src/HOL/Library/RBT.thy

 
 (*<*)
 theory RBT
 imports Main AssocList
 begin
+
+subsection {* Datatype of RB trees *}
 
 datatype color = R | B
 datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"
 
 lemma rbt_cases:

   case Empty with that show thesis by blast
 next
   case (Branch c) with that show thesis by (cases c) blast+
 qed
 
-text {* Content of a tree *}
-
-primrec entries
+subsection {* Tree properties *}
+
+subsubsection {* Content of a tree *}
+
+primrec entries :: "('a, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
 where 
   "entries Empty = []"
 | "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"
 
-text {* Search tree properties *}
-
-primrec entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
-where
-  "entry_in_tree k v Empty = False"
-| "entry_in_tree k v (Branch c l x y r) \<longleftrightarrow> k = x \<and> v = y \<or> entry_in_tree k v l \<or> entry_in_tree k v r"
-
-primrec keys :: "('k, 'v) rbt \<Rightarrow> 'k set"
-where
-  "keys Empty = {}"
-| "keys (Branch _ l k _ r) = { k } \<union> keys l \<union> keys r"
+abbreviation (input) entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
+where
+  "entry_in_tree k v t \<equiv> (k, v) \<in> set (entries t)"
+
+definition keys :: "('a, 'b) rbt \<Rightarrow> 'a list" where
+  "keys t = map fst (entries t)"
+
+lemma keys_simps [simp, code]:
+  "keys Empty = []"
+  "keys (Branch c l k v r) = keys l @ k # keys r"
+  by (simp_all add: keys_def)
 
 lemma entry_in_tree_keys:
-  "entry_in_tree k v t \<Longrightarrow> k \<in> keys t"
-  by (induct t) auto
+  assumes "(k, v) \<in> set (entries t)"
+  shows "k \<in> set (keys t)"
+proof -
+  from assms have "fst (k, v) \<in> fst ` set (entries t)" by (rule imageI)
+  then show ?thesis by (simp add: keys_def)
+qed
+
+
+subsubsection {* Search tree properties *}
 
 definition tree_less :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
 where
-  tree_less_prop: "tree_less k t \<longleftrightarrow> (\<forall>x\<in>keys t. x < k)"
+  tree_less_prop: "tree_less k t \<longleftrightarrow> (\<forall>x\<in>set (keys t). x < k)"
 
 abbreviation tree_less_symbol (infix "|\<guillemotleft>" 50)
 where "t |\<guillemotleft> x \<equiv> tree_less x t"
 
 definition tree_greater :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
 where
-  tree_greater_prop: "tree_greater k t = (\<forall>x\<in>keys t. k < x)"
+  tree_greater_prop: "tree_greater k t = (\<forall>x\<in>set (keys t). k < x)"
 
 lemma tree_less_simps [simp]:
   "tree_less k Empty = True"
   "tree_less k (Branch c lt kt v rt) \<longleftrightarrow> kt < k \<and> tree_less k lt \<and> tree_less k rt"
   by (auto simp add: tree_less_prop)

 lemmas tree_ord_props = tree_less_prop tree_greater_prop
 
 lemmas tree_greater_nit = tree_greater_prop entry_in_tree_keys
 lemmas tree_less_nit = tree_less_prop entry_in_tree_keys
 
-lemma tree_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
+lemma tree_less_eq_trans: "l |\<guillemotleft> u \<Longrightarrow> u \<le> v \<Longrightarrow> l |\<guillemotleft> v"
+  and tree_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
+  and tree_greater_eq_trans: "u \<le> v \<Longrightarrow> v \<guillemotleft>| r \<Longrightarrow> u \<guillemotleft>| r"
   and tree_greater_trans: "x < y \<Longrightarrow> y \<guillemotleft>| t \<Longrightarrow> x \<guillemotleft>| t"
-by (auto simp: tree_ord_props)
+  by (auto simp: tree_ord_props)
 
 primrec sorted :: "('a::linorder, 'b) rbt \<Rightarrow> bool"
 where
   "sorted Empty = True"
 | "sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> sorted l \<and> sorted r)"
 
+lemma sorted_entries:
+  "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
+by (induct t) 
+  (force simp: sorted_append sorted_Cons tree_ord_props 
+      dest!: entry_in_tree_keys)+
+
+lemma distinct_entries:
+  "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
+by (induct t) 
+  (force simp: sorted_append sorted_Cons tree_ord_props 
+      dest!: entry_in_tree_keys)+
+
+
+subsubsection {* Tree lookup *}
+
 primrec lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
 where
   "lookup Empty k = None"
 | "lookup (Branch _ l x y r) k = (if k < x then lookup l k else if x < k then lookup r k else Some y)"
 
+lemma lookup_keys: "sorted t \<Longrightarrow> dom (lookup t) = set (keys t)"
+  by (induct t) (auto simp: dom_def tree_greater_prop tree_less_prop)
+
+lemma dom_lookup_Branch: 
+  "sorted (Branch c t1 k v t2) \<Longrightarrow> 
+    dom (lookup (Branch c t1 k v t2)) 
+    = Set.insert k (dom (lookup t1) \<union> dom (lookup t2))"
+proof -
+  assume "sorted (Branch c t1 k v t2)"
+  moreover from this have "sorted t1" "sorted t2" by simp_all
+  ultimately show ?thesis by (simp add: lookup_keys)
+qed
+
+lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
+proof (induct t)
+  case Empty then show ?case by simp
+next
+  case (Branch color t1 a b t2)
+  let ?A = "Set.insert a (dom (lookup t1) \<union> dom (lookup t2))"
+  have "dom (lookup (Branch color t1 a b t2)) \<subseteq> ?A" by (auto split: split_if_asm)
+  moreover from Branch have "finite (insert a (dom (lookup t1) \<union> dom (lookup t2)))" by simp
+  ultimately show ?case by (rule finite_subset)
+qed 
+
 lemma lookup_tree_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> lookup t k = None" 
 by (induct t) auto
 
 lemma lookup_tree_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> lookup t k = None"
 by (induct t) auto
 
-lemma lookup_keys: "sorted t \<Longrightarrow> dom (lookup t) = keys t"
-by (induct t) (auto simp: dom_def tree_greater_prop tree_less_prop)
-
-lemma lookup_pit: "sorted t \<Longrightarrow> (lookup t k = Some v) = entry_in_tree k v t"
+lemma lookup_in_tree: "sorted t \<Longrightarrow> (lookup t k = Some v) = entry_in_tree k v t"
 by (induct t) (auto simp: tree_less_prop tree_greater_prop entry_in_tree_keys)
 
