# HG changeset patch
# User Andreas Lochbihler
# Date 1334310330 -7200
# Node ID 2ada2be850cb521052764aecaa90030a5f17d6ea
# Parent  11a0aa6cc677e84c97ba0a2e098b59b9e6b38f32
move RBT implementation into type class contexts

diff -r 11a0aa6cc677 -r 2ada2be850cb src/HOL/Library/RBT.thy
--- a/src/HOL/Library/RBT.thy	Thu Apr 12 13:47:21 2012 +0200
+++ b/src/HOL/Library/RBT.thy	Fri Apr 13 11:45:30 2012 +0200
@@ -35,7 +35,7 @@
 subsection {* Primitive operations *}
 
 definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" where
-  [code]: "lookup t = RBT_Impl.lookup (impl_of t)"
+  [code]: "lookup t = rbt_lookup (impl_of t)"
 
 definition empty :: "('a\<Colon>linorder, 'b) rbt" where
   "empty = RBT RBT_Impl.Empty"
@@ -45,17 +45,17 @@
   by (simp add: empty_def RBT_inverse)
 
 definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "insert k v t = RBT (RBT_Impl.insert k v (impl_of t))"
+  "insert k v t = RBT (rbt_insert k v (impl_of t))"
 
 lemma impl_of_insert [code abstract]:
-  "impl_of (insert k v t) = RBT_Impl.insert k v (impl_of t)"
+  "impl_of (insert k v t) = rbt_insert k v (impl_of t)"
   by (simp add: insert_def RBT_inverse)
 
 definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "delete k t = RBT (RBT_Impl.delete k (impl_of t))"
+  "delete k t = RBT (rbt_delete k (impl_of t))"
 
 lemma impl_of_delete [code abstract]:
-  "impl_of (delete k t) = RBT_Impl.delete k (impl_of t)"
+  "impl_of (delete k t) = rbt_delete k (impl_of t)"
   by (simp add: delete_def RBT_inverse)
 
 definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" where
@@ -65,17 +65,17 @@
   [code]: "keys t = RBT_Impl.keys (impl_of t)"
 
 definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where
-  "bulkload xs = RBT (RBT_Impl.bulkload xs)"
+  "bulkload xs = RBT (rbt_bulkload xs)"
 
 lemma impl_of_bulkload [code abstract]:
-  "impl_of (bulkload xs) = RBT_Impl.bulkload xs"
+  "impl_of (bulkload xs) = rbt_bulkload xs"
   by (simp add: bulkload_def RBT_inverse)
 
 definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "map_entry k f t = RBT (RBT_Impl.map_entry k f (impl_of t))"
+  "map_entry k f t = RBT (rbt_map_entry k f (impl_of t))"
 
 lemma impl_of_map_entry [code abstract]:
-  "impl_of (map_entry k f t) = RBT_Impl.map_entry k f (impl_of t)"
+  "impl_of (map_entry k f t) = rbt_map_entry k f (impl_of t)"
   by (simp add: map_entry_def RBT_inverse)
 
 definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
@@ -98,11 +98,11 @@
 subsection {* Abstract lookup properties *}
 
 lemma lookup_RBT:
-  "is_rbt t \<Longrightarrow> lookup (RBT t) = RBT_Impl.lookup t"
+  "is_rbt t \<Longrightarrow> lookup (RBT t) = rbt_lookup t"
   by (simp add: lookup_def RBT_inverse)
 
 lemma lookup_impl_of:
-  "RBT_Impl.lookup (impl_of t) = lookup t"
+  "rbt_lookup (impl_of t) = lookup t"
   by (simp add: lookup_def)
 
 lemma entries_impl_of:
@@ -119,11 +119,11 @@
 
 lemma lookup_insert [simp]:
   "lookup (insert k v t) = (lookup t)(k \<mapsto> v)"
-  by (simp add: insert_def lookup_RBT lookup_insert lookup_impl_of)
+  by (simp add: insert_def lookup_RBT rbt_lookup_rbt_insert lookup_impl_of)
 
 lemma lookup_delete [simp]:
   "lookup (delete k t) = (lookup t)(k := None)"
-  by (simp add: delete_def lookup_RBT RBT_Impl.lookup_delete lookup_impl_of restrict_complement_singleton_eq)
+  by (simp add: delete_def lookup_RBT rbt_lookup_rbt_delete lookup_impl_of restrict_complement_singleton_eq)
 
 lemma map_of_entries [simp]:
   "map_of (entries t) = lookup t"
@@ -131,19 +131,19 @@
 
