakk>\<And>v. map_of ys (snd x) = Some v \<Longrightarrow> P xs ys;
+      map_of ys (snd x) = None \<Longrightarrow> P (delete (fst x) xs) ys\<rbrakk>
+     \<Longrightarrow> P (x # xs) ys"
+  shows "P xs ys"
+apply (rule compose.induct [where ?P="\<lambda>xs. P xs ys"])
+apply (rule Nil)
+apply  (rule Cons)
+apply (erule allE, erule allE, erule impE, assumption,assumption)
+apply (erule allE, erule impE,assumption,assumption)
+done
+
+lemma compose_first_None [simp]: 
+  assumes "map_of xs k = None" 
+  shows "map_of (compose xs ys) k = None"
+using prems
+by (induct xs ys rule: compose_induct) (auto split: option.splits split_if_asm)
+
+
+lemma compose_conv: 
+  shows "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
+proof (induct xs ys rule: compose_induct )
+  case Nil thus ?case by simp
+next
+  case (Cons x xs)
+  show ?case
+  proof (cases "map_of ys (snd x)")
+    case None
+    with Cons
+    have hyp: "map_of (compose (delete (fst x) xs) ys) k =
+               (map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k"
+      by simp
+    show ?thesis
+    proof (cases "fst x = k")
+      case True
+      from True delete_notin_dom [of k xs]
+      have "map_of (delete (fst x) xs) k = None"
+	by (simp add: map_of_eq_None_iff)
+      with hyp show ?thesis
+	using True None
+	by simp
+    next
+      case False
+      from False have "map_of (delete (fst x) xs) k = map_of xs k"
+	by simp
+      with hyp show ?thesis
+	using False None
+	by (simp add: map_comp_def)
+    qed
+  next
+    case (Some v)
+    with Cons
+    have "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
+      by simp
+    with Some show ?thesis
+      by (auto simp add: map_comp_def)
+  qed
+qed
+   
+lemma compose_conv': 
+  shows "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)"
+  by (rule ext) (rule compose_conv)
+
+lemma compose_first_Some [simp]:
+  assumes "map_of xs k = Some v" 
+  shows "map_of (compose xs ys) k = map_of ys v"
+using prems by (simp add: compose_conv)
+
+lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs"
+proof (induct xs ys rule: compose_induct )
+  case Nil thus ?case by simp
+next
+  case (Cons x xs)
+  show ?case
+  proof (cases "map_of ys (snd x)")
+    case None
+    with Cons.hyps
+    have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)"
+      by simp
+    also
+    have "\<dots> \<subseteq> fst ` set xs"
+      by (rule dom_delete_subset)
+    finally show ?thesis
+      using None
+      by auto
+  next
+    case (Some v)
+    with Cons.hyps
+    have "fst ` set (compose xs ys) \<subseteq> fst ` set xs"
+      by simp
+    with Some show ?thesis
+      by auto
+  qed
+qed
+
+lemma distinct_compose:
+ assumes "distinct (map fst xs)"
+ shows "distinct (map fst (compose xs ys))"
+using prems
+proof (induct xs ys rule: compose_induct)
+  case Nil thus ?case by simp
+next
+  case (Cons x xs)
+  show ?case
+  proof (cases "map_of ys (snd x)")
+    case None
+    with Cons show ?thesis by simp
+  next
+    case (Some v)
+    with Cons dom_compose [of xs ys] show ?thesis 
+      by (auto)
+  qed
+qed
+
+lemma compose_delete_twist: "(compose (delete k xs) ys) = delete k (compose xs ys)"
+proof (induct xs ys rule: compose_induct)
+  case Nil thus ?case by simp
+next
+  case (Cons x xs)
+  show ?case
+  proof (cases "map_of ys (snd x)")
+    case None
+    with Cons have 
+      hyp: "compose (delete k (delete (fst x) xs)) ys =
+            delete k (compose (delete (fst x) xs) ys)"
+      by simp
+    show ?thesis
+    proof (cases "fst x = k")
+      case True
+      with None hyp
+      show ?thesis
+	by (simp add: delete_idem)
+    next
+      case False
+      from None False hyp
+      show ?thesis
+	by (simp add: delete_twist)
+    qed
+  next
+    case (Some v)
+    with Cons have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" by simp
+    with Some show ?thesis
+      by simp
+  qed
+qed
+
+lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys"
+  by (induct xs ys rule: compose_induct) 
+     (auto simp add: map_of_clearjunk split: option.splits)
+   
+lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys"
+  by (induct xs rule: clearjunk.induct)
+     (auto split: option.splits simp add: clearjunk_delete delete_idem
+               compose_delete_twist)
+   
+lemma compose_empty [simp]:
+ "compose xs [] = []"
+  by (induct xs rule: compose_induct [where ys="[]"]) auto
+
+
+lemma compose_Some_iff:
+  "(map_of (compose xs ys) k = Some v) = 
