
diff -r 5d88ffdb290c -r 14d8d72f8b1f src/HOL/Library/RBT.thy
--- a/src/HOL/Library/RBT.thy	Tue Mar 02 15:39:15 2010 +0100
+++ b/src/HOL/Library/RBT.thy	Wed Mar 03 08:28:33 2010 +0100
@@ -11,135 +11,151 @@
 begin
 
 datatype color = R | B
-datatype ('a,'b)"rbt" = Empty | Tr color "('a,'b)rbt" 'a 'b "('a,'b)rbt"
+datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"
+
+lemma rbt_cases:
+  obtains (Empty) "t = Empty" 
+  | (Red) l k v r where "t = Branch R l k v r" 
+  | (Black) l k v r where "t = Branch B l k v r"
+proof (cases t)
+  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
+where 
+  "entries Empty = []"
+| "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"
 
 text {* Search tree properties *}
 
-primrec
-  pin_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> bool"
+primrec entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
 where
-  "pin_tree k v Empty = False"
-| "pin_tree k v (Tr c l x y r) = (k = x \<and> v = y \<or> pin_tree k v l \<or> pin_tree k v r)"
+  "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"
+primrec keys :: "('k, 'v) rbt \<Rightarrow> 'k set"
 where
   "keys Empty = {}"
-| "keys (Tr _ l k _ r) = { k } \<union> keys l \<union> keys r"
+| "keys (Branch _ l k _ r) = { k } \<union> keys l \<union> keys r"
 
-lemma pint_keys: "pin_tree k v t \<Longrightarrow> k \<in> keys t" by (induct t) auto
+lemma entry_in_tree_keys:
+  "entry_in_tree k v t \<Longrightarrow> k \<in> keys t"
+  by (induct t) auto
 
-primrec tlt :: "'a\<Colon>order \<Rightarrow> ('a,'b) rbt \<Rightarrow> bool"
+definition tree_less :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
 where
-  "tlt k Empty = True"
-| "tlt k (Tr c lt kt v rt) = (kt < k \<and> tlt k lt \<and> tlt k rt)"
+  tree_less_prop: "tree_less k t \<longleftrightarrow> (\<forall>x\<in>keys t. x < k)"
+
+abbreviation tree_less_symbol (infix "|\<guillemotleft>" 50)
+where "t |\<guillemotleft> x \<equiv> tree_less x t"
 
-abbreviation tllt (infix "|\<guillemotleft>" 50)
-where "t |\<guillemotleft> x == tlt 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)"
 
-primrec tgt :: "'a\<Colon>order \<Rightarrow> ('a,'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
-where
-  "tgt k Empty = True"
-| "tgt k (Tr c lt kt v rt) = (k < kt \<and> tgt k lt \<and> tgt k rt)"
+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)
 
-lemma tlt_prop: "(t |\<guillemotleft> k) = (\<forall>x\<in>keys t. x < k)" by (induct t) auto
-lemma tgt_prop: "(k \<guillemotleft>| t) = (\<forall>x\<in>keys t. k < x)" by (induct t) auto
-lemmas tlgt_props = tlt_prop tgt_prop
+lemma tree_greater_simps [simp]:
+  "tree_greater k Empty = True"
+  "tree_greater k (Branch c lt kt v rt) \<longleftrightarrow> k < kt \<and> tree_greater k lt \<and> tree_greater k rt"
+  by (auto simp add: tree_greater_prop)
 
-lemmas tgt_nit = tgt_prop pint_keys
-lemmas tlt_nit = tlt_prop pint_keys
+lemmas tree_ord_props = tree_less_prop tree_greater_prop
 
-lemma tlt_trans: "\<lbrakk> t |\<guillemotleft> x; x < y \<rbrakk> \<Longrightarrow> t |\<guillemotleft> y"
-  and tgt_trans: "\<lbrakk> x < y; y \<guillemotleft>| t\<rbrakk> \<Longrightarrow> x \<guillemotleft>| t"
-by (auto simp: tlgt_props)
-
+lemmas tree_greater_nit = tree_greater_prop entry_in_tree_keys
+lemmas tree_less_nit = tree_less_prop entry_in_tree_keys
 
-primrec st :: "('a::linorder, 'b) rbt \<Rightarrow> bool"
-where
-  "st Empty = True"
-| "st (Tr c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> st l \<and> st r)"
+lemma tree_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
+  and tree_greater_trans: "x < y \<Longrightarrow> y \<guillemotleft>| t \<Longrightarrow> x \<guillemotleft>| t"
+by (auto simp: tree_ord_props)
 
-primrec map_of :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
+primrec sorted :: "('a::linorder, 'b) rbt \<Rightarrow> bool"
 where
-  "map_of Empty k = None"
-| "map_of (Tr _ l x y r) k = (if k < x then map_of l k else if x < k then map_of r k else Some y)"
+  "sorted Empty = True"
+| "sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> sorted l \<and> sorted r)"
 
-lemma map_of_tlt[simp]: "t |\<guillemotleft> k \<Longrightarrow> map_of t k = None" 
+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_tree_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> lookup t k = None" 
 by (induct t) auto
 
-lemma map_of_tgt[simp]: "k \<guillemotleft>| t \<Longrightarrow> map_of t k = None"
+lemma lookup_tree_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> lookup t k = None"
 by (induct t) auto
 
-lemma mapof_keys: "st t \<Longrightarrow> dom (map_of t) = keys t"
-by (induct t) (auto simp: dom_def tgt_prop tlt_prop)
+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 mapof_pit: "st t \<Longrightarrow> (map_of t k = Some v) = pin_tree k v t"
-by (induct t) (auto simp: tlt_prop tgt_prop pint_keys)
+lemma lookup_pit: "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 map_of_Empty: "map_of Empty = empty"
+lemma lookup_Empty: "lookup Empty = empty"
 by (rule ext) simp
 
 (* a kind of extensionality *)
-lemma mapof_from_pit: 
-  assumes st: "st t1" "st t2" 
-  and eq: "\<And>v. pin_tree (k\<Colon>'a\<Colon>linorder) v t1 = pin_tree k v t2" 
-  shows "map_of t1 k = map_of t2 k"
-proof (cases "map_of t1 k")
+lemma lookup_from_pit: 
+  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> pin_tree k v t1"
-    by (simp add: mapof_pit[symmetric] st)
+  then have "\<And>v. \<not> entry_in_tree k v t1"
+    by (simp add: lookup_pit[symmetric] sorted)
   with None show ?thesis
-    by (cases "map_of t2 k") (auto simp: mapof_pit st eq)
+    by (cases "lookup t2 k") (auto simp: lookup_pit sorted eq)
 next
   case (Some a)
   then show ?thesis
-    apply (cases "map_of t2 k")
-    apply (auto simp: mapof_pit st eq)
-    by (auto simp add: mapof_pit[symmetric] st Some)
+    apply (cases "lookup t2 k")
+    apply (auto simp: lookup_pit sorted eq)
+    by (auto simp add: lookup_pit[symmetric] sorted Some)
 qed
 
 subsection {* Red-black properties *}
 
-primrec treec :: "('a,'b) rbt \<Rightarrow> color"
+primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"
 where
-  "treec Empty = B"
-| "treec (Tr c _ _ _ _) = c"
+  "color_of Empty = B"
+| "color_of (Branch c _ _ _ _) = c"
 
-primrec inv1 :: "('a,'b) rbt \<Rightarrow> bool"
+primrec bheight :: "('a,'b) rbt \<Rightarrow> nat"
+where
+  "bheight Empty = 0"
+| "bheight (Branch c lt k v rt) = (if c = B then Suc (bheight lt) else bheight lt)"
+
+primrec inv1 :: "('a, 'b) rbt \<Rightarrow> bool"
 where
   "inv1 Empty = True"
-| "inv1 (Tr c lt k v rt) = (inv1 lt \<and> inv1 rt \<and> (c = B \<or> treec lt = B \<and> treec rt = B))"
+| "inv1 (Branch c lt k v rt) \<longleftrightarrow> inv1 lt \<and> inv1 rt \<and> (c = B \<or> color_of lt = B \<and> color_of rt = B)"
 
