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