+     (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = Some v)" 
+  by (simp add: compose_conv map_comp_Some_iff)
+
+lemma map_comp_None_iff:
+  "(map_of (compose xs ys) k = None) = 
+    (map_of xs k = None \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None)) " 
+  by (simp add: compose_conv map_comp_None_iff)
+
+
+(* ******************************************************************************** *)
+subsection {* @{const restrict} *}
+(* ******************************************************************************** *)
+
+lemma restrict_simps [simp]: 
+  "restrict A [] = []"
+  "restrict A (p#ps) = (if fst p \<in> A then p#restrict A ps else restrict A ps)"
+  by (auto simp add: restrict_def)
+
+lemma distinct_restr: "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))"
+  by (induct al) (auto simp add: restrict_def)
+
+lemma restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k"
+  apply (induct al)
+  apply  (simp add: restrict_def)
+  apply (cases "k\<in>A")
+  apply (auto simp add: restrict_def)
+  done
+
+lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)"
+  by (rule ext) (rule restr_conv)
+
+lemma restr_empty [simp]: 
+  "restrict {} al = []" 
+  "restrict A [] = []"
+  by (induct al) (auto simp add: restrict_def)
+
+lemma restr_in [simp]: "x \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x"
+  by (simp add: restr_conv')
+
+lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None"
+  by (simp add: restr_conv')
+
+lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> A"
+  by (induct al) (auto simp add: restrict_def)
+
+lemma restr_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) al"
+  by (induct al) (auto simp add: restrict_def)
+
+lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al"
+  by (induct al) (auto simp add: restrict_def)
+
+lemma restr_update[simp]:
+ "map_of (restrict D (update x y al)) = 
+  map_of ((if x \<in> D then (update x y (restrict (D-{x}) al)) else restrict D al))"
+  by (simp add: restr_conv' update_conv')
+
+lemma restr_delete [simp]:
+  "(delete x (restrict D al)) = 
+    (if x\<in> D then restrict (D - {x}) al else restrict D al)"
+proof (induct al)
+  case Nil thus ?case by simp
+next
+  case (Cons a al)
+  show ?case
+  proof (cases "x \<in> D")
+    case True
+    note x_D = this
+    with Cons have hyp: "delete x (restrict D al) = restrict (D - {x}) al"
+      by simp
+    show ?thesis
+    proof (cases "fst a = x")
+      case True
+      from Cons.hyps
+      show ?thesis
+	using x_D True
+	by simp
+    next
+      case False
+      note not_fst_a_x = this
+      show ?thesis
+      proof (cases "fst a \<in> D")
+	case True 
+	with not_fst_a_x 
+	have "delete x (restrict D (a#al)) = a#(delete x (restrict D al))"
+	  by (cases a) (simp add: restrict_def)
+	also from not_fst_a_x True hyp have "\<dots> = restrict (D - {x}) (a # al)"
+	  by (cases a) (simp add: restrict_def)
+	finally show ?thesis
+	  using x_D by simp
+      next
+	case False
+	hence "delete x (restrict D (a#al)) = delete x (restrict D al)"
+	  by (cases a) (simp add: restrict_def)
+	moreover from False not_fst_a_x
+	have "restrict (D - {x}) (a # al) = restrict (D - {x}) al"
+	  by (cases a) (simp add: restrict_def)
+	ultimately
+	show ?thesis using x_D hyp by simp
+      qed
+    qed
+  next
+    case False
+    from False Cons show ?thesis
+      by simp
+  qed
+qed
+
+lemma update_restr:
+ "map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
+  by (simp add: update_conv' restr_conv') (rule fun_upd_restrict)
+
+lemma upate_restr_conv[simp]:
+ "x \<in> D \<Longrightarrow> 
+ map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
+  by (simp add: update_conv' restr_conv')
+
+lemma restr_updates[simp]: "
+ \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
+ \<Longrightarrow> map_of (restrict D (updates xs ys al)) = 
+     map_of (updates xs ys (restrict (D - set xs) al))"
+  by (simp add: updates_conv' restr_conv')
+
+lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)"
+  by (induct ps) auto
+
+lemma clearjunk_restrict:
+ "clearjunk (restrict A al) = restrict A (clearjunk al)"
+  by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist)
+
+end
diff -r 77ca20b0ed77 -r 054332e39e0a src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Fri Mar 10 15:33:48 2006 +0100
+++ b/src/HOL/Library/Library.thy	Fri Mar 10 16:05:34 2006 +0100
@@ -22,6 +22,7 @@
   Commutative_Ring
   Coinductive_List
   ASeries
+  AssocList
 begin
 end
 (*>*)