# HG changeset patch
# User wenzelm
# Date 1396207499 -7200
# Node ID 3e62e68fb342cce5d42dc266d3d3741322e375ea
# Parent  c3d7b3bb2708e17d2a0abd695e2cb97467814830
tuned proofs;

diff -r c3d7b3bb2708 -r 3e62e68fb342 src/HOL/Library/AList.thy
--- a/src/HOL/Library/AList.thy	Sun Mar 30 21:03:40 2014 +0200
+++ b/src/HOL/Library/AList.thy	Sun Mar 30 21:24:59 2014 +0200
@@ -9,17 +9,18 @@
 begin
 
 text {*
-  The operations preserve distinctness of keys and 
-  function @{term "clearjunk"} distributes over them. Since 
+  The operations preserve distinctness of keys and
+  function @{term "clearjunk"} distributes over them. Since
   @{term clearjunk} enforces distinctness of keys it can be used
   to establish the invariant, e.g. for inductive proofs.
 *}
 
 subsection {* @{text update} and @{text updates} *}
 
-primrec update :: "'key \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
-    "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 update :: "'key \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+where
+  "update k v [] = [(k, v)]"
+| "update k v (p # ps) = (if fst p = k then (k, v) # ps else p # update k v ps)"
 
 lemma update_conv': "map_of (update k v al)  = (map_of al)(k\<mapsto>v)"
   by (induct al) (auto simp add: fun_eq_iff)
@@ -36,12 +37,12 @@
   by (induct al) simp_all
 
 lemma distinct_update:
-  assumes "distinct (map fst al)" 
+  assumes "distinct (map fst al)"
   shows "distinct (map fst (update k v al))"
   using assms by (simp add: update_keys)
 
-lemma update_filter: 
-  "a\<noteq>k \<Longrightarrow> update k v [q\<leftarrow>ps . fst q \<noteq> a] = [q\<leftarrow>update k v ps . fst q \<noteq> a]"
+lemma update_filter:
+  "a \<noteq> k \<Longrightarrow> update k v [q\<leftarrow>ps. fst q \<noteq> a] = [q\<leftarrow>update k v ps. fst q \<noteq> a]"
   by (induct ps) auto
 
 lemma update_triv: "map_of al k = Some v \<Longrightarrow> update k v al = al"
@@ -51,11 +52,13 @@
   by (induct al) auto
 
 lemma update_eqD: "update k v al = update k v' al' \<Longrightarrow> v = v'"
-proof (induct al arbitrary: al') 
-  case Nil thus ?case 
+proof (induct al arbitrary: al')
+  case Nil
+  then show ?case
     by (cases al') (auto split: split_if_asm)
 next
-  case Cons thus ?case
+  case Cons
+  then show ?case
     by (cases al') (auto split: split_if_asm)
 qed
 
@@ -63,13 +66,15 @@
   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)]"} 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))"
+
+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 (simp add: update_conv' fun_eq_iff)
 
-lemma update_Some_unfold: 
+lemma update_Some_unfold:
   "map_of (update k v al) x = Some y \<longleftrightarrow>
     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)
@@ -78,8 +83,8 @@
   "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A"
   by (simp add: update_conv')
 
-definition updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
-  "updates ks vs = fold (case_prod update) (zip ks vs)"
+definition updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+  where "updates ks vs = fold (case_prod update) (zip ks vs)"
 
 lemma updates_simps [simp]:
   "updates [] vs ps = ps"
@@ -95,9 +100,10 @@
 lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)"
 proof -
   have "map_of \<circ> fold (case_prod update) (zip ks vs) =
-    fold (\<lambda>(k, v) f. f(k \<mapsto> v)) (zip ks vs) \<circ> map_of"
+      fold (\<lambda>(k, v) f. f(k \<mapsto> v)) (zip ks vs) \<circ> map_of"
     by (rule fold_commute) (auto simp add: fun_eq_iff update_conv')
-  then show ?thesis by (auto simp add: updates_def fun_eq_iff map_upds_fold_map_upd foldl_conv_fold split_def)
+  then show ?thesis
+    by (auto simp add: updates_def fun_eq_iff map_upds_fold_map_upd foldl_conv_fold split_def)
 qed
 
 lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\<mapsto>]vs)) k"
@@ -112,52 +118,55 @@
        (zip ks vs) (map fst al))"
     by (rule fold_invariant [of "zip ks vs" "\<lambda>_. True"]) (auto intro: assms)
   moreover have "map fst \<circ> fold (case_prod update) (zip ks vs) =
-    fold (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k]) (zip ks vs) \<circ> map fst"
+      fold (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k]) (zip ks vs) \<circ> map fst"
     by (rule fold_commute) (simp add: update_keys split_def case_prod_beta comp_def)
-  ultimately show ?thesis by (simp add: updates_def fun_eq_iff)
+  ultimately show ?thesis
+    by (simp add: updates_def fun_eq_iff)
 qed
 
 lemma updates_append1[simp]: "size ks < size vs \<Longrightarrow>
-  updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)"
+    updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)"
   by (induct ks arbitrary: 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 arbitrary: al vs i) (auto split:list.splits nat.splits)
+  "size ks \<le> i \<Longrightarrow> i < size vs \<Longrightarrow>
+    updates ks (vs[i:=v]) al = updates ks vs al"
+  by (induct ks arbitrary: 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)))"
+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))"
+  "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')
 
-lemma updates_apply_notin[simp]:
- "k \<notin> set ks ==> map_of (updates ks vs al) k = map_of al k"
+lemma updates_apply_notin [simp]:
+  "k \<notin> set ks \<Longrightarrow> 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"
+lemma updates_append_drop [simp]:
+  "size xs = size ys \<Longrightarrow> updates (xs @ zs) ys al = updates xs ys al"
   by (induct xs arbitrary: 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"
+lemma updates_append2_drop [simp]:
+  "size xs = size ys \<Longrightarrow> updates xs (ys @ zs) al = updates xs ys al"
   by (induct xs arbitrary: ys al) (auto split: list.splits)
 
 
 subsection {* @{text delete} *}
 
-definition delete :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
-  delete_eq: "delete k = filter (\<lambda>(k', _). k \<noteq> k')"
+definition delete :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+  where delete_eq: "delete k = filter (\<lambda>(k', _). k \<noteq> k')"
 
 lemma delete_simps [simp]:
   "delete k [] = []"
-  "delete k (p#ps) = (if fst p = k then delete k ps else p # delete k ps)"
+  "delete k (p # ps) = (if fst p = k then delete k ps else p # delete k ps)"
   by (auto simp add: delete_eq)
 
 lemma delete_conv': "map_of (delete k al) = (map_of al)(k := None)"
@@ -166,12 +175,11 @@
 corollary delete_conv: "map_of (delete k al) k' = ((map_of al)(k := None)) k'"
   by (simp add: delete_conv')
 
-lemma delete_keys:
-  "map fst (delete k al) = removeAll k (map fst al)"
+lemma delete_keys: "map fst (delete k al) = removeAll k (map fst al)"
   by (simp add: delete_eq removeAll_filter_not_eq filter_map split_def comp_def)
 
 lemma distinct_delete:
-  assumes "distinct (map fst al)" 
+  assumes "distinct (map fst al)"
   shows "distinct (map fst (delete k al))"
   using assms by (simp add: delete_keys distinct_removeAll)
 
@@ -181,8 +189,7 @@
 lemma delete_idem: "delete k (delete k al) = delete k al"
   by (simp add: delete_eq)
 
-lemma map_of_delete [simp]:
-    "k' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'"
+lemma map_of_delete [simp]: "k' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'"
   by (simp add: delete_conv')
 
 lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)"
@@ -191,12 +198,10 @@
 lemma dom_delete_subset: "fst ` set (delete k al) \<subseteq> fst ` set al"
   by (auto simp add: delete_eq)
 
-lemma delete_update_same:
-  "delete k (update k v al) = delete k al"
+lemma delete_update_same: "delete k (update k v al) = delete k al"
   by (induct al) simp_all
 