 lemma lookup_Empty: "lookup Empty = empty"
 by (rule ext) simp
 
+lemma lookup_map_of_entries:
+  "sorted t \<Longrightarrow> lookup t = map_of (entries t)"
+proof (induct t)
+  case Empty thus ?case by (simp add: lookup_Empty)
+next
+  case (Branch c t1 k v t2)
+  have "lookup (Branch c t1 k v t2) = lookup t2 ++ [k\<mapsto>v] ++ lookup t1"
+  proof (rule ext)
+    fix x
+    from Branch have SORTED: "sorted (Branch c t1 k v t2)" by simp
+    let ?thesis = "lookup (Branch c t1 k v t2) x = (lookup t2 ++ [k \<mapsto> v] ++ lookup t1) x"
+
+    have DOM_T1: "!!k'. k'\<in>dom (lookup t1) \<Longrightarrow> k>k'"
+    proof -
+      fix k'
+      from SORTED have "t1 |\<guillemotleft> k" by simp
+      with tree_less_prop have "\<forall>k'\<in>set (keys t1). k>k'" by auto
+      moreover assume "k'\<in>dom (lookup t1)"
+      ultimately show "k>k'" using lookup_keys SORTED by auto
+    qed
+    
+    have DOM_T2: "!!k'. k'\<in>dom (lookup t2) \<Longrightarrow> k<k'"
+    proof -
+      fix k'
+      from SORTED have "k \<guillemotleft>| t2" by simp
+      with tree_greater_prop have "\<forall>k'\<in>set (keys t2). k<k'" by auto
+      moreover assume "k'\<in>dom (lookup t2)"
+      ultimately show "k<k'" using lookup_keys SORTED by auto
+    qed
+    
+    {
+      assume C: "x<k"
+      hence "lookup (Branch c t1 k v t2) x = lookup t1 x" by simp
+      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
+      moreover have "x\<notin>dom (lookup t2)" proof
+        assume "x\<in>dom (lookup t2)"
+        with DOM_T2 have "k<x" by blast
+        with C show False by simp
+      qed
+      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+    } moreover {
+      assume [simp]: "x=k"
+      hence "lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
+      moreover have "x\<notin>dom (lookup t1)" proof
+        assume "x\<in>dom (lookup t1)"
+        with DOM_T1 have "k>x" by blast
+        thus False by simp
+      qed
+      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+    } moreover {
+      assume C: "x>k"
+      hence "lookup (Branch c t1 k v t2) x = lookup t2 x" by (simp add: less_not_sym[of k x])
+      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
+      moreover have "x\<notin>dom (lookup t1)" proof
+        assume "x\<in>dom (lookup t1)"
+        with DOM_T1 have "k>x" by simp
+        with C show False by simp
+      qed
+      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+    } ultimately show ?thesis using less_linear by blast
+  qed
+  also from Branch have "lookup t2 ++ [k \<mapsto> v] ++ lookup t1 = map_of (entries (Branch c t1 k v t2))" by simp
+  finally show ?case .
+qed
+
+(*lemma map_of_inject:
+  assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
+  shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys"
+
+lemma entries_eqI:
+  assumes sorted: "sorted t1" "sorted t2" 
+  assumes lookup: "lookup t1 = lookup t2"
+  shows entries: "entries t1 = entries t2"
+proof -
+  from sorted lookup have "map_of (entries t1) = map_of (entries t2)"
+    by (simp_all add: lookup_map_of_entries)
+qed*)
+
 (* a kind of extensionality *)
-lemma lookup_from_pit: 
+lemma lookup_from_in_tree: 
   assumes sorted: "sorted t1" "sorted t2" 
   and eq: "\<And>v. entry_in_tree (k\<Colon>'a\<Colon>linorder) v t1 = entry_in_tree k v t2" 
   shows "lookup t1 k = lookup t2 k"
 proof (cases "lookup t1 k")
   case None
   then have "\<And>v. \<not> entry_in_tree k v t1"
-    by (simp add: lookup_pit[symmetric] sorted)
+    by (simp add: lookup_in_tree[symmetric] sorted)
   with None show ?thesis
-    by (cases "lookup t2 k") (auto simp: lookup_pit sorted eq)
+    by (cases "lookup t2 k") (auto simp: lookup_in_tree sorted eq)
 next
   case (Some a)
   then show ?thesis
     apply (cases "lookup t2 k")
-    apply (auto simp: lookup_pit sorted eq)
-    by (auto simp add: lookup_pit[symmetric] sorted Some)
+    apply (auto simp: lookup_in_tree sorted eq)
+    by (auto simp add: lookup_in_tree[symmetric] sorted Some)
 qed
 
-subsection {* Red-black properties *}
+
+subsubsection {* Red-black properties *}
 
 primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"
 where
   "color_of Empty = B"
 | "color_of (Branch c _ _ _ _) = c"

   from "5_4" have "y < z \<and> tree_greater z (Branch B vd ve vii vf)" by simp
   hence "tree_greater y (Branch B vd ve vii vf)" by (blast dest: tree_greater_trans)
   with 1 "5_4" show ?case by simp
 qed simp+
 
-lemma keys_balance[simp]: 
-  "keys (balance l k v r) = { k } \<union> keys l \<union> keys r"
-by (induct l k v r rule: balance.induct) auto
-
-lemma balance_pit:  
-  "entry_in_tree k x (balance l v y r) = (entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r)" 
-by (induct l v y r rule: balance.induct) auto
+lemma entries_balance [simp]:
+  "entries (balance l k v r) = entries l @ (k, v) # entries r"
+  by (induct l k v r rule: balance.induct) auto
+
+lemma keys_balance [simp]: 
+  "keys (balance l k v r) = keys l @ k # keys r"
+  by (simp add: keys_def)
+
+lemma balance_in_tree:  
+  "entry_in_tree k x (balance l v y r) \<longleftrightarrow> entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r"
+  by (auto simp add: keys_def)
 
 lemma lookup_balance[simp]: 
 fixes k :: "'a::linorder"
 assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
 shows "lookup (balance l k v r) x = lookup (Branch B l k v r) x"
-by (rule lookup_from_pit) (auto simp:assms balance_pit balance_sorted)
+by (rule lookup_from_in_tree) (auto simp:assms balance_in_tree balance_sorted)
 
 primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
   "paint c Empty = Empty"
 | "paint c (Branch _ l k v r) = Branch c l k v r"

 lemma paint_inv1l[simp]: "inv1l t \<Longrightarrow> inv1l (paint c t)" by (cases t) auto
 lemma paint_inv1[simp]: "inv1l t \<Longrightarrow> inv1 (paint B t)" by (cases t) auto
 lemma paint_inv2[simp]: "inv2 t \<Longrightarrow> inv2 (paint c t)" by (cases t) auto
 lemma paint_color_of[simp]: "color_of (paint B t) = B" by (cases t) auto
 lemma paint_sorted[simp]: "sorted t \<Longrightarrow> sorted (paint c t)" by (cases t) auto
-lemma paint_pit[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
+lemma paint_in_tree[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
 lemma paint_lookup[simp]: "lookup (paint c t) = lookup t" by (rule ext) (cases t, auto)
 lemma paint_tree_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
 lemma paint_tree_less[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
 
 fun

 lemma ins_tree_less[simp]: "(ins f k x t |\<guillemotleft> v) = (t |\<guillemotleft> v \<and> k < v)"
   by (induct f k x t rule: ins.induct) auto
 lemma ins_sorted[simp]: "sorted t \<Longrightarrow> sorted (ins f k x t)"
   by (induct f k x t rule: ins.induct) (auto simp: balance_sorted)
 
-lemma keys_ins: "keys (ins f k v t) = { k } \<union> keys t"
-by (induct f k v t rule: ins.induct) auto
+lemma keys_ins: "set (keys (ins f k v t)) = { k } \<union> set (keys t)"
+  by (induct f k v t rule: ins.induct) auto
 
 lemma lookup_ins: 
   fixes k :: "'a::linorder"
   assumes "sorted t"
   shows "lookup (ins f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v 
                                                        | Some w \<Rightarrow> f k w v)) x"
 using assms by (induct f k v t rule: ins.induct) auto
 
 definition
-  insertwithkey :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "insertwithkey f k v t = paint B (ins f k v t)"
-
-lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insertwithkey f k x t)"
-  by (auto simp: insertwithkey_def)
+  insert_with_key :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "insert_with_key f k v t = paint B (ins f k v t)"
+
+lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insert_with_key f k x t)"
+  by (auto simp: insert_with_key_def)
 
 theorem insertwk_is_rbt: 
   assumes inv: "is_rbt t" 
-  shows "is_rbt (insertwithkey f k x t)"
+  shows "is_rbt (insert_with_key f k x t)"
 using assms
-unfolding insertwithkey_def is_rbt_def
+unfolding insert_with_key_def is_rbt_def
 by (auto simp: ins_inv1_inv2)
 