-(* Weaker version *)
-primrec inv1l :: "('a,'b) rbt \<Rightarrow> bool"
+primrec inv1l :: "('a, 'b) rbt \<Rightarrow> bool" -- {* Weaker version *}
 where
   "inv1l Empty = True"
-| "inv1l (Tr c l k v r) = (inv1 l \<and> inv1 r)"
+| "inv1l (Branch c l k v r) = (inv1 l \<and> inv1 r)"
 lemma [simp]: "inv1 t \<Longrightarrow> inv1l t" by (cases t) simp+
 
-primrec bh :: "('a,'b) rbt \<Rightarrow> nat"
-where
-  "bh Empty = 0"
-| "bh (Tr c lt k v rt) = (if c = B then Suc (bh lt) else bh lt)"
-
-primrec inv2 :: "('a,'b) rbt \<Rightarrow> bool"
+primrec inv2 :: "('a, 'b) rbt \<Rightarrow> bool"
 where
   "inv2 Empty = True"
-| "inv2 (Tr c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bh lt = bh rt)"
+| "inv2 (Branch c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bheight lt = bheight rt)"
 
-definition
-  "isrbt t = (inv1 t \<and> inv2 t \<and> treec t = B \<and> st t)"
-
-lemma isrbt_st[simp]: "isrbt t \<Longrightarrow> st t" by (simp add: isrbt_def)
+definition is_rbt :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where
+  "is_rbt t \<longleftrightarrow> inv1 t \<and> inv2 t \<and> color_of t = B \<and> sorted t"
 
-lemma rbt_cases:
-  obtains (Empty) "t = Empty" 
-  | (Red) l k v r where "t = Tr R l k v r" 
-  | (Black) l k v r where "t = Tr B l k v r" 
-by (cases t, simp) (case_tac "color", auto)
+lemma is_rbt_sorted [simp]:
+  "is_rbt t \<Longrightarrow> sorted t" by (simp add: is_rbt_def)
 
-theorem Empty_isrbt[simp]: "isrbt Empty"
-unfolding isrbt_def by simp
+theorem Empty_is_rbt [simp]:
+  "is_rbt Empty" by (simp add: is_rbt_def)
 
 
 subsection {* Insertion *}
@@ -147,80 +163,80 @@
 fun (* slow, due to massive case splitting *)
   balance :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
-  "balance (Tr R a w x b) s t (Tr R c y z d) = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
-  "balance (Tr R (Tr R a w x b) s t c) y z d = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
-  "balance (Tr R a w x (Tr R b s t c)) y z d = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
-  "balance a w x (Tr R b s t (Tr R c y z d)) = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
-  "balance a w x (Tr R (Tr R b s t c) y z d) = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
-  "balance a s t b = Tr B a s t b"
+  "balance (Branch R a w x b) s t (Branch R c y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+  "balance (Branch R (Branch R a w x b) s t c) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+  "balance (Branch R a w x (Branch R b s t c)) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+  "balance a w x (Branch R b s t (Branch R c y z d)) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+  "balance a w x (Branch R (Branch R b s t c) y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+  "balance a s t b = Branch B a s t b"
 
 lemma balance_inv1: "\<lbrakk>inv1l l; inv1l r\<rbrakk> \<Longrightarrow> inv1 (balance l k v r)" 
   by (induct l k v r rule: balance.induct) auto
 
-lemma balance_bh: "bh l = bh r \<Longrightarrow> bh (balance l k v r) = Suc (bh l)"
+lemma balance_bheight: "bheight l = bheight r \<Longrightarrow> bheight (balance l k v r) = Suc (bheight l)"
   by (induct l k v r rule: balance.induct) auto
 
 lemma balance_inv2: 
-  assumes "inv2 l" "inv2 r" "bh l = bh r"
+  assumes "inv2 l" "inv2 r" "bheight l = bheight r"
   shows "inv2 (balance l k v r)"
   using assms
   by (induct l k v r rule: balance.induct) auto
 
-lemma balance_tgt[simp]: "(v \<guillemotleft>| balance a k x b) = (v \<guillemotleft>| a \<and> v \<guillemotleft>| b \<and> v < k)" 
+lemma balance_tree_greater[simp]: "(v \<guillemotleft>| balance a k x b) = (v \<guillemotleft>| a \<and> v \<guillemotleft>| b \<and> v < k)" 
   by (induct a k x b rule: balance.induct) auto
 
-lemma balance_tlt[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"
+lemma balance_tree_less[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"
   by (induct a k x b rule: balance.induct) auto
 
-lemma balance_st: 
+lemma balance_sorted: 
   fixes k :: "'a::linorder"
-  assumes "st l" "st r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-  shows "st (balance l k v r)"
+  assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+  shows "sorted (balance l k v r)"
 using assms proof (induct l k v r rule: balance.induct)
   case ("2_2" a x w b y t c z s va vb vd vc)
-  hence "y < z \<and> z \<guillemotleft>| Tr B va vb vd vc" 
-    by (auto simp add: tlgt_props)
-  hence "tgt y (Tr B va vb vd vc)" by (blast dest: tgt_trans)
+  hence "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" 
+    by (auto simp add: tree_ord_props)
+  hence "tree_greater y (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
   with "2_2" show ?case by simp
 next
   case ("3_2" va vb vd vc x w b y s c z)
-  from "3_2" have "x < y \<and> tlt x (Tr B va vb vd vc)" 
-    by (simp add: tlt.simps tgt.simps)
-  hence "tlt y (Tr B va vb vd vc)" by (blast dest: tlt_trans)
+  from "3_2" have "x < y \<and> tree_less x (Branch B va vb vd vc)" 
+    by simp
+  hence "tree_less y (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
   with "3_2" show ?case by simp
 next
   case ("3_3" x w b y s c z t va vb vd vc)
-  from "3_3" have "y < z \<and> tgt z (Tr B va vb vd vc)" by simp
-  hence "tgt y (Tr B va vb vd vc)" by (blast dest: tgt_trans)
+  from "3_3" have "y < z \<and> tree_greater z (Branch B va vb vd vc)" by simp
+  hence "tree_greater y (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
   with "3_3" show ?case by simp
 next
   case ("3_4" vd ve vg vf x w b y s c z t va vb vii vc)
-  hence "x < y \<and> tlt x (Tr B vd ve vg vf)" by simp
-  hence 1: "tlt y (Tr B vd ve vg vf)" by (blast dest: tlt_trans)
-  from "3_4" have "y < z \<and> tgt z (Tr B va vb vii vc)" by simp
-  hence "tgt y (Tr B va vb vii vc)" by (blast dest: tgt_trans)
+  hence "x < y \<and> tree_less x (Branch B vd ve vg vf)" by simp
+  hence 1: "tree_less y (Branch B vd ve vg vf)" by (blast dest: tree_less_trans)
+  from "3_4" have "y < z \<and> tree_greater z (Branch B va vb vii vc)" by simp
+  hence "tree_greater y (Branch B va vb vii vc)" by (blast dest: tree_greater_trans)
   with 1 "3_4" show ?case by simp
 next
   case ("4_2" va vb vd vc x w b y s c z t dd)
-  hence "x < y \<and> tlt x (Tr B va vb vd vc)" by simp
-  hence "tlt y (Tr B va vb vd vc)" by (blast dest: tlt_trans)
+  hence "x < y \<and> tree_less x (Branch B va vb vd vc)" by simp
+  hence "tree_less y (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
   with "4_2" show ?case by simp
 next
   case ("5_2" x w b y s c z t va vb vd vc)
-  hence "y < z \<and> tgt z (Tr B va vb vd vc)" by simp
-  hence "tgt y (Tr B va vb vd vc)" by (blast dest: tgt_trans)
+  hence "y < z \<and> tree_greater z (Branch B va vb vd vc)" by simp
+  hence "tree_greater y (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
   with "5_2" show ?case by simp
 next
   case ("5_3" va vb vd vc x w b y s c z t)
-  hence "x < y \<and> tlt x (Tr B va vb vd vc)" by simp
-  hence "tlt y (Tr B va vb vd vc)" by (blast dest: tlt_trans)
+  hence "x < y \<and> tree_less x (Branch B va vb vd vc)" by simp
+  hence "tree_less y (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
   with "5_3" show ?case by simp
 next
   case ("5_4" va vb vg vc x w b y s c z t vd ve vii vf)
-  hence "x < y \<and> tlt x (Tr B va vb vg vc)" by simp
-  hence 1: "tlt y (Tr B va vb vg vc)" by (blast dest: tlt_trans)
-  from "5_4" have "y < z \<and> tgt z (Tr B vd ve vii vf)" by simp
-  hence "tgt y (Tr B vd ve vii vf)" by (blast dest: tgt_trans)
+  hence "x < y \<and> tree_less x (Branch B va vb vg vc)" by simp
+  hence 1: "tree_less y (Branch B va vb vg vc)" by (blast dest: tree_less_trans)
+  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+
 