-lemma delete_update:
-  "k \<noteq> l \<Longrightarrow> delete l (update k v al) = update k v (delete l al)"
+lemma delete_update: "k \<noteq> l \<Longrightarrow> delete l (update k v al) = update k v (delete l al)"
   by (induct al) simp_all
 
 lemma delete_twist: "delete x (delete y al) = delete y (delete x al)"
@@ -208,8 +213,8 @@
 
 subsection {* @{text restrict} *}
 
-definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
-  restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)"
+definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+  where restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)"
 
 lemma restr_simps [simp]:
   "restrict A [] = []"
@@ -230,8 +235,8 @@
   "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))"
   by (induct al) (auto simp add: restrict_eq)
 
-lemma restr_empty [simp]: 
-  "restrict {} al = []" 
+lemma restr_empty [simp]:
+  "restrict {} al = []"
   "restrict A [] = []"
   by (induct al) (auto simp add: restrict_eq)
 
@@ -251,38 +256,39 @@
   by (induct al) (auto simp add: restrict_eq)
 
 lemma restr_update[simp]:
- "map_of (restrict D (update x y al)) = 
+ "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)"
-apply (simp add: delete_eq restrict_eq)
-apply (auto simp add: split_def)
+  "delete x (restrict D al) = (if x \<in> D then restrict (D - {x}) al else restrict D al)"
+  apply (simp add: delete_eq restrict_eq)
+  apply (auto simp add: split_def)
 proof -
-  have "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y" by auto
+  have "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y"
+    by auto
   then show "[p \<leftarrow> al. fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al. fst p \<in> D \<and> fst p \<noteq> x]"
     by simp
   assume "x \<notin> D"
-  then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" by auto
+  then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y"
+    by auto
   then show "[p \<leftarrow> al . fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al . fst p \<in> D]"
     by simp
 qed
 
 lemma update_restr:
- "map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
+  "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 update_restr_conv [simp]:
- "x \<in> D \<Longrightarrow> 
- map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
+  "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))"
+lemma restr_updates [simp]:
+  "length xs = length ys \<Longrightarrow> set xs \<subseteq> D \<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)"
@@ -291,38 +297,30 @@
 
 subsection {* @{text clearjunk} *}
 
-function clearjunk  :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
-    "clearjunk [] = []"  
-  | "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
+function clearjunk  :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+where
+  "clearjunk [] = []"
+| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
   by pat_completeness auto
-termination by (relation "measure length")
-  (simp_all add: less_Suc_eq_le length_delete_le)
+termination
+  by (relation "measure length") (simp_all add: less_Suc_eq_le length_delete_le)
 
-lemma map_of_clearjunk:
-  "map_of (clearjunk al) = map_of al"
-  by (induct al rule: clearjunk.induct)
-    (simp_all add: fun_eq_iff)
+lemma map_of_clearjunk: "map_of (clearjunk al) = map_of al"
+  by (induct al rule: clearjunk.induct) (simp_all add: fun_eq_iff)
 
-lemma clearjunk_keys_set:
-  "set (map fst (clearjunk al)) = set (map fst al)"
-  by (induct al rule: clearjunk.induct)
-    (simp_all add: delete_keys)
+lemma clearjunk_keys_set: "set (map fst (clearjunk al)) = set (map fst al)"
+  by (induct al rule: clearjunk.induct) (simp_all add: delete_keys)
 
-lemma dom_clearjunk:
-  "fst ` set (clearjunk al) = fst ` set al"
+lemma dom_clearjunk: "fst ` set (clearjunk al) = fst ` set al"
   using clearjunk_keys_set by simp
 
-lemma distinct_clearjunk [simp]:
-  "distinct (map fst (clearjunk al))"
-  by (induct al rule: clearjunk.induct)
-    (simp_all del: set_map add: clearjunk_keys_set delete_keys)
+lemma distinct_clearjunk [simp]: "distinct (map fst (clearjunk al))"
+  by (induct al rule: clearjunk.induct) (simp_all del: set_map add: clearjunk_keys_set delete_keys)
 
-lemma ran_clearjunk:
-  "ran (map_of (clearjunk al)) = ran (map_of al)"
+lemma ran_clearjunk: "ran (map_of (clearjunk al)) = ran (map_of al)"
   by (simp add: map_of_clearjunk)
 
-lemma ran_map_of:
-  "ran (map_of al) = snd ` set (clearjunk al)"
+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)
@@ -331,45 +329,42 @@
   finally show ?thesis .
 qed
 