 lemma lookup_insertwk: 
   assumes "sorted t"
-  shows "lookup (insertwithkey f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v 
+  shows "lookup (insert_with_key f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v 
                                                        | Some w \<Rightarrow> f k w v)) x"
-unfolding insertwithkey_def using assms
+unfolding insert_with_key_def using assms
 by (simp add:lookup_ins)
 
 definition
-  insertw_def: "insertwith f = insertwithkey (\<lambda>_. f)"
-
-lemma insertw_sorted: "sorted t \<Longrightarrow> sorted (insertwith f k v t)" by (simp add: insertwk_sorted insertw_def)
-theorem insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insertwith f k v t)" by (simp add: insertwk_is_rbt insertw_def)
+  insertw_def: "insert_with f = insert_with_key (\<lambda>_. f)"
+
+lemma insertw_sorted: "sorted t \<Longrightarrow> sorted (insert_with f k v t)" by (simp add: insertwk_sorted insertw_def)
+theorem insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insert_with f k v t)" by (simp add: insertwk_is_rbt insertw_def)
 
 lemma lookup_insertw:
   assumes "is_rbt t"
-  shows "lookup (insertwith f k v t) = (lookup t)(k \<mapsto> (if k:dom (lookup t) then f (the (lookup t k)) v else v))"
+  shows "lookup (insert_with f k v t) = (lookup t)(k \<mapsto> (if k:dom (lookup t) then f (the (lookup t k)) v else v))"
 using assms
 unfolding insertw_def
 by (rule_tac ext) (cases "lookup t k", auto simp:lookup_insertwk dom_def)
 
 definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "insert k v t = insertwithkey (\<lambda>_ _ nv. nv) k v t"
+  "insert = insert_with_key (\<lambda>_ _ nv. nv)"
 
 lemma insert_sorted: "sorted t \<Longrightarrow> sorted (insert k v t)" by (simp add: insertwk_sorted insert_def)
-theorem insert_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insert k v t)" by (simp add: insertwk_is_rbt insert_def)
+theorem insert_is_rbt [simp]: "is_rbt t \<Longrightarrow> is_rbt (insert k v t)" by (simp add: insertwk_is_rbt insert_def)
 
 lemma lookup_insert: 
   assumes "is_rbt t"
   shows "lookup (insert k v t) = (lookup t)(k\<mapsto>v)"
 unfolding insert_def

 
 lemma bheight_paintR'[simp]: "color_of t = B \<Longrightarrow> bheight (paint R t) = bheight t - 1"
 by (cases t rule: rbt_cases) auto
 
 fun
-  balleft :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "balleft (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
-  "balleft bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
-  "balleft bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
-  "balleft t k x s = Empty"
-
-lemma balleft_inv2_with_inv1:
+  balance_left :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "balance_left (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
+  "balance_left bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
+  "balance_left bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
+  "balance_left t k x s = Empty"
+
+lemma balance_left_inv2_with_inv1:
   assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"
-  shows "bheight (balleft lt k v rt) = bheight lt + 1"
-  and   "inv2 (balleft lt k v rt)"
+  shows "bheight (balance_left lt k v rt) = bheight lt + 1"
+  and   "inv2 (balance_left lt k v rt)"
 using assms 
-by (induct lt k v rt rule: balleft.induct) (auto simp: balance_inv2 balance_bheight)
-
-lemma balleft_inv2_app: 
+by (induct lt k v rt rule: balance_left.induct) (auto simp: balance_inv2 balance_bheight)
+
+lemma balance_left_inv2_app: 
   assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"
-  shows "inv2 (balleft lt k v rt)" 
-        "bheight (balleft lt k v rt) = bheight rt"
+  shows "inv2 (balance_left lt k v rt)" 
+        "bheight (balance_left lt k v rt) = bheight rt"
 using assms 
-by (induct lt k v rt rule: balleft.induct) (auto simp add: balance_inv2 balance_bheight)+ 
-
-lemma balleft_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balleft a k x b)"
-  by (induct a k x b rule: balleft.induct) (simp add: balance_inv1)+
-
-lemma balleft_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balleft lt k x rt)"
-by (induct lt k x rt rule: balleft.induct) (auto simp: balance_inv1)
-
-lemma balleft_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balleft l k v r)"
-apply (induct l k v r rule: balleft.induct)
+by (induct lt k v rt rule: balance_left.induct) (auto simp add: balance_inv2 balance_bheight)+ 
+
+lemma balance_left_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balance_left a k x b)"
+  by (induct a k x b rule: balance_left.induct) (simp add: balance_inv1)+
+
+lemma balance_left_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balance_left lt k x rt)"
+by (induct lt k x rt rule: balance_left.induct) (auto simp: balance_inv1)
+
+lemma balance_left_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_left l k v r)"
+apply (induct l k v r rule: balance_left.induct)
 apply (auto simp: balance_sorted)
 apply (unfold tree_greater_prop tree_less_prop)
 by force+
 
-lemma balleft_tree_greater: 
+lemma balance_left_tree_greater: 
   fixes k :: "'a::order"
   assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
-  shows "k \<guillemotleft>| balleft a x t b"
+  shows "k \<guillemotleft>| balance_left a x t b"
 using assms 
-by (induct a x t b rule: balleft.induct) auto
-
-lemma balleft_tree_less: 
+by (induct a x t b rule: balance_left.induct) auto
+
+lemma balance_left_tree_less: 
   fixes k :: "'a::order"
   assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
-  shows "balleft a x t b |\<guillemotleft> k"
+  shows "balance_left a x t b |\<guillemotleft> k"
 using assms
-by (induct a x t b rule: balleft.induct) auto
-
-lemma balleft_pit: 
+by (induct a x t b rule: balance_left.induct) auto
+
+lemma balance_left_in_tree: 
   assumes "inv1l l" "inv1 r" "bheight l + 1 = bheight r"
-  shows "entry_in_tree k v (balleft l a b r) = (entry_in_tree k v l \<or> k = a \<and> v = b \<or> entry_in_tree k v r)"
+  shows "entry_in_tree k v (balance_left l a b r) = (entry_in_tree k v l \<or> k = a \<and> v = b \<or> entry_in_tree k v r)"
 using assms 
-by (induct l k v r rule: balleft.induct) (auto simp: balance_pit)
+by (induct l k v r rule: balance_left.induct) (auto simp: balance_in_tree)
 
 fun
-  balright :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "balright a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
-  "balright (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
-  "balright (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
-  "balright t k x s = Empty"
-
-lemma balright_inv2_with_inv1:
+  balance_right :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "balance_right a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
+  "balance_right (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
+  "balance_right (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
+  "balance_right t k x s = Empty"
+
+lemma balance_right_inv2_with_inv1:
   assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt + 1" "inv1 lt"
-  shows "inv2 (balright lt k v rt) \<and> bheight (balright lt k v rt) = bheight lt"
+  shows "inv2 (balance_right lt k v rt) \<and> bheight (balance_right lt k v rt) = bheight lt"
 using assms
-by (induct lt k v rt rule: balright.induct) (auto simp: balance_inv2 balance_bheight)
-
-lemma balright_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balright a k x b)"
-by (induct a k x b rule: balright.induct) (simp add: balance_inv1)+
-
-lemma balright_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balright lt k x rt)"
-by (induct lt k x rt rule: balright.induct) (auto simp: balance_inv1)
-
-lemma balright_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balright l k v r)"
-apply (induct l k v r rule: balright.induct)
+by (induct lt k v rt rule: balance_right.induct) (auto simp: balance_inv2 balance_bheight)
+
+lemma balance_right_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balance_right a k x b)"
+by (induct a k x b rule: balance_right.induct) (simp add: balance_inv1)+
+
+lemma balance_right_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balance_right lt k x rt)"
+by (induct lt k x rt rule: balance_right.induct) (auto simp: balance_inv1)
+
+lemma balance_right_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_right l k v r)"
+apply (induct l k v r rule: balance_right.induct)
 apply (auto simp:balance_sorted)
 apply (unfold tree_less_prop tree_greater_prop)
 by force+
 
