(* Title: HOL/Library/AssocList.thy
ID: $Id$
Author: Norbert Schirmer, Tobias Nipkow, Martin Wildmoser
*)
header {* Map operations implemented on association lists*}
theory AssocList
imports Plain "~~/src/HOL/Map"
begin
text {*
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.
*}
primrec
delete :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"delete k [] = []"
| "delete k (p#ps) = (if fst p = k then delete k ps else p # delete k 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)"
primrec
updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"updates [] vs ps = ps"
| "updates (k#ks) vs ps = (case vs
of [] \<Rightarrow> ps
| (v#vs') \<Rightarrow> updates ks vs' (update k v ps))"
primrec
merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"merge qs [] = qs"
| "merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)"
lemma length_delete_le: "length (delete k al) \<le> length al"
proof (induct al)
case Nil thus ?case by simp
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\<leftarrow>al . fst p \<noteq> fst a] \<le> Suc (length al)" .
with Cons show ?case
by auto
qed
lemma compose_hint [simp]:
"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
fun
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)"
primrec
restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"restrict A [] = []"
| "restrict A (p#ps) = (if fst p \<in> A then p#restrict A ps else restrict A ps)"
primrec
map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"map_ran f [] = []"
| "map_ran f (p#ps) = (fst p, f (fst p) (snd p)) # map_ran f ps"
fun
clearjunk :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"clearjunk [] = []"
| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
lemmas [simp del] = compose_hint
subsection {* @{const delete} *}
lemma delete_eq:
"delete k xs = filter (\<lambda>p. fst p \<noteq> k) xs"
by (induct xs) auto
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 assms
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
with Cons 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_eq)
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_eq)
lemma map_of_filter: "k \<noteq> a \<Longrightarrow> map_of [q\<leftarrow>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 p ps)
from Cons have "length (clearjunk [q\<leftarrow>ps . fst q \<noteq> fst p]) \<le> length [q\<leftarrow>ps . fst q \<noteq> fst p]"
by (simp add: delete_eq)
also have "\<dots> \<le> length ps"
by simp
finally show ?case
by (simp add: delete_eq)
qed
lemma notin_fst_filter: "a \<notin> fst ` set ps \<Longrightarrow> [q\<leftarrow>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 assms
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\<leftarrow>ps . fst q \<noteq> a] = [q\<leftarrow>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_eq)
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 arbitrary: 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 arbitrary: 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 assms
by (induct ks arbitrary: 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 arbitrary: 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 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)
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 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"
by (induct xs arbitrary: ys al) (auto split: list.splits)
subsection {* @{const map_ran} *}
lemma map_ran_conv: "map_of (map_ran f al) k = option_map (f k) (map_of al k)"
by (induct al) auto
lemma dom_map_ran: "fst ` set (map_ran f al) = fst ` set al"
by (induct al) auto
lemma distinct_map_ran: "distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))"
by (induct al) (auto simp add: dom_map_ran)
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 (induct ps) auto
lemma clearjunk_map_ran: "clearjunk (map_ran f al) = map_ran f (clearjunk al)"
by (induct al rule: clearjunk.induct) (auto simp add: delete_eq map_ran_filter)
subsection {* @{const merge} *}
lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys"
by (induct ys arbitrary: xs) (auto simp add: dom_update)
lemma distinct_merge:
assumes "distinct (map fst xs)"
shows "distinct (map fst (merge xs ys))"
using assms
by (induct ys arbitrary: 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_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)
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 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: "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 2
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 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
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with "2.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 "2.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 assms
proof (induct xs ys rule: compose.induct)
case 1 thus ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with 2 show ?thesis by simp
next
case (Some v)
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)"
proof (induct xs ys rule: compose.induct)
case 1 thus ?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)"
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 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)
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) (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)"
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_eq:
"restrict A = filter (\<lambda>p. fst p \<in> A)"
proof
fix xs
show "restrict A xs = filter (\<lambda>p. fst p \<in> A) xs"
by (induct xs) auto
qed
lemma distinct_restr: "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))"
by (induct al) (auto simp add: restrict_eq)
lemma restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k"
apply (induct al)
apply (simp add: restrict_eq)
apply (cases "k\<in>A")
apply (auto simp add: restrict_eq)
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_eq)
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_eq)
lemma restr_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) al"
by (induct al) (auto simp add: restrict_eq)
lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al"
by (induct al) (auto simp add: restrict_eq)
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_eq)
also from not_fst_a_x True hyp have "\<dots> = restrict (D - {x}) (a # al)"
by (cases a) (simp add: restrict_eq)
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_eq)
moreover from False not_fst_a_x
have "restrict (D - {x}) (a # al) = restrict (D - {x}) al"
by (cases a) (simp add: restrict_eq)
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