 lemma entries_lookup:
   "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
-  by (simp add: entries_def lookup_def entries_lookup)
+  by (simp add: entries_def lookup_def entries_rbt_lookup)
 
 lemma lookup_bulkload [simp]:
   "lookup (bulkload xs) = map_of xs"
-  by (simp add: bulkload_def lookup_RBT RBT_Impl.lookup_bulkload)
+  by (simp add: bulkload_def lookup_RBT rbt_lookup_rbt_bulkload)
 
 lemma lookup_map_entry [simp]:
   "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
-  by (simp add: map_entry_def lookup_RBT RBT_Impl.lookup_map_entry lookup_impl_of)
+  by (simp add: map_entry_def lookup_RBT rbt_lookup_rbt_map_entry lookup_impl_of)
 
 lemma lookup_map [simp]:
   "lookup (map f t) k = Option.map (f k) (lookup t k)"
-  by (simp add: map_def lookup_RBT RBT_Impl.lookup_map lookup_impl_of)
+  by (simp add: map_def lookup_RBT rbt_lookup_map lookup_impl_of)
 
 lemma fold_fold:
   "fold f t = List.fold (prod_case f) (entries t)"
@@ -154,7 +154,7 @@
   by (simp add: rbt_eq_iff is_empty_def impl_of_empty split: rbt.split)
 
 lemma RBT_lookup_empty [simp]: (*FIXME*)
-  "RBT_Impl.lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
+  "rbt_lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
   by (cases t) (auto simp add: fun_eq_iff)
 
 lemma lookup_empty_empty [simp]:
@@ -163,7 +163,7 @@
 
 lemma sorted_keys [iff]:
   "sorted (keys t)"
-  by (simp add: keys_def RBT_Impl.keys_def sorted_entries)
+  by (simp add: keys_def RBT_Impl.keys_def rbt_sorted_entries)
 
 lemma distinct_keys [iff]:
   "distinct (keys t)"
diff -r 11a0aa6cc677 -r 2ada2be850cb src/HOL/Library/RBT_Impl.thy
--- a/src/HOL/Library/RBT_Impl.thy	Thu Apr 12 13:47:21 2012 +0200
+++ b/src/HOL/Library/RBT_Impl.thy	Fri Apr 13 11:45:30 2012 +0200
@@ -1,4 +1,4 @@
-(*  Title:      HOL/Library/RBT_Impl.thy
+(*  Title:      RBT_Impl.thy
     Author:     Markus Reiter, TU Muenchen
     Author:     Alexander Krauss, TU Muenchen
 *)
@@ -65,202 +65,221 @@
 
 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>set (keys t). x < k)"
+context ord begin
 
-abbreviation tree_less_symbol (infix "|\<guillemotleft>" 50)
-where "t |\<guillemotleft> x \<equiv> tree_less x t"
+definition rbt_less :: "'a \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
+where
+  rbt_less_prop: "rbt_less k t \<longleftrightarrow> (\<forall>x\<in>set (keys t). x < k)"
 
-definition tree_greater :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
+abbreviation rbt_less_symbol (infix "|\<guillemotleft>" 50)
+where "t |\<guillemotleft> x \<equiv> rbt_less x t"
+
+definition rbt_greater :: "'a \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
 where
-  tree_greater_prop: "tree_greater k t = (\<forall>x\<in>set (keys t). k < x)"
+  rbt_greater_prop: "rbt_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)
+lemma rbt_less_simps [simp]:
+  "Empty |\<guillemotleft> k = True"
+  "Branch c lt kt v rt |\<guillemotleft> k \<longleftrightarrow> kt < k \<and> lt |\<guillemotleft> k \<and> rt |\<guillemotleft> k"
+  by (auto simp add: rbt_less_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)
+lemma rbt_greater_simps [simp]:
+  "k \<guillemotleft>| Empty = True"
+  "k \<guillemotleft>| (Branch c lt kt v rt) \<longleftrightarrow> k < kt \<and> k \<guillemotleft>| lt \<and> k \<guillemotleft>| rt"
+  by (auto simp add: rbt_greater_prop)
 
-lemmas tree_ord_props = tree_less_prop tree_greater_prop
+lemmas rbt_ord_props = rbt_less_prop rbt_greater_prop
+
+lemmas rbt_greater_nit = rbt_greater_prop entry_in_tree_keys
+lemmas rbt_less_nit = rbt_less_prop entry_in_tree_keys
 
