author wenzelm Sun, 30 Mar 2014 21:24:59 +0200 changeset 56327 3e62e68fb342 parent 56326 c3d7b3bb2708 child 56328 b3551501424e child 56333 38f1422ef473
tuned proofs;
 src/HOL/Library/AList.thy file | annotate | diff | comparison | revisions
```--- 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))"

-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"
@@ -78,8 +83,8 @@
"x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A"

-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)"

"updates [] vs ps = ps"
@@ -95,9 +100,10 @@
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
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
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)

- "\<lbrakk>size ks \<le> i; i < size vs\<rbrakk>
-  by (induct ks arbitrary: al vs i) (auto split:list.splits nat.splits)
+  "size ks \<le> i \<Longrightarrow> i < size vs \<Longrightarrow>
+  by (induct ks arbitrary: al vs i) (auto split: list.splits nat.splits)

- 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)))"
+  "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)))"

- "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))"

- "k \<notin> set ks ==> map_of (updates ks vs al) k = map_of al k"
+  "k \<notin> set ks \<Longrightarrow> map_of (updates ks vs al) k = map_of al k"

-  "size xs = size ys \<Longrightarrow> updates (xs@zs) ys al = updates xs ys al"
+  "size xs = size ys \<Longrightarrow> updates (xs @ zs) ys al = updates xs ys al"
by (induct xs arbitrary: ys al) (auto split: list.splits)

-  "size xs = size ys \<Longrightarrow> updates xs (ys@zs) al = updates xs ys al"
+  "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)"

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'"

-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"

-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'"

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"

-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))"

lemma restr_delete [simp]:
-  "(delete x (restrict D al)) =
-    (if x \<in> D then restrict (D - {x}) al else restrict D al)"
+  "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))"

- \<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))"
+  "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))"

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")
+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)
+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)
+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)"

-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))"
@@ -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)
+lemma clearjunk_update: "clearjunk (update k v al) = update k v (clearjunk al)"
+  by (induct al rule: clearjunk.induct) (simp_all add: delete_update)

-  "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
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"

-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)"

-  "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"

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))"

-lemma clearjunk_merge:
-  "clearjunk (merge xs ys) = merge (clearjunk xs) ys"
+lemma clearjunk_merge: "clearjunk (merge xs ys) = merge (clearjunk xs) ys"

-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
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"

lemma merge_empty: "map_of (merge [] ys) = map_of ys"

-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)"

-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"

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"

-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)"

-lemma merge_upd[simp]:
+lemma merge_upd [simp]:
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))"

"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))"

-lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)"
+lemma merge_append: "map_of (xs @ ys) = map_of (merge ys xs)"

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)")
+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
+        using False None by (simp add: map_comp_def)
qed
next
case (Some v)
@@ -542,19 +528,19 @@
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
next
case False
-      from None False hyp
-      show ?thesis
+      from None False hyp show ?thesis
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)"

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))"

+
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
```