src/HOL/Library/AList.thy

fixed typo in theorem name in AList theory

(*  Title:      HOL/Library/AList.thy
    Author:     Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen
*)

header {* Implementation of Association Lists *}

theory AList
imports Main More_List
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.
*}

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)"

lemma update_conv': "map_of (update k v al)  = (map_of al)(k\<mapsto>v)"
  by (induct al) (auto simp add: fun_eq_iff)

corollary update_conv: "map_of (update k v al) k' = ((map_of al)(k\<mapsto>v)) k'"
  by (simp add: update_conv')

lemma dom_update: "fst ` set (update k v al) = {k} \<union> fst ` set al"
  by (induct al) auto

lemma update_keys:
  "map fst (update k v al) =
    (if k \<in> set (map fst al) then map fst al else map fst al @ [k])"
  by (induct al) simp_all

lemma distinct_update:
  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]"
  by (induct ps) auto

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 (simp add: update_conv' fun_eq_iff)

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)

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)

definition updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
  "updates ks vs = More_List.fold (prod_case update) (zip ks vs)"

lemma updates_simps [simp]:
  "updates [] vs ps = ps"
  "updates ks [] ps = ps"
  "updates (k#ks) (v#vs) ps = updates ks vs (update k v ps)"
  by (simp_all add: updates_def)

lemma updates_key_simp [simp]:
  "updates (k # ks) vs ps =
    (case vs of [] \<Rightarrow> ps | v # vs \<Rightarrow> updates ks vs (update k v ps))"
  by (cases vs) simp_all

lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)"
proof -
  have "map_of \<circ> More_List.fold (prod_case update) (zip ks vs) =
    More_List.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_fold split_def)
qed

lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\<mapsto>]vs)) k"
  by (simp add: updates_conv')

lemma distinct_updates:
  assumes "distinct (map fst al)"
  shows "distinct (map fst (updates ks vs al))"
proof -
  have "distinct (More_List.fold
       (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k])
       (zip ks vs) (map fst al))"
    by (rule fold_invariant [of "zip ks vs" "\<lambda>_. True"]) (auto intro: assms)
  moreover have "map fst \<circ> More_List.fold (prod_case update) (zip ks vs) =
    More_List.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 prod_case_beta comp_def)
  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)"
  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 {* @{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')"

lemma delete_simps [simp]:
  "delete k [] = []"
  "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)"
  by (induct al) (auto simp add: fun_eq_iff)

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)"
  by (simp add: delete_eq removeAll_filter_not_eq filter_map split_def comp_def)

lemma distinct_delete:
  assumes "distinct (map fst al)" 
  shows "distinct (map fst (delete k al))"
  using assms by (simp add: delete_keys distinct_removeAll)

lemma delete_id [simp]: "k \<notin> fst ` set al \<Longrightarrow> delete k al = al"
  by (auto simp add: image_iff delete_eq filter_id_conv)

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'"
  by (simp add: delete_conv')

lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)"
  by (auto simp add: delete_eq)

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"
  by (induct al) simp_all

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)"
  by (simp add: delete_eq conj_commute)

lemma length_delete_le: "length (delete k al) \<le> length al"
  by (simp add: delete_eq)


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)"

lemma restr_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_eq)

lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)"
proof
  fix k
  show "map_of (restrict A al) k = ((map_of al)|` A) k"
    by (induct al) (simp, cases "k \<in> A", auto)
qed

corollary restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k"
  by (simp add: restr_conv')

lemma distinct_restr:
  "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 = []" 
  "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)"
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
  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 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))"
  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))"
  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


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)"
  by pat_completeness auto
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 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"
  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 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)"
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

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)"
proof -
  have "clearjunk \<circ> More_List.fold (prod_case update) (zip ks vs) =
    More_List.fold (prod_case update) (zip ks vs) \<circ> clearjunk"
    by (rule fold_commute) (simp add: clearjunk_update prod_case_beta o_def)
  then show ?thesis by (simp add: updates_def fun_eq_iff)
qed

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)"
  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"
  by (induct al rule: clearjunk.induct) auto

lemma clearjunk_idem:
  "clearjunk (clearjunk al) = clearjunk al"
  by simp

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
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
qed

lemma delete_map:
  assumes "\<And>kv. fst (f kv) = fst kv"
  shows "delete k (map f ps) = map f (delete k ps)"
  by (simp add: delete_eq filter_map comp_def split_def assms)

lemma clearjunk_map:
  assumes "\<And>kv. fst (f kv) = fst kv"
  shows "clearjunk (map f ps) = map f (clearjunk ps)"
  by (induct ps rule: clearjunk.induct [case_names Nil Cons])
    (simp_all add: clearjunk_delete delete_map assms)


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))"

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"
  by (simp add: map_ran_def image_image split_def)
  
lemma map_ran_conv:
  "map_of (map_ran f al) k = Option.map (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))"
  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]"
  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)"
  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"

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"
  by (simp add: merge_def updates_def foldr_fold_rev zip_rev zip_map_fst_snd)

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 (simp add: merge_updates distinct_updates)

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"
proof -
  have "map_of \<circ> More_List.fold (prod_case update) (rev ys) =
    More_List.fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> map_of"
    by (rule fold_commute) (simp add: update_conv' prod_case_beta split_def fun_eq_iff)
  then show ?thesis
    by (simp add: merge_def map_add_map_of_foldr foldr_fold_rev fun_eq_iff)
qed

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)"
  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 [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"
  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 {* @{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)"
  by pat_completeness auto
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" 
  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)

end