@@ -229,62 +245,62 @@
 by (induct l k v r rule: balance.induct) auto
 
 lemma balance_pit:  
-  "pin_tree k x (balance l v y r) = (pin_tree k x l \<or> k = v \<and> x = y \<or> pin_tree k x r)" 
+  "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 map_of_balance[simp]: 
+lemma lookup_balance[simp]: 
 fixes k :: "'a::linorder"
-assumes "st l" "st r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-shows "map_of (balance l k v r) x = map_of (Tr B l k v r) x"
-by (rule mapof_from_pit) (auto simp:assms balance_pit balance_st)
+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)
 
 primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
   "paint c Empty = Empty"
-| "paint c (Tr _ l k v r) = Tr c l k v r"
+| "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_treec[simp]: "treec (paint B t) = B" by (cases t) auto
-lemma paint_st[simp]: "st t \<Longrightarrow> st (paint c t)" by (cases t) auto
-lemma paint_pit[simp]: "pin_tree k x (paint c t) = pin_tree k x t" by (cases t) auto
-lemma paint_mapof[simp]: "map_of (paint c t) = map_of t" by (rule ext) (cases t, auto)
-lemma paint_tgt[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
-lemma paint_tlt[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" 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_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
   ins :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
-  "ins f k v Empty = Tr R Empty k v Empty" |
-  "ins f k v (Tr B l x y r) = (if k < x then balance (ins f k v l) x y r
+  "ins f k v Empty = Branch R Empty k v Empty" |
+  "ins f k v (Branch B l x y r) = (if k < x then balance (ins f k v l) x y r
                                else if k > x then balance l x y (ins f k v r)
-                               else Tr B l x (f k y v) r)" |
-  "ins f k v (Tr R l x y r) = (if k < x then Tr R (ins f k v l) x y r
-                               else if k > x then Tr R l x y (ins f k v r)
-                               else Tr R l x (f k y v) r)"
+                               else Branch B l x (f k y v) r)" |
+  "ins f k v (Branch R l x y r) = (if k < x then Branch R (ins f k v l) x y r
+                               else if k > x then Branch R l x y (ins f k v r)
+                               else Branch R l x (f k y v) r)"
 
 lemma ins_inv1_inv2: 
   assumes "inv1 t" "inv2 t"
-  shows "inv2 (ins f k x t)" "bh (ins f k x t) = bh t" 
-  "treec t = B \<Longrightarrow> inv1 (ins f k x t)" "inv1l (ins f k x t)"
+  shows "inv2 (ins f k x t)" "bheight (ins f k x t) = bheight t" 
+  "color_of t = B \<Longrightarrow> inv1 (ins f k x t)" "inv1l (ins f k x t)"
   using assms
-  by (induct f k x t rule: ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bh)
+  by (induct f k x t rule: ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
 
-lemma ins_tgt[simp]: "(v \<guillemotleft>| ins f k x t) = (v \<guillemotleft>| t \<and> k > v)"
+lemma ins_tree_greater[simp]: "(v \<guillemotleft>| ins f k x t) = (v \<guillemotleft>| t \<and> k > v)"
   by (induct f k x t rule: ins.induct) auto
-lemma ins_tlt[simp]: "(ins f k x t |\<guillemotleft> v) = (t |\<guillemotleft> v \<and> k < v)"
+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_st[simp]: "st t \<Longrightarrow> st (ins f k x t)"
-  by (induct f k x t rule: ins.induct) (auto simp: balance_st)
+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 map_of_ins: 
+lemma lookup_ins: 
   fixes k :: "'a::linorder"
-  assumes "st t"
-  shows "map_of (ins f k v t) x = ((map_of t)(k |-> case map_of t k of None \<Rightarrow> v 
+  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
 
@@ -293,98 +309,97 @@
 where
   "insertwithkey f k v t = paint B (ins f k v t)"
 
-lemma insertwk_st: "st t \<Longrightarrow> st (insertwithkey f k x t)"
+lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insertwithkey f k x t)"
   by (auto simp: insertwithkey_def)
 
-theorem insertwk_isrbt: 
-  assumes inv: "isrbt t" 
-  shows "isrbt (insertwithkey f k x t)"
+theorem insertwk_is_rbt: 
+  assumes inv: "is_rbt t" 
+  shows "is_rbt (insertwithkey f k x t)"
 using assms
-unfolding insertwithkey_def isrbt_def
+unfolding insertwithkey_def is_rbt_def
 by (auto simp: ins_inv1_inv2)
 
-lemma map_of_insertwk: 
-  assumes "st t"
-  shows "map_of (insertwithkey f k v t) x = ((map_of t)(k |-> case map_of t k of None \<Rightarrow> v 
+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 
                                                        | Some w \<Rightarrow> f k w v)) x"
 unfolding insertwithkey_def using assms
-by (simp add:map_of_ins)
+by (simp add:lookup_ins)
 
 definition
   insertw_def: "insertwith f = insertwithkey (\<lambda>_. f)"
 
-lemma insertw_st: "st t \<Longrightarrow> st (insertwith f k v t)" by (simp add: insertwk_st insertw_def)
-theorem insertw_isrbt: "isrbt t \<Longrightarrow> isrbt (insertwith f k v t)" by (simp add: insertwk_isrbt insertw_def)
+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)
 
-lemma map_of_insertw:
-  assumes "isrbt t"
-  shows "map_of (insertwith f k v t) = (map_of t)(k \<mapsto> (if k:dom (map_of t) then f (the (map_of t k)) v else v))"
+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))"
 using assms
 unfolding insertw_def
-by (rule_tac ext) (cases "map_of t k", auto simp:map_of_insertwk dom_def)
-
+by (rule_tac ext) (cases "lookup t k", auto simp:lookup_insertwk dom_def)
 
-definition
-  "insrt k v t = insertwithkey (\<lambda>_ _ nv. nv) k v t"
+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"
 
-lemma insrt_st: "st t \<Longrightarrow> st (insrt k v t)" by (simp add: insertwk_st insrt_def)
-theorem insrt_isrbt: "isrbt t \<Longrightarrow> isrbt (insrt k v t)" by (simp add: insertwk_isrbt insrt_def)
+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)
 