-lemma balright_tree_greater: 
+lemma balance_right_tree_greater: 
   fixes k :: "'a::order"
   assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
-  shows "k \<guillemotleft>| balright a x t b"
-using assms by (induct a x t b rule: balright.induct) auto
-
-lemma balright_tree_less: 
+  shows "k \<guillemotleft>| balance_right a x t b"
+using assms by (induct a x t b rule: balance_right.induct) auto
+
+lemma balance_right_tree_less: 
   fixes k :: "'a::order"
   assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
-  shows "balright a x t b |\<guillemotleft> k"
-using assms by (induct a x t b rule: balright.induct) auto
-
-lemma balright_pit:
+  shows "balance_right a x t b |\<guillemotleft> k"
+using assms by (induct a x t b rule: balance_right.induct) auto
+
+lemma balance_right_in_tree:
   assumes "inv1 l" "inv1l r" "bheight l = bheight r + 1" "inv2 l" "inv2 r"
-  shows "entry_in_tree x y (balright l k v r) = (entry_in_tree x y l \<or> x = k \<and> y = v \<or> entry_in_tree x y r)"
-using assms by (induct l k v r rule: balright.induct) (auto simp: balance_pit)
-
-
-text {* app *}
+  shows "entry_in_tree x y (balance_right l k v r) = (entry_in_tree x y l \<or> x = k \<and> y = v \<or> entry_in_tree x y r)"
+using assms by (induct l k v r rule: balance_right.induct) (auto simp: balance_in_tree)
 
 fun
-  app :: "('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "app Empty x = x" 
-| "app x Empty = x" 
-| "app (Branch R a k x b) (Branch R c s y d) = (case (app b c) of
+  combine :: "('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "combine Empty x = x" 
+| "combine x Empty = x" 
+| "combine (Branch R a k x b) (Branch R c s y d) = (case (combine b c) of
                                       Branch R b2 t z c2 \<Rightarrow> (Branch R (Branch R a k x b2) t z (Branch R c2 s y d)) |
                                       bc \<Rightarrow> Branch R a k x (Branch R bc s y d))" 
-| "app (Branch B a k x b) (Branch B c s y d) = (case (app b c) of
+| "combine (Branch B a k x b) (Branch B c s y d) = (case (combine b c) of
                                       Branch R b2 t z c2 \<Rightarrow> Branch R (Branch B a k x b2) t z (Branch B c2 s y d) |
-                                      bc \<Rightarrow> balleft a k x (Branch B bc s y d))" 
-| "app a (Branch R b k x c) = Branch R (app a b) k x c" 
-| "app (Branch R a k x b) c = Branch R a k x (app b c)" 
-
-lemma app_inv2:
+                                      bc \<Rightarrow> balance_left a k x (Branch B bc s y d))" 
+| "combine a (Branch R b k x c) = Branch R (combine a b) k x c" 
+| "combine (Branch R a k x b) c = Branch R a k x (combine b c)" 
+
+lemma combine_inv2:
   assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
-  shows "bheight (app lt rt) = bheight lt" "inv2 (app lt rt)"
+  shows "bheight (combine lt rt) = bheight lt" "inv2 (combine lt rt)"
 using assms 
-by (induct lt rt rule: app.induct) 
-   (auto simp: balleft_inv2_app split: rbt.splits color.splits)
-
-lemma app_inv1: 
+by (induct lt rt rule: combine.induct) 
+   (auto simp: balance_left_inv2_app split: rbt.splits color.splits)
+
+lemma combine_inv1: 
   assumes "inv1 lt" "inv1 rt"
-  shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (app lt rt)"
-         "inv1l (app lt rt)"
+  shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (combine lt rt)"
+         "inv1l (combine lt rt)"
 using assms 
-by (induct lt rt rule: app.induct)
-   (auto simp: balleft_inv1 split: rbt.splits color.splits)
-
-lemma app_tree_greater[simp]: 
+by (induct lt rt rule: combine.induct)
+   (auto simp: balance_left_inv1 split: rbt.splits color.splits)
+
+lemma combine_tree_greater[simp]: 
   fixes k :: "'a::linorder"
   assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r" 
-  shows "k \<guillemotleft>| app l r"
+  shows "k \<guillemotleft>| combine l r"
 using assms 
-by (induct l r rule: app.induct)
-   (auto simp: balleft_tree_greater split:rbt.splits color.splits)
-
-lemma app_tree_less[simp]: 
+by (induct l r rule: combine.induct)
+   (auto simp: balance_left_tree_greater split:rbt.splits color.splits)
+
+lemma combine_tree_less[simp]: 
   fixes k :: "'a::linorder"
   assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k" 
-  shows "app l r |\<guillemotleft> k"
+  shows "combine l r |\<guillemotleft> k"
 using assms 
-by (induct l r rule: app.induct)
-   (auto simp: balleft_tree_less split:rbt.splits color.splits)
-
-lemma app_sorted: 
+by (induct l r rule: combine.induct)
+   (auto simp: balance_left_tree_less split:rbt.splits color.splits)
+
+lemma combine_sorted: 
   fixes k :: "'a::linorder"
   assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-  shows "sorted (app l r)"
-using assms proof (induct l r rule: app.induct)
+  shows "sorted (combine l r)"
+using assms proof (induct l r rule: combine.induct)
   case (3 a x v b c y w d)
   hence ineqs: "a |\<guillemotleft> x" "x \<guillemotleft>| b" "b |\<guillemotleft> k" "k \<guillemotleft>| c" "c |\<guillemotleft> y" "y \<guillemotleft>| d"
     by auto
   with 3
   show ?case
-    apply (cases "app b c" rule: rbt_cases)
-    apply auto
-    by (metis app_tree_greater app_tree_less ineqs ineqs tree_less_simps(2) tree_greater_simps(2) tree_greater_trans tree_less_trans)+
+    by (cases "combine b c" rule: rbt_cases)
+      (auto, (metis combine_tree_greater combine_tree_less ineqs ineqs tree_less_simps(2) tree_greater_simps(2) tree_greater_trans tree_less_trans)+)
 next
   case (4 a x v b c y w d)
   hence "x < k \<and> tree_greater k c" by simp
   hence "tree_greater x c" by (blast dest: tree_greater_trans)
-  with 4 have 2: "tree_greater x (app b c)" by (simp add: app_tree_greater)
+  with 4 have 2: "tree_greater x (combine b c)" by (simp add: combine_tree_greater)
   from 4 have "k < y \<and> tree_less k b" by simp
   hence "tree_less y b" by (blast dest: tree_less_trans)
-  with 4 have 3: "tree_less y (app b c)" by (simp add: app_tree_less)
+  with 4 have 3: "tree_less y (combine b c)" by (simp add: combine_tree_less)
   show ?case
-  proof (cases "app b c" rule: rbt_cases)
+  proof (cases "combine b c" rule: rbt_cases)
     case Empty
     from 4 have "x < y \<and> tree_greater y d" by auto
     hence "tree_greater x d" by (blast dest: tree_greater_trans)
     with 4 Empty have "sorted a" and "sorted (Branch B Empty y w d)" and "tree_less x a" and "tree_greater x (Branch B Empty y w d)" by auto
-    with Empty show ?thesis by (simp add: balleft_sorted)
+    with Empty show ?thesis by (simp add: balance_left_sorted)
   next
     case (Red lta va ka rta)
     with 2 4 have "x < va \<and> tree_less x a" by simp
     hence 5: "tree_less va a" by (blast dest: tree_less_trans)
     from Red 3 4 have "va < y \<and> tree_greater y d" by simp