-lemma clearjunk_update:
-  "clearjunk (update k v al) = update k v (clearjunk al)"
-  by (induct al rule: clearjunk.induct)
-    (simp_all add: delete_update)
+lemma clearjunk_update: "clearjunk (update k v al) = update k v (clearjunk al)"
+  by (induct al rule: clearjunk.induct) (simp_all add: delete_update)
 
-lemma clearjunk_updates:
-  "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
+lemma clearjunk_updates: "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
 proof -
   have "clearjunk \<circ> fold (case_prod update) (zip ks vs) =
     fold (case_prod update) (zip ks vs) \<circ> clearjunk"
     by (rule fold_commute) (simp add: clearjunk_update case_prod_beta o_def)
-  then show ?thesis by (simp add: updates_def fun_eq_iff)
+  then show ?thesis
+    by (simp add: updates_def fun_eq_iff)
 qed
 
-lemma clearjunk_delete:
-  "clearjunk (delete x al) = delete x (clearjunk al)"
+lemma clearjunk_delete: "clearjunk (delete x al) = delete x (clearjunk al)"
   by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist)
 
-lemma clearjunk_restrict:
- "clearjunk (restrict A al) = restrict A (clearjunk al)"
+lemma clearjunk_restrict: "clearjunk (restrict A al) = restrict A (clearjunk al)"
   by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist)
 
-lemma distinct_clearjunk_id [simp]:
-  "distinct (map fst al) \<Longrightarrow> clearjunk al = al"
+lemma distinct_clearjunk_id [simp]: "distinct (map fst al) \<Longrightarrow> clearjunk al = al"
   by (induct al rule: clearjunk.induct) auto
 
-lemma clearjunk_idem:
-  "clearjunk (clearjunk al) = clearjunk al"
+lemma clearjunk_idem: "clearjunk (clearjunk al) = clearjunk al"
   by simp
 
-lemma length_clearjunk:
-  "length (clearjunk al) \<le> length al"
+lemma length_clearjunk: "length (clearjunk al) \<le> length al"
 proof (induct al rule: clearjunk.induct [case_names Nil Cons])
-  case Nil then show ?case by simp
+  case Nil
+  then show ?case by simp
 next
   case (Cons kv al)
-  moreover have "length (delete (fst kv) al) \<le> length al" by (fact length_delete_le)
-  ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al" by (rule order_trans)
-  then show ?case by simp
+  moreover have "length (delete (fst kv) al) \<le> length al"
+    by (fact length_delete_le)
+  ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al"
+    by (rule order_trans)
+  then show ?case
+    by simp
 qed
 
 lemma delete_map:
@@ -386,47 +381,41 @@
 
 subsection {* @{text map_ran} *}
 
-definition map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
-  "map_ran f = map (\<lambda>(k, v). (k, f k v))"
+definition map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+  where "map_ran f = map (\<lambda>(k, v). (k, f k v))"
 
 lemma map_ran_simps [simp]:
   "map_ran f [] = []"
   "map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps"
   by (simp_all add: map_ran_def)
 
-lemma dom_map_ran:
-  "fst ` set (map_ran f al) = fst ` set al"
+lemma dom_map_ran: "fst ` set (map_ran f al) = fst ` set al"
   by (simp add: map_ran_def image_image split_def)
-  
-lemma map_ran_conv:
-  "map_of (map_ran f al) k = map_option (f k) (map_of al k)"
+
+lemma map_ran_conv: "map_of (map_ran f al) k = map_option (f k) (map_of al k)"
   by (induct al) auto
 
-lemma distinct_map_ran:
-  "distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))"
+lemma distinct_map_ran: "distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))"
   by (simp add: map_ran_def split_def comp_def)
 
-lemma map_ran_filter:
-  "map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]"
+lemma map_ran_filter: "map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]"
   by (simp add: map_ran_def filter_map split_def comp_def)
 
-lemma clearjunk_map_ran:
-  "clearjunk (map_ran f al) = map_ran f (clearjunk al)"
+lemma clearjunk_map_ran: "clearjunk (map_ran f al) = map_ran f (clearjunk al)"
   by (simp add: map_ran_def split_def clearjunk_map)
 
 
 subsection {* @{text merge} *}
 
-definition merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
-  "merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs"
+definition merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+  where "merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs"
 
 lemma merge_simps [simp]:
   "merge qs [] = qs"
   "merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)"
   by (simp_all add: merge_def split_def)
 