-lemma map_of_insert: 
-  assumes "isrbt t"
-  shows "map_of (insrt k v t) = (map_of t)(k\<mapsto>v)"
-unfolding insrt_def
+lemma lookup_insert: 
+  assumes "is_rbt t"
+  shows "lookup (insert k v t) = (lookup t)(k\<mapsto>v)"
+unfolding insert_def
 using assms
-by (rule_tac ext) (simp add: map_of_insertwk split:option.split)
+by (rule_tac ext) (simp add: lookup_insertwk split:option.split)
 
 
 subsection {* Deletion *}
 
-lemma bh_paintR'[simp]: "treec t = B \<Longrightarrow> bh (paint R t) = bh t - 1"
+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 (Tr R a k x b) s y c = Tr R (Tr B a k x b) s y c" |
-  "balleft bl k x (Tr B a s y b) = balance bl k x (Tr R a s y b)" |
-  "balleft bl k x (Tr R (Tr B a s y b) t z c) = Tr R (Tr B bl k x a) s y (balance b t z (paint R c))" |
+  "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:
-  assumes "inv2 lt" "inv2 rt" "bh lt + 1 = bh rt" "inv1 rt"
-  shows "bh (balleft lt k v rt) = bh lt + 1"
+  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)"
 using assms 
-by (induct lt k v rt rule: balleft.induct) (auto simp: balance_inv2 balance_bh)
+by (induct lt k v rt rule: balleft.induct) (auto simp: balance_inv2 balance_bheight)
 
 lemma balleft_inv2_app: 
-  assumes "inv2 lt" "inv2 rt" "bh lt + 1 = bh rt" "treec rt = B"
+  assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"
   shows "inv2 (balleft lt k v rt)" 
-        "bh (balleft lt k v rt) = bh rt"
+        "bheight (balleft lt k v rt) = bheight rt"
 using assms 
-by (induct lt k v rt rule: balleft.induct) (auto simp add: balance_inv2 balance_bh)+ 
+by (induct lt k v rt rule: balleft.induct) (auto simp add: balance_inv2 balance_bheight)+ 
 
-lemma balleft_inv1: "\<lbrakk>inv1l a; inv1 b; treec b = B\<rbrakk> \<Longrightarrow> inv1 (balleft a k x b)"
+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_st: "\<lbrakk> st l; st r; tlt k l; tgt k r \<rbrakk> \<Longrightarrow> st (balleft l k v r)"
+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)
-apply (auto simp: balance_st)
-apply (unfold tgt_prop tlt_prop)
+apply (auto simp: balance_sorted)
+apply (unfold tree_greater_prop tree_less_prop)
 by force+
 
-lemma balleft_tgt: 
+lemma balleft_tree_greater: 
   fixes k :: "'a::order"
   assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
   shows "k \<guillemotleft>| balleft a x t b"
 using assms 
 by (induct a x t b rule: balleft.induct) auto
 
-lemma balleft_tlt: 
+lemma balleft_tree_less: 
   fixes k :: "'a::order"
   assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
   shows "balleft a x t b |\<guillemotleft> k"
@@ -392,52 +407,52 @@
 by (induct a x t b rule: balleft.induct) auto
 
 lemma balleft_pit: 
-  assumes "inv1l l" "inv1 r" "bh l + 1 = bh r"
-  shows "pin_tree k v (balleft l a b r) = (pin_tree k v l \<or> k = a \<and> v = b \<or> pin_tree k v r)"
+  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)"
 using assms 
 by (induct l k v r rule: balleft.induct) (auto simp: balance_pit)
 
 fun
   balright :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
-  "balright a k x (Tr R b s y c) = Tr R a k x (Tr B b s y c)" |
-  "balright (Tr B a k x b) s y bl = balance (Tr R a k x b) s y bl" |
-  "balright (Tr R a k x (Tr B b s y c)) t z bl = Tr R (balance (paint R a) k x b) s y (Tr B c t z bl)" |
+  "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:
-  assumes "inv2 lt" "inv2 rt" "bh lt = bh rt + 1" "inv1 lt"
-  shows "inv2 (balright lt k v rt) \<and> bh (balright lt k v rt) = bh lt"
+  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"
 using assms
-by (induct lt k v rt rule: balright.induct) (auto simp: balance_inv2 balance_bh)
+by (induct lt k v rt rule: balright.induct) (auto simp: balance_inv2 balance_bheight)
 
-lemma balright_inv1: "\<lbrakk>inv1 a; inv1l b; treec a = B\<rbrakk> \<Longrightarrow> inv1 (balright a k x b)"
+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_st: "\<lbrakk> st l; st r; tlt k l; tgt k r \<rbrakk> \<Longrightarrow> st (balright l k v r)"
+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)
-apply (auto simp:balance_st)
-apply (unfold tlt_prop tgt_prop)
+apply (auto simp:balance_sorted)
+apply (unfold tree_less_prop tree_greater_prop)
 by force+
 
-lemma balright_tgt: 
+lemma balright_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_tlt: 
+lemma balright_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:
-  assumes "inv1 l" "inv1l r" "bh l = bh r + 1" "inv2 l" "inv2 r"
-  shows "pin_tree x y (balright l k v r) = (pin_tree x y l \<or> x = k \<and> y = v \<or> pin_tree x y r)"
+  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)
 
 
@@ -448,50 +463,50 @@
 where
   "app Empty x = x" 
 | "app x Empty = x" 
-| "app (Tr R a k x b) (Tr R c s y d) = (case (app b c) of
-                                      Tr R b2 t z c2 \<Rightarrow> (Tr R (Tr R a k x b2) t z (Tr R c2 s y d)) |
-                                      bc \<Rightarrow> Tr R a k x (Tr R bc s y d))" 
-| "app (Tr B a k x b) (Tr B c s y d) = (case (app b c) of
-                                      Tr R b2 t z c2 \<Rightarrow> Tr R (Tr B a k x b2) t z (Tr B c2 s y d) |
-                                      bc \<Rightarrow> balleft a k x (Tr B bc s y d))" 
-| "app a (Tr R b k x c) = Tr R (app a b) k x c" 
-| "app (Tr R a k x b) c = Tr R a k x (app b c)" 
+| "app (Branch R a k x b) (Branch R c s y d) = (case (app 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
+                                      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:
-  assumes "inv2 lt" "inv2 rt" "bh lt = bh rt"
-  shows "bh (app lt rt) = bh lt" "inv2 (app lt rt)"
+  assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
+  shows "bheight (app lt rt) = bheight lt" "inv2 (app lt rt)"
 using assms 
 by (induct lt rt rule: app.induct) 
    (auto simp: balleft_inv2_app split: rbt.splits color.splits)
 
 lemma app_inv1: 
   assumes "inv1 lt" "inv1 rt"
-  shows "treec lt = B \<Longrightarrow> treec rt = B \<Longrightarrow> inv1 (app lt rt)"
+  shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (app lt rt)"
          "inv1l (app lt rt)"
 using assms 
 by (induct lt rt rule: app.induct)
    (auto simp: balleft_inv1 split: rbt.splits color.splits)
 
-lemma app_tgt[simp]: 
+lemma app_tree_greater[simp]: 
   fixes k :: "'a::linorder"
   assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r" 
   shows "k \<guillemotleft>| app l r"
 using assms 
 by (induct l r rule: app.induct)
-   (auto simp: balleft_tgt split:rbt.splits color.splits)
+   (auto simp: balleft_tree_greater split:rbt.splits color.splits)
 
-lemma app_tlt[simp]: 
+lemma app_tree_less[simp]: 
   fixes k :: "'a::linorder"
   assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k" 
   shows "app l r |\<guillemotleft> k"
 using assms 
 by (induct l r rule: app.induct)
-   (auto simp: balleft_tlt split:rbt.splits color.splits)
+   (auto simp: balleft_tree_less split:rbt.splits color.splits)
 