     with Red 2 3 4 5 show ?thesis by simp
   next
     case (Black lta va ka rta)
     from 4 have "x < y \<and> tree_greater y d" by auto
     hence "tree_greater x d" by (blast dest: tree_greater_trans)
-    with Black 2 3 4 have "sorted a" and "sorted (Branch B (app b c) y w d)" and "tree_less x a" and "tree_greater x (Branch B (app b c) y w d)" by auto
-    with Black show ?thesis by (simp add: balleft_sorted)
+    with Black 2 3 4 have "sorted a" and "sorted (Branch B (combine b c) y w d)" and "tree_less x a" and "tree_greater x (Branch B (combine b c) y w d)" by auto
+    with Black show ?thesis by (simp add: balance_left_sorted)
   qed
 next
   case (5 va vb vd vc b x w c)
   hence "k < x \<and> tree_less k (Branch B va vb vd vc)" by simp
   hence "tree_less x (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
-  with 5 show ?case by (simp add: app_tree_less)
+  with 5 show ?case by (simp add: combine_tree_less)
 next
   case (6 a x v b va vb vd vc)
   hence "x < k \<and> tree_greater k (Branch B va vb vd vc)" by simp
   hence "tree_greater x (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
-  with 6 show ?case by (simp add: app_tree_greater)
+  with 6 show ?case by (simp add: combine_tree_greater)
 qed simp+
 
-lemma app_pit: 
+lemma combine_in_tree: 
   assumes "inv2 l" "inv2 r" "bheight l = bheight r" "inv1 l" "inv1 r"
-  shows "entry_in_tree k v (app l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
+  shows "entry_in_tree k v (combine l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
 using assms 
-proof (induct l r rule: app.induct)
+proof (induct l r rule: combine.induct)
   case (4 _ _ _ b c)
-  hence a: "bheight (app b c) = bheight b" by (simp add: app_inv2)
-  from 4 have b: "inv1l (app b c)" by (simp add: app_inv1)
+  hence a: "bheight (combine b c) = bheight b" by (simp add: combine_inv2)
+  from 4 have b: "inv1l (combine b c)" by (simp add: combine_inv1)
 
   show ?case
-  proof (cases "app b c" rule: rbt_cases)
+  proof (cases "combine b c" rule: rbt_cases)
     case Empty
-    with 4 a show ?thesis by (auto simp: balleft_pit)
+    with 4 a show ?thesis by (auto simp: balance_left_in_tree)
   next
     case (Red lta ka va rta)
     with 4 show ?thesis by auto
   next
     case (Black lta ka va rta)
-    with a b 4  show ?thesis by (auto simp: balleft_pit)
+    with a b 4  show ?thesis by (auto simp: balance_left_in_tree)
   qed 
 qed (auto split: rbt.splits color.splits)
 
 fun
-  delformLeft :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
-  delformRight :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
+  del_from_left :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
+  del_from_right :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
   del :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
   "del x Empty = Empty" |
-  "del x (Branch c a y s b) = (if x < y then delformLeft x a y s b else (if x > y then delformRight x a y s b else app a b))" |
-  "delformLeft x (Branch B lt z v rt) y s b = balleft (del x (Branch B lt z v rt)) y s b" |
-  "delformLeft x a y s b = Branch R (del x a) y s b" |
-  "delformRight x a y s (Branch B lt z v rt) = balright a y s (del x (Branch B lt z v rt))" | 
-  "delformRight x a y s b = Branch R a y s (del x b)"
+  "del x (Branch c a y s b) = (if x < y then del_from_left x a y s b else (if x > y then del_from_right x a y s b else combine a b))" |
+  "del_from_left x (Branch B lt z v rt) y s b = balance_left (del x (Branch B lt z v rt)) y s b" |
+  "del_from_left x a y s b = Branch R (del x a) y s b" |
+  "del_from_right x a y s (Branch B lt z v rt) = balance_right a y s (del x (Branch B lt z v rt))" | 
+  "del_from_right x a y s b = Branch R a y s (del x b)"
 
 lemma 
   assumes "inv2 lt" "inv1 lt"
   shows
   "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
-  inv2 (delformLeft x lt k v rt) \<and> bheight (delformLeft x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (delformLeft x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (delformLeft x lt k v rt))"
+  inv2 (del_from_left x lt k v rt) \<and> bheight (del_from_left x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (del_from_left x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (del_from_left x lt k v rt))"
   and "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
-  inv2 (delformRight x lt k v rt) \<and> bheight (delformRight x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (delformRight x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (delformRight x lt k v rt))"
+  inv2 (del_from_right x lt k v rt) \<and> bheight (del_from_right x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (del_from_right x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (del_from_right x lt k v rt))"
   and del_inv1_inv2: "inv2 (del x lt) \<and> (color_of lt = R \<and> bheight (del x lt) = bheight lt \<and> inv1 (del x lt) 
   \<or> color_of lt = B \<and> bheight (del x lt) = bheight lt - 1 \<and> inv1l (del x lt))"
 using assms
-proof (induct x lt k v rt and x lt k v rt and x lt rule: delformLeft_delformRight_del.induct)
+proof (induct x lt k v rt and x lt k v rt and x lt rule: del_from_left_del_from_right_del.induct)
 case (2 y c _ y')
   have "y = y' \<or> y < y' \<or> y > y'" by auto
   thus ?case proof (elim disjE)
     assume "y = y'"
-    with 2 show ?thesis by (cases c) (simp add: app_inv2 app_inv1)+
+    with 2 show ?thesis by (cases c) (simp add: combine_inv2 combine_inv1)+
   next
     assume "y < y'"
     with 2 show ?thesis by (cases c) auto
   next
     assume "y' < y"
     with 2 show ?thesis by (cases c) auto
   qed
 next
   case (3 y lt z v rta y' ss bb) 
-  thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balleft_inv2_with_inv1 balleft_inv1 balleft_inv1l)+
+  thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balance_left_inv2_with_inv1 balance_left_inv1 balance_left_inv1l)+
 next
   case (5 y a y' ss lt z v rta)
-  thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balright_inv2_with_inv1 balright_inv1 balright_inv1l)+
+  thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balance_right_inv2_with_inv1 balance_right_inv1 balance_right_inv1l)+
 next
   case ("6_1" y a y' ss) thus ?case by (cases "color_of a = B \<and> color_of Empty = B") simp+
 qed auto
 
