--- 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