-lemma app_st: 
+lemma app_sorted: 
   fixes k :: "'a::linorder"
-  assumes "st l" "st r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-  shows "st (app l r)"
+  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)
   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"
@@ -500,55 +515,55 @@
   show ?case
     apply (cases "app b c" rule: rbt_cases)
     apply auto
-    by (metis app_tgt app_tlt ineqs ineqs tlt.simps(2) tgt.simps(2) tgt_trans tlt_trans)+
+    by (metis app_tree_greater app_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> tgt k c" by simp
-  hence "tgt x c" by (blast dest: tgt_trans)
-  with 4 have 2: "tgt x (app b c)" by (simp add: app_tgt)
-  from 4 have "k < y \<and> tlt k b" by simp
-  hence "tlt y b" by (blast dest: tlt_trans)
-  with 4 have 3: "tlt y (app b c)" by (simp add: app_tlt)
+  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)
+  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)
   show ?case
   proof (cases "app b c" rule: rbt_cases)
     case Empty
-    from 4 have "x < y \<and> tgt y d" by auto
-    hence "tgt x d" by (blast dest: tgt_trans)
-    with 4 Empty have "st a" and "st (Tr B Empty y w d)" and "tlt x a" and "tgt x (Tr B Empty y w d)" by auto
-    with Empty show ?thesis by (simp add: balleft_st)
+    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)
   next
     case (Red lta va ka rta)
-    with 2 4 have "x < va \<and> tlt x a" by simp
-    hence 5: "tlt va a" by (blast dest: tlt_trans)
-    from Red 3 4 have "va < y \<and> tgt y d" by simp
-    hence "tgt va d" by (blast dest: tgt_trans)
+    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
+    hence "tree_greater va d" by (blast dest: tree_greater_trans)
     with Red 2 3 4 5 show ?thesis by simp
   next
     case (Black lta va ka rta)
-    from 4 have "x < y \<and> tgt y d" by auto
-    hence "tgt x d" by (blast dest: tgt_trans)
-    with Black 2 3 4 have "st a" and "st (Tr B (app b c) y w d)" and "tlt x a" and "tgt x (Tr B (app b c) y w d)" by auto
-    with Black show ?thesis by (simp add: balleft_st)
+    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)
   qed
 next
   case (5 va vb vd vc b x w c)
-  hence "k < x \<and> tlt k (Tr B va vb vd vc)" by simp
-  hence "tlt x (Tr B va vb vd vc)" by (blast dest: tlt_trans)
-  with 5 show ?case by (simp add: app_tlt)
+  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)
 next
   case (6 a x v b va vb vd vc)
-  hence "x < k \<and> tgt k (Tr B va vb vd vc)" by simp
-  hence "tgt x (Tr B va vb vd vc)" by (blast dest: tgt_trans)
-  with 6 show ?case by (simp add: app_tgt)
+  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)
 qed simp+
 
 lemma app_pit: 
-  assumes "inv2 l" "inv2 r" "bh l = bh r" "inv1 l" "inv1 r"
-  shows "pin_tree k v (app l r) = (pin_tree k v l \<or> pin_tree k v r)"
+  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)"
 using assms 
 proof (induct l r rule: app.induct)
   case (4 _ _ _ b c)
-  hence a: "bh (app b c) = bh b" by (simp add: app_inv2)
+  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)
 
   show ?case
@@ -570,21 +585,21 @@
   del :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
   "del x Empty = Empty" |
-  "del x (Tr 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 (Tr B lt z v rt) y s b = balleft (del x (Tr B lt z v rt)) y s b" |
-  "delformLeft x a y s b = Tr R (del x a) y s b" |
-  "delformRight x a y s (Tr B lt z v rt) = balright a y s (del x (Tr B lt z v rt))" | 
-  "delformRight x a y s b = Tr R a y s (del x b)"
+  "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)"
 
 lemma 
   assumes "inv2 lt" "inv1 lt"
   shows
-  "\<lbrakk>inv2 rt; bh lt = bh rt; inv1 rt\<rbrakk> \<Longrightarrow>
-  inv2 (delformLeft x lt k v rt) \<and> bh (delformLeft x lt k v rt) = bh lt \<and> (treec lt = B \<and> treec rt = B \<and> inv1 (delformLeft x lt k v rt) \<or> (treec lt \<noteq> B \<or> treec rt \<noteq> B) \<and> inv1l (delformLeft x lt k v rt))"
-  and "\<lbrakk>inv2 rt; bh lt = bh rt; inv1 rt\<rbrakk> \<Longrightarrow>
-  inv2 (delformRight x lt k v rt) \<and> bh (delformRight x lt k v rt) = bh lt \<and> (treec lt = B \<and> treec rt = B \<and> inv1 (delformRight x lt k v rt) \<or> (treec lt \<noteq> B \<or> treec rt \<noteq> B) \<and> inv1l (delformRight x lt k v rt))"
-  and del_inv1_inv2: "inv2 (del x lt) \<and> (treec lt = R \<and> bh (del x lt) = bh lt \<and> inv1 (del x lt) 
-  \<or> treec lt = B \<and> bh (del x lt) = bh lt - 1 \<and> inv1l (del x lt))"
+  "\<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))"
+  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))"
+  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)
 case (2 y c _ y')
@@ -601,55 +616,55 @@
   qed
 next
   case (3 y lt z v rta y' ss bb) 
-  thus ?case by (cases "treec (Tr B lt z v rta) = B \<and> treec 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: balleft_inv2_with_inv1 balleft_inv1 balleft_inv1l)+
 next
   case (5 y a y' ss lt z v rta)
-  thus ?case by (cases "treec a = B \<and> treec (Tr 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: balright_inv2_with_inv1 balright_inv1 balright_inv1l)+
 next
-  case ("6_1" y a y' ss) thus ?case by (cases "treec a = B \<and> treec Empty = B") simp+
+  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_tlt: "\<lbrakk>tlt v lt; tlt v rt; k < v\<rbrakk> \<Longrightarrow> tlt v (delformLeft x lt k y rt)"
-  and delformRight_tlt: "\<lbrakk>tlt v lt; tlt v rt; k < v\<rbrakk> \<Longrightarrow> tlt v (delformRight x lt k y rt)"
-  and del_tlt: "tlt v lt \<Longrightarrow> tlt v (del x lt)"
+  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)"
+  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_tlt balright_tlt)
+   (auto simp: balleft_tree_less balright_tree_less)
 
-lemma delformLeft_tgt: "\<lbrakk>tgt v lt; tgt v rt; k > v\<rbrakk> \<Longrightarrow> tgt v (delformLeft x lt k y rt)"
-  and delformRight_tgt: "\<lbrakk>tgt v lt; tgt v rt; k > v\<rbrakk> \<Longrightarrow> tgt v (delformRight x lt k y rt)"
-  and del_tgt: "tgt v lt \<Longrightarrow> tgt v (del x lt)"
+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)"
+  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_tgt balright_tgt)
+   (auto simp: balleft_tree_greater balright_tree_greater)
 
-lemma "\<lbrakk>st lt; st rt; tlt k lt; tgt k rt\<rbrakk> \<Longrightarrow> st (delformLeft x lt k y rt)"
-  and "\<lbrakk>st lt; st rt; tlt k lt; tgt k rt\<rbrakk> \<Longrightarrow> st (delformRight x lt k y rt)"
-  and del_st: "st lt \<Longrightarrow> st (del x lt)"
+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)"
+  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)
   case (3 x lta zz v rta yy ss bb)
-  from 3 have "tlt yy (Tr B lta zz v rta)" by simp
-  hence "tlt yy (del x (Tr B lta zz v rta))" by (rule del_tlt)
-  with 3 show ?case by (simp add: balleft_st)
+  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)
 next
   case ("4_2" x vaa vbb vdd vc yy ss bb)
-  hence "tlt yy (Tr R vaa vbb vdd vc)" by simp
-  hence "tlt yy (del x (Tr R vaa vbb vdd vc))" by (rule del_tlt)
+  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 "tgt yy (Tr B lta zz v rta)" by simp
-  hence "tgt yy (del x (Tr B lta zz v rta))" by (rule del_tgt)
-  with 5 show ?case by (simp add: balright_st)
+  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)
 next
   case ("6_2" x aa yy ss vaa vbb vdd vc)
-  hence "tgt yy (Tr R vaa vbb vdd vc)" by simp
-  hence "tgt yy (del x (Tr R vaa vbb vdd vc))" by (rule del_tgt)
+  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_st)
+qed (auto simp: app_sorted)
 