 lemma 
-  delformLeft_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (delformLeft x lt k y rt)"
-  and delformRight_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (delformRight x lt k y rt)"
+  del_from_left_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (del_from_left x lt k y rt)"
+  and del_from_right_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (del_from_right x lt k y rt)"
   and del_tree_less: "tree_less v lt \<Longrightarrow> tree_less v (del x lt)"
-by (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct) 
-   (auto simp: balleft_tree_less balright_tree_less)
-
-lemma delformLeft_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (delformLeft x lt k y rt)"
-  and delformRight_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (delformRight x lt k y rt)"
+by (induct x lt k y rt and x lt k y rt and x lt rule: del_from_left_del_from_right_del.induct) 
+   (auto simp: balance_left_tree_less balance_right_tree_less)
+
+lemma del_from_left_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (del_from_left x lt k y rt)"
+  and del_from_right_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (del_from_right x lt k y rt)"
   and del_tree_greater: "tree_greater v lt \<Longrightarrow> tree_greater v (del x lt)"
-by (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
-   (auto simp: balleft_tree_greater balright_tree_greater)
-
-lemma "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (delformLeft x lt k y rt)"
-  and "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (delformRight x lt k y rt)"
+by (induct x lt k y rt and x lt k y rt and x lt rule: del_from_left_del_from_right_del.induct)
+   (auto simp: balance_left_tree_greater balance_right_tree_greater)
+
+lemma "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (del_from_left x lt k y rt)"
+  and "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (del_from_right x lt k y rt)"
   and del_sorted: "sorted lt \<Longrightarrow> sorted (del x lt)"
-proof (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
+proof (induct x lt k y rt and x lt k y rt and x lt rule: del_from_left_del_from_right_del.induct)
   case (3 x lta zz v rta yy ss bb)
   from 3 have "tree_less yy (Branch B lta zz v rta)" by simp
   hence "tree_less yy (del x (Branch B lta zz v rta))" by (rule del_tree_less)
-  with 3 show ?case by (simp add: balleft_sorted)
+  with 3 show ?case by (simp add: balance_left_sorted)
 next
   case ("4_2" x vaa vbb vdd vc yy ss bb)
   hence "tree_less yy (Branch R vaa vbb vdd vc)" by simp
   hence "tree_less yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_less)
   with "4_2" show ?case by simp
 next
   case (5 x aa yy ss lta zz v rta) 
   hence "tree_greater yy (Branch B lta zz v rta)" by simp
   hence "tree_greater yy (del x (Branch B lta zz v rta))" by (rule del_tree_greater)
-  with 5 show ?case by (simp add: balright_sorted)
+  with 5 show ?case by (simp add: balance_right_sorted)
 next
   case ("6_2" x aa yy ss vaa vbb vdd vc)
   hence "tree_greater yy (Branch R vaa vbb vdd vc)" by simp
   hence "tree_greater yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_greater)
   with "6_2" show ?case by simp
-qed (auto simp: app_sorted)
-
-lemma "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (delformLeft x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
-  and "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (delformRight x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
-  and del_pit: "\<lbrakk>sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
-proof (induct x lt kt y rt and x lt kt y rt and x t rule: delformLeft_delformRight_del.induct)
+qed (auto simp: combine_sorted)
+
+lemma "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_left x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+  and "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_right x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+  and del_in_tree: "\<lbrakk>sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
+proof (induct x lt kt y rt and x lt kt y rt and x t rule: del_from_left_del_from_right_del.induct)
   case (2 xx c aa yy ss bb)
   have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
   from this 2 show ?case proof (elim disjE)
     assume "xx = yy"
     with 2 show ?thesis proof (cases "xx = k")
       case True
       from 2 `xx = yy` `xx = k` have "sorted (Branch c aa yy ss bb) \<and> k = yy" by simp
       hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: tree_less_nit tree_greater_prop)
-      with `xx = yy` 2 `xx = k` show ?thesis by (simp add: app_pit)
-    qed (simp add: app_pit)
+      with `xx = yy` 2 `xx = k` show ?thesis by (simp add: combine_in_tree)
+    qed (simp add: combine_in_tree)
   qed simp+
 next    
   case (3 xx lta zz vv rta yy ss bb)
   def mt[simp]: mt == "Branch B lta zz vv rta"
   from 3 have "inv2 mt \<and> inv1 mt" by simp
   hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
-  with 3 have 4: "entry_in_tree k v (delformLeft xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> entry_in_tree k v mt \<or> (k = yy \<and> v = ss) \<or> entry_in_tree k v bb)" by (simp add: balleft_pit)
+  with 3 have 4: "entry_in_tree k v (del_from_left xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> entry_in_tree k v mt \<or> (k = yy \<and> v = ss) \<or> entry_in_tree k v bb)" by (simp add: balance_left_in_tree)
   thus ?case proof (cases "xx = k")
     case True
     from 3 True have "tree_greater yy bb \<and> yy > k" by simp
     hence "tree_greater k bb" by (blast dest: tree_greater_trans)
     with 3 4 True show ?thesis by (auto simp: tree_greater_nit)

   thus ?case proof (cases "xx = k")
     case True
     with "4_2" have "k < yy \<and> tree_greater yy bb" by simp
     hence "tree_greater k bb" by (blast dest: tree_greater_trans)
     with True "4_2" show ?thesis by (auto simp: tree_greater_nit)
-  qed simp
+  qed auto
 next
   case (5 xx aa yy ss lta zz vv rta)
   def mt[simp]: mt == "Branch B lta zz vv rta"
   from 5 have "inv2 mt \<and> inv1 mt" by simp
   hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
-  with 5 have 3: "entry_in_tree k v (delformRight xx aa yy ss mt) = (entry_in_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> entry_in_tree k v mt)" by (simp add: balright_pit)
+  with 5 have 3: "entry_in_tree k v (del_from_right xx aa yy ss mt) = (entry_in_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> entry_in_tree k v mt)" by (simp add: balance_right_in_tree)
   thus ?case proof (cases "xx = k")
     case True
     from 5 True have "tree_less yy aa \<and> yy < k" by simp
     hence "tree_less k aa" by (blast dest: tree_less_trans)
     with 3 5 True show ?thesis by (auto simp: tree_less_nit)

   thus ?case proof (cases "xx = k")
     case True
     with "6_2" have "k > yy \<and> tree_less yy aa" by simp
     hence "tree_less k aa" by (blast dest: tree_less_trans)
     with True "6_2" show ?thesis by (auto simp: tree_less_nit)
-  qed simp
+  qed auto
 qed simp
 
 
 definition delete where
   delete_def: "delete k t = paint B (del k t)"
 
-theorem delete_is_rbt[simp]: assumes "is_rbt t" shows "is_rbt (delete k t)"
+theorem delete_is_rbt [simp]: assumes "is_rbt t" shows "is_rbt (delete k t)"
 proof -
   from assms have "inv2 t" and "inv1 t" unfolding is_rbt_def by auto 
   hence "inv2 (del k t) \<and> (color_of t = R \<and> bheight (del k t) = bheight t \<and> inv1 (del k t) \<or> color_of t = B \<and> bheight (del k t) = bheight t - 1 \<and> inv1l (del k t))" by (rule del_inv1_inv2)
   hence "inv2 (del k t) \<and> inv1l (del k t)" by (cases "color_of t") auto
   with assms show ?thesis
     unfolding is_rbt_def delete_def
     by (auto intro: paint_sorted del_sorted)
 qed
 
-lemma delete_pit: 
+lemma delete_in_tree: 
   assumes "is_rbt t" 
   shows "entry_in_tree k v (delete x t) = (x \<noteq> k \<and> entry_in_tree k v t)"
   using assms unfolding is_rbt_def delete_def
-  by (auto simp: del_pit)
+  by (auto simp: del_in_tree)
 
 lemma lookup_delete:
   assumes is_rbt: "is_rbt t"
   shows "lookup (delete k t) = (lookup t)|`(-{k})"
 proof
   fix x
   show "lookup (delete k t) x = (lookup t |` (-{k})) x" 
   proof (cases "x = k")
     assume "x = k" 
     with is_rbt show ?thesis
-      by (cases "lookup (delete k t) k") (auto simp: lookup_pit delete_pit)
+      by (cases "lookup (delete k t) k") (auto simp: lookup_in_tree delete_in_tree)
   next
     assume "x \<noteq> k"
     thus ?thesis
-      by auto (metis is_rbt delete_is_rbt delete_pit is_rbt_sorted lookup_from_pit)
+      by auto (metis is_rbt delete_is_rbt delete_in_tree is_rbt_sorted lookup_from_in_tree)
   qed
 qed
 