-lemma merge_updates:
-  "merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs"
+lemma merge_updates: "merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs"
   by (simp add: merge_def updates_def foldr_conv_fold zip_rev zip_map_fst_snd)
 
 lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys"
@@ -435,86 +424,84 @@
 lemma distinct_merge:
   assumes "distinct (map fst xs)"
   shows "distinct (map fst (merge xs ys))"
-using assms by (simp add: merge_updates distinct_updates)
+  using assms by (simp add: merge_updates distinct_updates)
 
-lemma clearjunk_merge:
-  "clearjunk (merge xs ys) = merge (clearjunk xs) ys"
+lemma clearjunk_merge: "clearjunk (merge xs ys) = merge (clearjunk xs) ys"
   by (simp add: merge_updates clearjunk_updates)
 
-lemma merge_conv':
-  "map_of (merge xs ys) = map_of xs ++ map_of ys"
+lemma merge_conv': "map_of (merge xs ys) = map_of xs ++ map_of ys"
 proof -
   have "map_of \<circ> fold (case_prod update) (rev ys) =
-    fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> map_of"
+      fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> map_of"
     by (rule fold_commute) (simp add: update_conv' case_prod_beta split_def fun_eq_iff)
   then show ?thesis
     by (simp add: merge_def map_add_map_of_foldr foldr_conv_fold fun_eq_iff)
 qed
 
-corollary merge_conv:
-  "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"
+corollary merge_conv: "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"
   by (simp add: merge_conv')
 
 lemma merge_empty: "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)"
+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)"
+lemma merge_Some_iff:
+  "map_of (merge m n) k = Some x \<longleftrightarrow>
+    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 [dest!] = merge_Some_iff [THEN iffD1]
 
-lemma merge_find_right[simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v"
+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]: 
+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]: 
+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]: 
+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)"
+lemma merge_append: "map_of (xs @ ys) = map_of (merge ys xs)"
   by (simp add: merge_conv')
 
 
 subsection {* @{text compose} *}
 
-function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" where
-    "compose [] ys = []"
-  | "compose (x#xs) 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)"
+function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list"
+where
+  "compose [] ys = []"
+| "compose (x # xs) 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)"
   by pat_completeness auto
-termination by (relation "measure (length \<circ> fst)")
-  (simp_all add: less_Suc_eq_le length_delete_le)
+termination
+  by (relation "measure (length \<circ> fst)") (simp_all add: less_Suc_eq_le length_delete_le)
 
-lemma compose_first_None [simp]: 
-  assumes "map_of xs k = None" 
+lemma compose_first_None [simp]:
+  assumes "map_of xs k = None"
   shows "map_of (compose xs ys) k = None"
-using assms by (induct xs ys rule: compose.induct)
-  (auto split: option.splits split_if_asm)
+  using assms 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"
+lemma compose_conv: "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
 proof (induct xs ys rule: compose.induct)
-  case 1 then show ?case by simp
+  case 1
+  then show ?case by simp
 next
-  case (2 x xs ys) show ?case
+  case (2 x xs ys)
+  show ?case
   proof (cases "map_of ys (snd x)")
-    case None with 2
-    have hyp: "map_of (compose (delete (fst x) xs) ys) k =
-               (map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k"
+    case None
+    with 2 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")
@@ -530,8 +517,7 @@
       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)
+        using False None by (simp add: map_comp_def)
     qed
   next
     case (Some v)
@@ -542,19 +528,19 @@
       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)"
+
+lemma compose_conv': "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" 
+  assumes "map_of xs k = Some v"
   shows "map_of (compose xs ys) k = map_of ys v"
-using assms by (simp add: compose_conv)
+  using assms 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 1 thus ?case by simp
+  case 1
+  then show ?case by simp
 next
   case (2 x xs ys)
   show ?case
@@ -580,11 +566,12 @@
 qed
 