-lemma "\<lbrakk>st lt; st rt; tlt kt lt; tgt kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bh lt = bh rt; x < kt\<rbrakk> \<Longrightarrow> pin_tree k v (delformLeft x lt kt y rt) = (False \<or> (x \<noteq> k \<and> pin_tree k v (Tr c lt kt y rt)))"
-  and "\<lbrakk>st lt; st rt; tlt kt lt; tgt kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bh lt = bh rt; x > kt\<rbrakk> \<Longrightarrow> pin_tree k v (delformRight x lt kt y rt) = (False \<or> (x \<noteq> k \<and> pin_tree k v (Tr c lt kt y rt)))"
-  and del_pit: "\<lbrakk>st t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> pin_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> pin_tree k v t))"
+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)
   case (2 xx c aa yy ss bb)
   have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
@@ -657,68 +672,68 @@
     assume "xx = yy"
     with 2 show ?thesis proof (cases "xx = k")
       case True
-      from 2 `xx = yy` `xx = k` have "st (Tr c aa yy ss bb) \<and> k = yy" by simp
-      hence "\<not> pin_tree k v aa" "\<not> pin_tree k v bb" by (auto simp: tlt_nit tgt_prop)
+      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)
   qed simp+
 next    
   case (3 xx lta zz vv rta yy ss bb)
-  def mt[simp]: mt == "Tr B lta zz vv rta"
+  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> (treec mt = R \<and> bh (del xx mt) = bh mt \<and> inv1 (del xx mt) \<or> treec mt = B \<and> bh (del xx mt) = bh mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
-  with 3 have 4: "pin_tree k v (delformLeft xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> pin_tree k v mt \<or> (k = yy \<and> v = ss) \<or> pin_tree k v bb)" by (simp add: balleft_pit)
+  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)
   thus ?case proof (cases "xx = k")
     case True
-    from 3 True have "tgt yy bb \<and> yy > k" by simp
-    hence "tgt k bb" by (blast dest: tgt_trans)
-    with 3 4 True show ?thesis by (auto simp: tgt_nit)
+    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)
   qed auto
 next
   case ("4_1" xx yy ss bb)
   show ?case proof (cases "xx = k")
     case True
-    with "4_1" have "tgt yy bb \<and> k < yy" by simp
-    hence "tgt k bb" by (blast dest: tgt_trans)
+    with "4_1" have "tree_greater yy bb \<and> k < yy" by simp
+    hence "tree_greater k bb" by (blast dest: tree_greater_trans)
     with "4_1" `xx = k` 
-   have "pin_tree k v (Tr R Empty yy ss bb) = pin_tree k v Empty" by (auto simp: tgt_nit)
+   have "entry_in_tree k v (Branch R Empty yy ss bb) = entry_in_tree k v Empty" by (auto simp: tree_greater_nit)
     thus ?thesis by auto
   qed simp+
 next
   case ("4_2" xx vaa vbb vdd vc yy ss bb)
   thus ?case proof (cases "xx = k")
     case True
-    with "4_2" have "k < yy \<and> tgt yy bb" by simp
-    hence "tgt k bb" by (blast dest: tgt_trans)
-    with True "4_2" show ?thesis by (auto simp: tgt_nit)
+    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
 next
   case (5 xx aa yy ss lta zz vv rta)
-  def mt[simp]: mt == "Tr B 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> (treec mt = R \<and> bh (del xx mt) = bh mt \<and> inv1 (del xx mt) \<or> treec mt = B \<and> bh (del xx mt) = bh mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
-  with 5 have 3: "pin_tree k v (delformRight xx aa yy ss mt) = (pin_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> pin_tree k v mt)" by (simp add: balright_pit)
+  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)
   thus ?case proof (cases "xx = k")
     case True
-    from 5 True have "tlt yy aa \<and> yy < k" by simp
-    hence "tlt k aa" by (blast dest: tlt_trans)
-    with 3 5 True show ?thesis by (auto simp: tlt_nit)
+    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)
   qed auto
 next
   case ("6_1" xx aa yy ss)
   show ?case proof (cases "xx = k")
     case True
-    with "6_1" have "tlt yy aa \<and> k > yy" by simp
-    hence "tlt k aa" by (blast dest: tlt_trans)
-    with "6_1" `xx = k` show ?thesis by (auto simp: tlt_nit)
+    with "6_1" have "tree_less yy aa \<and> k > yy" by simp
+    hence "tree_less k aa" by (blast dest: tree_less_trans)
+    with "6_1" `xx = k` show ?thesis by (auto simp: tree_less_nit)
   qed simp
 next
   case ("6_2" xx aa yy ss vaa vbb vdd vc)
   thus ?case proof (cases "xx = k")
     case True
-    with "6_2" have "k > yy \<and> tlt yy aa" by simp
-    hence "tlt k aa" by (blast dest: tlt_trans)
-    with True "6_2" show ?thesis by (auto simp: tlt_nit)
+    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 simp
 
@@ -726,36 +741,36 @@
 definition delete where
   delete_def: "delete k t = paint B (del k t)"
 
-theorem delete_isrbt[simp]: assumes "isrbt t" shows "isrbt (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 isrbt_def by auto 
-  hence "inv2 (del k t) \<and> (treec t = R \<and> bh (del k t) = bh t \<and> inv1 (del k t) \<or> treec t = B \<and> bh (del k t) = bh t - 1 \<and> inv1l (del k t))" by (rule del_inv1_inv2)
-  hence "inv2 (del k t) \<and> inv1l (del k t)" by (cases "treec t") auto
+  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 isrbt_def delete_def
-    by (auto intro: paint_st del_st)
+    unfolding is_rbt_def delete_def
+    by (auto intro: paint_sorted del_sorted)
 qed
 
 lemma delete_pit: 
-  assumes "isrbt t" 
-  shows "pin_tree k v (delete x t) = (x \<noteq> k \<and> pin_tree k v t)"
-  using assms unfolding isrbt_def delete_def
+  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)
 
-lemma map_of_delete:
-  assumes isrbt: "isrbt t"
-  shows "map_of (delete k t) = (map_of t)|`(-{k})"
+lemma lookup_delete:
+  assumes is_rbt: "is_rbt t"
+  shows "lookup (delete k t) = (lookup t)|`(-{k})"
 proof
   fix x
-  show "map_of (delete k t) x = (map_of t |` (-{k})) x" 
+  show "lookup (delete k t) x = (lookup t |` (-{k})) x" 
   proof (cases "x = k")
     assume "x = k" 
-    with isrbt show ?thesis
-      by (cases "map_of (delete k t) k") (auto simp: mapof_pit delete_pit)
+    with is_rbt show ?thesis
+      by (cases "lookup (delete k t) k") (auto simp: lookup_pit delete_pit)
   next
     assume "x \<noteq> k"
     thus ?thesis
-      by auto (metis isrbt delete_isrbt delete_pit isrbt_st mapof_from_pit)
+      by auto (metis is_rbt delete_is_rbt delete_pit is_rbt_sorted lookup_from_pit)
   qed
 qed
 
@@ -765,43 +780,43 @@
   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 (Tr c lt k v rt) = unionwithkey f (unionwithkey f (insertwithkey f k v t) lt) rt"
+| "unionwithkey f t (Branch c lt k v rt) = unionwithkey f (unionwithkey f (insertwithkey f k v t) lt) rt"
 