+
 subsection {* Union *}
 
 primrec
-  unionwithkey :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "unionwithkey f t Empty = t"
-| "unionwithkey f t (Branch c lt k v rt) = unionwithkey f (unionwithkey f (insertwithkey f k v t) lt) rt"
-
-lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (unionwithkey f lt rt)" 
+  union_with_key :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "union_with_key f t Empty = t"
+| "union_with_key f t (Branch c lt k v rt) = union_with_key f (union_with_key f (insert_with_key f k v t) lt) rt"
+
+lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (union_with_key f lt rt)" 
   by (induct rt arbitrary: lt) (auto simp: insertwk_sorted)
-theorem unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (unionwithkey f lt rt)" 
+theorem unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (union_with_key f lt rt)" 
   by (induct rt arbitrary: lt) (simp add: insertwk_is_rbt)+
 
 definition
-  unionwith where
-  "unionwith f = unionwithkey (\<lambda>_. f)"
-
-theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (unionwith f lt rt)" unfolding unionwith_def by simp
+  union_with where
+  "union_with f = union_with_key (\<lambda>_. f)"
+
+theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union_with f lt rt)" unfolding union_with_def by simp
 
 definition union where
-  "union = unionwithkey (%_ _ rv. rv)"
+  "union = union_with_key (%_ _ rv. rv)"
 
 theorem union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union lt rt)" unfolding union_def by simp
 
 lemma union_Branch[simp]:
   "union t (Branch c lt k v rt) = union (union (insert k v t) lt) rt"

 using assms
 proof (induct t arbitrary: s)
   case Empty thus ?case by (auto simp: union_def)
 next
   case (Branch c l k v r s)
-  hence sortedrl: "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
+  then have "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
 
   have meq: "lookup s(k \<mapsto> v) ++ lookup l ++ lookup r =
     lookup s ++
     (\<lambda>a. if a < k then lookup l a
     else if k < a then lookup r a else Some v)" (is "?m1 = ?m2")

       with `a < k` show ?thesis
         by (auto simp: map_add_def split: option.splits)
     qed
   qed
 
-  from Branch
-  have IHs:
+  from Branch have is_rbt: "is_rbt (RBT.union (RBT.insert k v s) l)"
+    by (auto intro: union_is_rbt insert_is_rbt)
+  with Branch have IHs:
     "lookup (union (union (insert k v s) l) r) = lookup (union (insert k v s) l) ++ lookup r"
     "lookup (union (insert k v s) l) = lookup (insert k v s) ++ lookup l"
-    by (auto intro: union_is_rbt insert_is_rbt)
+    by auto
   
   with meq show ?case
     by (auto simp: lookup_insert[OF Branch(3)])
+
 qed
 
-subsection {* Adjust *}
+
+subsection {* Modifying existing entries *}
 
 primrec
-  adjustwithkey :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "adjustwithkey f k Empty = Empty"
-| "adjustwithkey f k (Branch c lt x v rt) = (if k < x then (Branch c (adjustwithkey f k lt) x v rt) else if k > x then (Branch c lt x v (adjustwithkey f k rt)) else (Branch c lt x (f x v) rt))"
-
-lemma adjustwk_color_of: "color_of (adjustwithkey f k t) = color_of t" by (induct t) simp+
-lemma adjustwk_inv1: "inv1 (adjustwithkey f k t) = inv1 t" by (induct t) (simp add: adjustwk_color_of)+
-lemma adjustwk_inv2: "inv2 (adjustwithkey f k t) = inv2 t" "bheight (adjustwithkey f k t) = bheight t" by (induct t) simp+
-lemma adjustwk_tree_greater: "tree_greater k (adjustwithkey f kk t) = tree_greater k t" by (induct t) simp+
-lemma adjustwk_tree_less: "tree_less k (adjustwithkey f kk t) = tree_less k t" by (induct t) simp+
-lemma adjustwk_sorted: "sorted (adjustwithkey f k t) = sorted t" by (induct t) (simp add: adjustwk_tree_less adjustwk_tree_greater)+
-
-theorem adjustwk_is_rbt[simp]: "is_rbt (adjustwithkey f k t) = is_rbt t" 
-unfolding is_rbt_def by (simp add: adjustwk_inv2 adjustwk_color_of adjustwk_sorted adjustwk_inv1 )
-
-theorem adjustwithkey_map[simp]:
-  "lookup (adjustwithkey f k t) x = 
+  map_entry :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "map_entry f k Empty = Empty"
+| "map_entry f k (Branch c lt x v rt) = (if k < x then (Branch c (map_entry f k lt) x v rt) else if k > x then (Branch c lt x v (map_entry f k rt)) else (Branch c lt x (f x v) rt))"
+
+lemma map_entrywk_color_of: "color_of (map_entry f k t) = color_of t" by (induct t) simp+
+lemma map_entrywk_inv1: "inv1 (map_entry f k t) = inv1 t" by (induct t) (simp add: map_entrywk_color_of)+
+lemma map_entrywk_inv2: "inv2 (map_entry f k t) = inv2 t" "bheight (map_entry f k t) = bheight t" by (induct t) simp+
+lemma map_entrywk_tree_greater: "tree_greater k (map_entry f kk t) = tree_greater k t" by (induct t) simp+
+lemma map_entrywk_tree_less: "tree_less k (map_entry f kk t) = tree_less k t" by (induct t) simp+
+lemma map_entrywk_sorted: "sorted (map_entry f k t) = sorted t" by (induct t) (simp add: map_entrywk_tree_less map_entrywk_tree_greater)+
+
+theorem map_entrywk_is_rbt [simp]: "is_rbt (map_entry f k t) = is_rbt t" 
+unfolding is_rbt_def by (simp add: map_entrywk_inv2 map_entrywk_color_of map_entrywk_sorted map_entrywk_inv1 )
+
+theorem map_entry_map [simp]:
+  "lookup (map_entry f k t) x = 
   (if x = k then case lookup t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f k y)
             else lookup t x)"
 by (induct t arbitrary: x) (auto split:option.splits)
 
-definition adjust where
-  "adjust f = adjustwithkey (\<lambda>_. f)"
-
-theorem adjust_is_rbt[simp]: "is_rbt (adjust f k t) = is_rbt t" unfolding adjust_def by simp
-
-theorem adjust_map[simp]:
-  "lookup (adjust f k t) x = 
-  (if x = k then case lookup t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f y)
-            else lookup t x)"
-unfolding adjust_def by simp
-
-subsection {* Map *}
+
+subsection {* Mapping all entries *}
 