 lemma distinct_compose:
- assumes "distinct (map fst xs)"
- shows "distinct (map fst (compose xs ys))"
-using assms
+  assumes "distinct (map fst xs)"
+  shows "distinct (map fst (compose xs ys))"
+  using assms
 proof (induct xs ys rule: compose.induct)
-  case 1 thus ?case by simp
+  case 1
+  then show ?case by simp
 next
   case (2 x xs ys)
   show ?case
@@ -593,105 +580,106 @@
     with 2 show ?thesis by simp
   next
     case (Some v)
-    with 2 dom_compose [of xs ys] show ?thesis 
-      by (auto)
+    with 2 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)"
+lemma compose_delete_twist: "compose (delete k xs) ys = delete k (compose xs ys)"
 proof (induct xs ys rule: compose.induct)
-  case 1 thus ?case by simp
+  case 1
+  then show ?case by simp
 next
   case (2 x xs ys)
   show ?case
   proof (cases "map_of ys (snd x)")
     case None
-    with 2 have 
-      hyp: "compose (delete k (delete (fst x) xs)) ys =
-            delete k (compose (delete (fst x) xs) ys)"
+    with 2 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
+      with None hyp show ?thesis
         by (simp add: delete_idem)
     next
       case False
-      from None False hyp
-      show ?thesis
+      from None False hyp show ?thesis
         by (simp add: delete_twist)
     qed
   next
     case (Some v)
-    with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" by simp
+    with 2 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)
-   
+  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 [] = []"
+    (auto split: option.splits simp add: clearjunk_delete delete_idem compose_delete_twist)
+
+lemma compose_empty [simp]: "compose xs [] = []"
   by (induct xs) (auto simp add: compose_delete_twist)
 
 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)" 
+  "(map_of (compose xs ys) k = Some v) \<longleftrightarrow>
+    (\<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)) " 
+  "map_of (compose xs ys) k = None \<longleftrightarrow>
+    (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 {* @{text map_entry} *}
 
 fun map_entry :: "'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
 where
   "map_entry k f [] = []"
-| "map_entry k f (p # ps) = (if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)"
+| "map_entry k f (p # ps) =
+    (if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)"
 
 lemma map_of_map_entry:
-  "map_of (map_entry k f xs) = (map_of xs)(k := case map_of xs k of None => None | Some v' => Some (f v'))"
-by (induct xs) auto
+  "map_of (map_entry k f xs) =
+    (map_of xs)(k := case map_of xs k of None \<Rightarrow> None | Some v' \<Rightarrow> Some (f v'))"
+  by (induct xs) auto
 
-lemma dom_map_entry:
-  "fst ` set (map_entry k f xs) = fst ` set xs"
-by (induct xs) auto
+lemma dom_map_entry: "fst ` set (map_entry k f xs) = fst ` set xs"
+  by (induct xs) auto
 
 lemma distinct_map_entry:
   assumes "distinct (map fst xs)"
   shows "distinct (map fst (map_entry k f xs))"
-using assms by (induct xs) (auto simp add: dom_map_entry)
+  using assms by (induct xs) (auto simp add: dom_map_entry)
+
 
 subsection {* @{text map_default} *}
 
 fun map_default :: "'key \<Rightarrow> 'val \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
 where
   "map_default k v f [] = [(k, v)]"
-| "map_default k v f (p # ps) = (if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)"
+| "map_default k v f (p # ps) =
+    (if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)"
 
 lemma map_of_map_default:
-  "map_of (map_default k v f xs) = (map_of xs)(k := case map_of xs k of None => Some v | Some v' => Some (f v'))"
-by (induct xs) auto
+  "map_of (map_default k v f xs) =
+    (map_of xs)(k := case map_of xs k of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (f v'))"
+  by (induct xs) auto
 
-lemma dom_map_default:
-  "fst ` set (map_default k v f xs) = insert k (fst ` set xs)" 
-by (induct xs) auto
+lemma dom_map_default: "fst ` set (map_default k v f xs) = insert k (fst ` set xs)"
+  by (induct xs) auto
 
 lemma distinct_map_default:
   assumes "distinct (map fst xs)"
   shows "distinct (map fst (map_default k v f xs))"
-using assms by (induct xs) (auto simp add: dom_map_default)
+  using assms by (induct xs) (auto simp add: dom_map_default)
 
 hide_const (open) update updates delete restrict clearjunk merge compose map_entry