-lemma unionwk_st: "st lt \<Longrightarrow> st (unionwithkey f lt rt)" 
-  by (induct rt arbitrary: lt) (auto simp: insertwk_st)
-theorem unionwk_isrbt[simp]: "isrbt lt \<Longrightarrow> isrbt (unionwithkey f lt rt)" 
-  by (induct rt arbitrary: lt) (simp add: insertwk_isrbt)+
+lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (unionwithkey 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)" 
+  by (induct rt arbitrary: lt) (simp add: insertwk_is_rbt)+
 
 definition
   unionwith where
   "unionwith f = unionwithkey (\<lambda>_. f)"
 
-theorem unionw_isrbt: "isrbt lt \<Longrightarrow> isrbt (unionwith f lt rt)" unfolding unionwith_def by simp
+theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (unionwith f lt rt)" unfolding unionwith_def by simp
 
 definition union where
   "union = unionwithkey (%_ _ rv. rv)"
 
-theorem union_isrbt: "isrbt lt \<Longrightarrow> isrbt (union lt rt)" unfolding union_def by simp
+theorem union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union lt rt)" unfolding union_def by simp
 
-lemma union_Tr[simp]:
-  "union t (Tr c lt k v rt) = union (union (insrt k v t) lt) rt"
-  unfolding union_def insrt_def
+lemma union_Branch[simp]:
+  "union t (Branch c lt k v rt) = union (union (insert k v t) lt) rt"
+  unfolding union_def insert_def
   by simp
 
-lemma map_of_union:
-  assumes "isrbt s" "st t"
-  shows "map_of (union s t) = map_of s ++ map_of t"
+lemma lookup_union:
+  assumes "is_rbt s" "sorted t"
+  shows "lookup (union s t) = lookup s ++ lookup t"
 using assms
 proof (induct t arbitrary: s)
   case Empty thus ?case by (auto simp: union_def)
 next
-  case (Tr c l k v r s)
-  hence strl: "st r" "st l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
+  case (Branch c l k v r s)
+  hence sortedrl: "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
 
-  have meq: "map_of s(k \<mapsto> v) ++ map_of l ++ map_of r =
-    map_of s ++
-    (\<lambda>a. if a < k then map_of l a
-    else if k < a then map_of r a else Some v)" (is "?m1 = ?m2")
+  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")
   proof (rule ext)
     fix a
 
@@ -809,7 +824,7 @@
     thus "?m1 a = ?m2 a"
     proof (elim disjE)
       assume "k < a"
-      with `l |\<guillemotleft> k` have "l |\<guillemotleft> a" by (rule tlt_trans)
+      with `l |\<guillemotleft> k` have "l |\<guillemotleft> a" by (rule tree_less_trans)
       with `k < a` show ?thesis
         by (auto simp: map_add_def split: option.splits)
     next
@@ -818,20 +833,20 @@
       show ?thesis by (auto simp: map_add_def)
     next
       assume "a < k"
-      from this `k \<guillemotleft>| r` have "a \<guillemotleft>| r" by (rule tgt_trans)
+      from this `k \<guillemotleft>| r` have "a \<guillemotleft>| r" by (rule tree_greater_trans)
       with `a < k` show ?thesis
         by (auto simp: map_add_def split: option.splits)
     qed
   qed
 
-  from Tr
+  from Branch
   have IHs:
-    "map_of (union (union (insrt k v s) l) r) = map_of (union (insrt k v s) l) ++ map_of r"
-    "map_of (union (insrt k v s) l) = map_of (insrt k v s) ++ map_of l"
-    by (auto intro: union_isrbt insrt_isrbt)
+    "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)
   
   with meq show ?case
-    by (auto simp: map_of_insert[OF Tr(3)])
+    by (auto simp: lookup_insert[OF Branch(3)])
 qed
 
 subsection {* Adjust *}
@@ -840,33 +855,33 @@
   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 (Tr c lt x v rt) = (if k < x then (Tr c (adjustwithkey f k lt) x v rt) else if k > x then (Tr c lt x v (adjustwithkey f k rt)) else (Tr c lt x (f x v) rt))"
+| "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_treec: "treec (adjustwithkey f k t) = treec t" by (induct t) simp+
-lemma adjustwk_inv1: "inv1 (adjustwithkey f k t) = inv1 t" by (induct t) (simp add: adjustwk_treec)+
-lemma adjustwk_inv2: "inv2 (adjustwithkey f k t) = inv2 t" "bh (adjustwithkey f k t) = bh t" by (induct t) simp+
-lemma adjustwk_tgt: "tgt k (adjustwithkey f kk t) = tgt k t" by (induct t) simp+
-lemma adjustwk_tlt: "tlt k (adjustwithkey f kk t) = tlt k t" by (induct t) simp+
-lemma adjustwk_st: "st (adjustwithkey f k t) = st t" by (induct t) (simp add: adjustwk_tlt adjustwk_tgt)+
+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_isrbt[simp]: "isrbt (adjustwithkey f k t) = isrbt t" 
-unfolding isrbt_def by (simp add: adjustwk_inv2 adjustwk_treec adjustwk_st adjustwk_inv1 )
+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]:
-  "map_of (adjustwithkey f k t) x = 
-  (if x = k then case map_of t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f k y)
-            else map_of t x)"
+  "lookup (adjustwithkey 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_isrbt[simp]: "isrbt (adjust f k t) = isrbt t" unfolding adjust_def by simp
+theorem adjust_is_rbt[simp]: "is_rbt (adjust f k t) = is_rbt t" unfolding adjust_def by simp
 
 theorem adjust_map[simp]:
-  "map_of (adjust f k t) x = 
-  (if x = k then case map_of t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f y)
-            else map_of t x)"
+  "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 *}
@@ -875,27 +890,27 @@
   mapwithkey :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'c) rbt"
 where
   "mapwithkey f Empty = Empty"
-| "mapwithkey f (Tr c lt k v rt) = Tr c (mapwithkey f lt) k (f k v) (mapwithkey f rt)"
+| "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_tgt: "tgt k (mapwithkey f t) = tgt k t" by (induct t) simp+
-lemma mapwk_tlt: "tlt k (mapwithkey f t) = tlt k t" by (induct t) simp+
-lemma mapwk_st: "st (mapwithkey f t) = st t"  by (induct t) (simp add: mapwk_tlt mapwk_tgt)+
-lemma mapwk_treec: "treec (mapwithkey f t) = treec t" by (induct t) simp+
-lemma mapwk_inv1: "inv1 (mapwithkey f t) = inv1 t" by (induct t) (simp add: mapwk_treec)+
-lemma mapwk_inv2: "inv2 (mapwithkey f t) = inv2 t" "bh (mapwithkey f t) = bh t" by (induct t) simp+
-theorem mapwk_isrbt[simp]: "isrbt (mapwithkey f t) = isrbt t" 
-unfolding isrbt_def by (simp add: mapwk_inv1 mapwk_inv2 mapwk_st mapwk_treec)
+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 map_of_mapwk[simp]: "map_of (mapwithkey f t) x = Option.map (f x) (map_of t x)"
+theorem lookup_mapwk[simp]: "lookup (mapwithkey 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_isrbt[simp]: "isrbt (map f t) = isrbt t" unfolding map_def by simp
-theorem map_of_map[simp]: "map_of (map f t) = Option.map f o map_of t"
+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 *}
@@ -906,62 +921,57 @@
   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 (Tr c lt k x rt) v = foldwithkey f rt (f k x (foldwithkey f lt v))"
+| "foldwithkey f (Branch c lt k x rt) v = foldwithkey f rt (f k x (foldwithkey f lt v))"
 