 primrec
-  mapwithkey :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'c) rbt"
-where
-  "mapwithkey f Empty = Empty"
-| "mapwithkey f (Branch c lt k v rt) = Branch c (mapwithkey f lt) k (f k v) (mapwithkey f rt)"
-
-theorem mapwk_keys[simp]: "keys (mapwithkey f t) = keys t" by (induct t) auto
-lemma mapwk_tree_greater: "tree_greater k (mapwithkey f t) = tree_greater k t" by (induct t) simp+
-lemma mapwk_tree_less: "tree_less k (mapwithkey f t) = tree_less k t" by (induct t) simp+
-lemma mapwk_sorted: "sorted (mapwithkey f t) = sorted t"  by (induct t) (simp add: mapwk_tree_less mapwk_tree_greater)+
-lemma mapwk_color_of: "color_of (mapwithkey f t) = color_of t" by (induct t) simp+
-lemma mapwk_inv1: "inv1 (mapwithkey f t) = inv1 t" by (induct t) (simp add: mapwk_color_of)+
-lemma mapwk_inv2: "inv2 (mapwithkey f t) = inv2 t" "bheight (mapwithkey f t) = bheight t" by (induct t) simp+
-theorem mapwk_is_rbt[simp]: "is_rbt (mapwithkey f t) = is_rbt t" 
-unfolding is_rbt_def by (simp add: mapwk_inv1 mapwk_inv2 mapwk_sorted mapwk_color_of)
-
-theorem lookup_mapwk[simp]: "lookup (mapwithkey f t) x = Option.map (f x) (lookup t x)"
+  map :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'c) rbt"
+where
+  "map f Empty = Empty"
+| "map f (Branch c lt k v rt) = Branch c (map f lt) k (f k v) (map f rt)"
+
+lemma map_entries [simp]: "entries (map f t) = List.map (\<lambda>(k, v). (k, f k v)) (entries t)"
+  by (induct t) auto
+lemma map_keys [simp]: "keys (map f t) = keys t" by (simp add: keys_def split_def)
+lemma map_tree_greater: "tree_greater k (map f t) = tree_greater k t" by (induct t) simp+
+lemma map_tree_less: "tree_less k (map f t) = tree_less k t" by (induct t) simp+
+lemma map_sorted: "sorted (map f t) = sorted t"  by (induct t) (simp add: map_tree_less map_tree_greater)+
+lemma map_color_of: "color_of (map f t) = color_of t" by (induct t) simp+
+lemma map_inv1: "inv1 (map f t) = inv1 t" by (induct t) (simp add: map_color_of)+
+lemma map_inv2: "inv2 (map f t) = inv2 t" "bheight (map f t) = bheight t" by (induct t) simp+
+theorem map_is_rbt [simp]: "is_rbt (map f t) = is_rbt t" 
+unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_sorted map_color_of)
+
+theorem lookup_map [simp]: "lookup (map f t) x = Option.map (f x) (lookup t x)"
 by (induct t) auto
 
-definition map
-where map_def: "map f == mapwithkey (\<lambda>_. f)"
-
-theorem map_keys[simp]: "keys (map f t) = keys t" unfolding map_def by simp
-theorem map_is_rbt[simp]: "is_rbt (map f t) = is_rbt t" unfolding map_def by simp
-theorem lookup_map[simp]: "lookup (map f t) = Option.map f o lookup t"
-  by (rule ext) (simp add:map_def)
-
-subsection {* Fold *}
-
-text {* The following is still incomplete... *}
-
-primrec
-  foldwithkey :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"
-where
-  "foldwithkey f Empty v = v"
-| "foldwithkey f (Branch c lt k x rt) v = foldwithkey f rt (f k x (foldwithkey f lt v))"
-
-lemma lookup_entries_aux: "sorted (Branch c t1 k v t2) \<Longrightarrow> RBT.lookup (Branch c t1 k v t2) = RBT.lookup t2 ++ [k\<mapsto>v] ++ RBT.lookup t1"
-proof (rule ext)
-  fix x
-  assume SORTED: "sorted (Branch c t1 k v t2)"
-  let ?thesis = "RBT.lookup (Branch c t1 k v t2) x = (RBT.lookup t2 ++ [k \<mapsto> v] ++ RBT.lookup t1) x"
-
-  have DOM_T1: "!!k'. k'\<in>dom (RBT.lookup t1) \<Longrightarrow> k>k'"
-  proof -
-    fix k'
-    from SORTED have "t1 |\<guillemotleft> k" by simp
-    with tree_less_prop have "\<forall>k'\<in>keys t1. k>k'" by auto
-    moreover assume "k'\<in>dom (RBT.lookup t1)"
-    ultimately show "k>k'" using RBT.lookup_keys SORTED by auto
-  qed
-
-  have DOM_T2: "!!k'. k'\<in>dom (RBT.lookup t2) \<Longrightarrow> k<k'"
-  proof -
-    fix k'
-    from SORTED have "k \<guillemotleft>| t2" by simp
-    with tree_greater_prop have "\<forall>k'\<in>keys t2. k<k'" by auto
-    moreover assume "k'\<in>dom (RBT.lookup t2)"
-    ultimately show "k<k'" using RBT.lookup_keys SORTED by auto
-  qed
-
-  {
-    assume C: "x<k"
-    hence "RBT.lookup (Branch c t1 k v t2) x = RBT.lookup t1 x" by simp
-    moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
-    moreover have "x\<notin>dom (RBT.lookup t2)" proof
-      assume "x\<in>dom (RBT.lookup t2)"
-      with DOM_T2 have "k<x" by blast
-      with C show False by simp
-    qed
-    ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
-  } moreover {
-    assume [simp]: "x=k"
-    hence "RBT.lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
-    moreover have "x\<notin>dom (RBT.lookup t1)" proof
-      assume "x\<in>dom (RBT.lookup t1)"
-      with DOM_T1 have "k>x" by blast
-      thus False by simp
-    qed
-    ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
-  } moreover {
-    assume C: "x>k"
-    hence "RBT.lookup (Branch c t1 k v t2) x = RBT.lookup t2 x" by (simp add: less_not_sym[of k x])
-    moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
-    moreover have "x\<notin>dom (RBT.lookup t1)" proof
-      assume "x\<in>dom (RBT.lookup t1)"
-      with DOM_T1 have "k>x" by simp
-      with C show False by simp
-    qed
-    ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
-  } ultimately show ?thesis using less_linear by blast
-qed
-
-lemma map_of_entries:
-  shows "sorted t \<Longrightarrow> map_of (entries t) = lookup t"
-proof (induct t)
-  case Empty thus ?case by (simp add: RBT.lookup_Empty)
-next
-  case (Branch c t1 k v t2)
-  hence "map_of (entries (Branch c t1 k v t2)) = RBT.lookup t2 ++ [k \<mapsto> v] ++ RBT.lookup t1" by simp
-  also note lookup_entries_aux [OF Branch.prems,symmetric]
-  finally show ?case .
-qed
-
-lemma fold_entries_fold:
-  "foldwithkey f t x = foldl (\<lambda>x (k,v). f k v x) x (entries t)"
-by (induct t arbitrary: x) auto
-
-lemma entries_pit[simp]: "(k, v) \<in> set (entries t) = entry_in_tree k v t"
-by (induct t) auto
-
-lemma sorted_entries:
-  "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
-by (induct t) 
-  (force simp: sorted_append sorted_Cons tree_ord_props 
-      dest!: entry_in_tree_keys)+
-
-lemma distinct_entries:
-  "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
-by (induct t) 
-  (force simp: sorted_append sorted_Cons tree_ord_props 
-      dest!: entry_in_tree_keys)+
-
-hide (open) const Empty insert delete entries lookup map fold union adjust sorted
-
+
+subsection {* Folding over entries *}
+
+definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
+  "fold f t s = foldl (\<lambda>s (k, v). f k v s) s (entries t)"
+
+lemma fold_simps [simp, code]:
+  "fold f Empty = id"
+  "fold f (Branch c lt k v rt) = fold f rt \<circ> f k v \<circ> fold f lt"
+  by (simp_all add: fold_def expand_fun_eq)
+
+hide (open) const Empty insert delete entries lookup map_entry map fold union sorted
 (*>*)
 
 text {* 
   This theory defines purely functional red-black trees which can be
   used as an efficient representation of finite maps.
 *}
+
 
 subsection {* Data type and invariant *}
 
 text {*
   The type @{typ "('k, 'v) rbt"} denotes red-black trees with keys of

   operations. Furthermore, it implements the lookup functionality for
   the data sortedructure: It is executable and the lookup is performed in
   $O(\log n)$.  
 *}
 
+
 subsection {* Operations *}
 
 text {*
   Currently, the following operations are supported:
 

   @{thm union_is_rbt}\hfill(@{text "union_is_rbt"})
 
   \noindent
   @{thm map_is_rbt}\hfill(@{text "map_is_rbt"})
 *}
+
 
 subsection {* Map Semantics *}
 
 text {*
   \noindent