-primrec alist_of
-where 
-  "alist_of Empty = []"
-| "alist_of (Tr _ l k v r) = alist_of l @ (k,v) # alist_of r"
-
-lemma map_of_alist_of_aux: "st (Tr c t1 k v t2) \<Longrightarrow> RBT.map_of (Tr c t1 k v t2) = RBT.map_of t2 ++ [k\<mapsto>v] ++ RBT.map_of t1"
+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 ST: "st (Tr c t1 k v t2)"
-  let ?thesis = "RBT.map_of (Tr c t1 k v t2) x = (RBT.map_of t2 ++ [k \<mapsto> v] ++ RBT.map_of t1) 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.map_of t1) \<Longrightarrow> k>k'"
+  have DOM_T1: "!!k'. k'\<in>dom (RBT.lookup t1) \<Longrightarrow> k>k'"
   proof -
     fix k'
-    from ST have "t1 |\<guillemotleft> k" by simp
-    with tlt_prop have "\<forall>k'\<in>keys t1. k>k'" by auto
-    moreover assume "k'\<in>dom (RBT.map_of t1)"
-    ultimately show "k>k'" using RBT.mapof_keys ST by auto
+    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.map_of t2) \<Longrightarrow> k<k'"
+  have DOM_T2: "!!k'. k'\<in>dom (RBT.lookup t2) \<Longrightarrow> k<k'"
   proof -
     fix k'
-    from ST have "k \<guillemotleft>| t2" by simp
-    with tgt_prop have "\<forall>k'\<in>keys t2. k<k'" by auto
-    moreover assume "k'\<in>dom (RBT.map_of t2)"
-    ultimately show "k<k'" using RBT.mapof_keys ST by auto
+    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.map_of (Tr c t1 k v t2) x = RBT.map_of t1 x" by simp
+    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.map_of t2)" proof
-      assume "x\<in>dom (RBT.map_of t2)"
+    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.map_of (Tr c t1 k v t2) x = [k \<mapsto> v] x" by simp
-    moreover have "x\<notin>dom (RBT.map_of t1)" proof
-      assume "x\<in>dom (RBT.map_of t1)"
+    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.map_of (Tr c t1 k v t2) x = RBT.map_of t2 x" by (simp add: less_not_sym[of k x])
+    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.map_of t1)" proof
-      assume "x\<in>dom (RBT.map_of t1)"
+    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
@@ -969,35 +979,38 @@
   } ultimately show ?thesis using less_linear by blast
 qed
 
-lemma map_of_alist_of:
-  shows "st t \<Longrightarrow> Map.map_of (alist_of t) = map_of t"
+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.map_of_Empty)
+  case Empty thus ?case by (simp add: RBT.lookup_Empty)
 next
-  case (Tr c t1 k v t2)
-  hence "Map.map_of (alist_of (Tr c t1 k v t2)) = RBT.map_of t2 ++ [k \<mapsto> v] ++ RBT.map_of t1" by simp
-  also note map_of_alist_of_aux[OF Tr.prems,symmetric]
+  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_alist_fold:
-  "foldwithkey f t x = foldl (\<lambda>x (k,v). f k v x) x (alist_of t)"
+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 alist_pit[simp]: "(k, v) \<in> set (alist_of t) = pin_tree k v t"
+lemma entries_pit[simp]: "(k, v) \<in> set (entries t) = entry_in_tree k v t"
 by (induct t) auto
 
-lemma sorted_alist:
-  "st t \<Longrightarrow> sorted (List.map fst (alist_of t))"
+lemma sorted_entries:
+  "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
 by (induct t) 
-  (force simp: sorted_append sorted_Cons tlgt_props 
-      dest!:pint_keys)+
+  (force simp: sorted_append sorted_Cons tree_ord_props 
+      dest!: entry_in_tree_keys)+
 
-lemma distinct_alist:
-  "st t \<Longrightarrow> distinct (List.map fst (alist_of t))"
+lemma distinct_entries:
+  "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
 by (induct t) 
-  (force simp: sorted_append sorted_Cons tlgt_props 
-      dest!:pint_keys)+
+  (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
+
 (*>*)
 
 text {* 
@@ -1010,20 +1023,20 @@
 text {*
   The type @{typ "('k, 'v) rbt"} denotes red-black trees with keys of
   type @{typ "'k"} and values of type @{typ "'v"}. To function
-  properly, the key type must belong to the @{text "linorder"} class.
+  properly, the key type musorted belong to the @{text "linorder"} class.
 
   A value @{term t} of this type is a valid red-black tree if it
-  satisfies the invariant @{text "isrbt t"}.
+  satisfies the invariant @{text "is_rbt t"}.
   This theory provides lemmas to prove that the invariant is
   satisfied throughout the computation.
 
-  The interpretation function @{const "map_of"} returns the partial
+  The interpretation function @{const "RBT.lookup"} returns the partial
   map represented by a red-black tree:
-  @{term_type[display] "map_of"}
+  @{term_type[display] "RBT.lookup"}
 
   This function should be used for reasoning about the semantics of the RBT
   operations. Furthermore, it implements the lookup functionality for
-  the data structure: It is executable and the lookup is performed in
+  the data sortedructure: It is executable and the lookup is performed in
   $O(\log n)$.  
 *}
 
@@ -1032,19 +1045,19 @@
 text {*
   Currently, the following operations are supported:
 
-  @{term_type[display] "Empty"}
+  @{term_type[display] "RBT.Empty"}
   Returns the empty tree. $O(1)$
 
-  @{term_type[display] "insrt"}
+  @{term_type[display] "RBT.insert"}
   Updates the map at a given position. $O(\log n)$
 
-  @{term_type[display] "delete"}
+  @{term_type[display] "RBT.delete"}
   Deletes a map entry at a given position. $O(\log n)$
 
-  @{term_type[display] "union"}
+  @{term_type[display] "RBT.union"}
   Forms the union of two trees, preferring entries from the first one.
 
-  @{term_type[display] "map"}
+  @{term_type[display] "RBT.map"}
   Maps a function over the values of a map. $O(n)$
 *}
 
@@ -1053,47 +1066,47 @@
 
 text {*
   \noindent
-  @{thm Empty_isrbt}\hfill(@{text "Empty_isrbt"})
+  @{thm Empty_is_rbt}\hfill(@{text "Empty_is_rbt"})
 
   \noindent
-  @{thm insrt_isrbt}\hfill(@{text "insrt_isrbt"})
+  @{thm insert_is_rbt}\hfill(@{text "insert_is_rbt"})
 
   \noindent
-  @{thm delete_isrbt}\hfill(@{text "delete_isrbt"})
+  @{thm delete_is_rbt}\hfill(@{text "delete_is_rbt"})
 
   \noindent
-  @{thm union_isrbt}\hfill(@{text "union_isrbt"})
+  @{thm union_is_rbt}\hfill(@{text "union_is_rbt"})
 
   \noindent
-  @{thm map_isrbt}\hfill(@{text "map_isrbt"})
+  @{thm map_is_rbt}\hfill(@{text "map_is_rbt"})
 *}
 
 subsection {* Map Semantics *}
 
 text {*
   \noindent
-  \underline{@{text "map_of_Empty"}}
-  @{thm[display] map_of_Empty}
+  \underline{@{text "lookup_Empty"}}
+  @{thm[display] lookup_Empty}
   \vspace{1ex}
 
   \noindent
-  \underline{@{text "map_of_insert"}}
-  @{thm[display] map_of_insert}
+  \underline{@{text "lookup_insert"}}
+  @{thm[display] lookup_insert}
   \vspace{1ex}
 
   \noindent
-  \underline{@{text "map_of_delete"}}
-  @{thm[display] map_of_delete}
+  \underline{@{text "lookup_delete"}}
+  @{thm[display] lookup_delete}
   \vspace{1ex}
 
   \noindent
-  \underline{@{text "map_of_union"}}
-  @{thm[display] map_of_union}
+  \underline{@{text "lookup_union"}}
+  @{thm[display] lookup_union}
   \vspace{1ex}
 
   \noindent
-  \underline{@{text "map_of_map"}}
-  @{thm[display] map_of_map}
+  \underline{@{text "lookup_map"}}
+  @{thm[display] lookup_map}
   \vspace{1ex}
 *}
 