-lemmas tree_greater_nit = tree_greater_prop entry_in_tree_keys
-lemmas tree_less_nit = tree_less_prop entry_in_tree_keys
+lemma (in order)
+  shows rbt_less_eq_trans: "l |\<guillemotleft> u \<Longrightarrow> u \<le> v \<Longrightarrow> l |\<guillemotleft> v"
+  and rbt_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
+  and rbt_greater_eq_trans: "u \<le> v \<Longrightarrow> v \<guillemotleft>| r \<Longrightarrow> u \<guillemotleft>| r"
+  and rbt_greater_trans: "x < y \<Longrightarrow> y \<guillemotleft>| t \<Longrightarrow> x \<guillemotleft>| t"
+  by (auto simp: rbt_ord_props)
 
-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)
-
-primrec sorted :: "('a::linorder, 'b) rbt \<Rightarrow> bool"
+primrec rbt_sorted :: "('a, '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)"
+  "rbt_sorted Empty = True"
+| "rbt_sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> rbt_sorted l \<and> rbt_sorted r)"
+
+end
 
-lemma sorted_entries:
-  "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
+context linorder begin
+
+lemma rbt_sorted_entries:
+  "rbt_sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
 by (induct t) 
-  (force simp: sorted_append sorted_Cons tree_ord_props 
+  (force simp: sorted_append sorted_Cons rbt_ord_props 
       dest!: entry_in_tree_keys)+
 
 lemma distinct_entries:
-  "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
+  "rbt_sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
 by (induct t) 
-  (force simp: sorted_append sorted_Cons tree_ord_props 
+  (force simp: sorted_append sorted_Cons rbt_ord_props 
       dest!: entry_in_tree_keys)+
 
-
 subsubsection {* Tree lookup *}
 
-primrec lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
+primrec (in ord) rbt_lookup :: "('a, '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)"
+  "rbt_lookup Empty k = None"
+| "rbt_lookup (Branch _ l x y r) k = 
+   (if k < x then rbt_lookup l k else if x < k then rbt_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 rbt_lookup_keys: "rbt_sorted t \<Longrightarrow> dom (rbt_lookup t) = set (keys t)"
+  by (induct t) (auto simp: dom_def rbt_greater_prop rbt_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))"
+lemma dom_rbt_lookup_Branch: 
+  "rbt_sorted (Branch c t1 k v t2) \<Longrightarrow> 
+    dom (rbt_lookup (Branch c t1 k v t2)) 
+    = Set.insert k (dom (rbt_lookup t1) \<union> dom (rbt_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)
+  assume "rbt_sorted (Branch c t1 k v t2)"
+  moreover from this have "rbt_sorted t1" "rbt_sorted t2" by simp_all
+  ultimately show ?thesis by (simp add: rbt_lookup_keys)
 qed
 
-lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
+lemma finite_dom_rbt_lookup [simp, intro!]: "finite (dom (rbt_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
+  let ?A = "Set.insert a (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2))"
+  have "dom (rbt_lookup (Branch color t1 a b t2)) \<subseteq> ?A" by (auto split: split_if_asm)
+  moreover from Branch have "finite (insert a (dom (rbt_lookup t1) \<union> dom (rbt_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" 
+end
+
+context ord begin
+
+lemma rbt_lookup_rbt_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> rbt_lookup t k = None" 
 by (induct t) auto
 
-lemma lookup_tree_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> lookup t k = None"
+lemma rbt_lookup_rbt_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> rbt_lookup t k = None"
 by (induct t) auto
 
-lemma lookup_Empty: "lookup Empty = empty"
+lemma rbt_lookup_Empty: "rbt_lookup Empty = empty"
 by (rule ext) simp
 
+end
+
+context linorder begin
+
 lemma map_of_entries:
-  "sorted t \<Longrightarrow> map_of (entries t) = lookup t"
+  "rbt_sorted t \<Longrightarrow> map_of (entries t) = rbt_lookup t"
 proof (induct t)
-  case Empty thus ?case by (simp add: lookup_Empty)
+  case Empty thus ?case by (simp add: rbt_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"
+  have "rbt_lookup (Branch c t1 k v t2) = rbt_lookup t2 ++ [k\<mapsto>v] ++ rbt_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"
+    from Branch have RBT_SORTED: "rbt_sorted (Branch c t1 k v t2)" by simp
+    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 (lookup t1) \<Longrightarrow> k>k'"
+    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>set (keys t1). k>k'" by auto
-      moreover assume "k'\<in>dom (lookup t1)"
-      ultimately show "k>k'" using lookup_keys SORTED by auto
+      from RBT_SORTED have "t1 |\<guillemotleft> k" by simp
+      with rbt_less_prop have "\<forall>k'\<in>set (keys t1). k>k'" by auto
+      moreover assume "k'\<in>dom (rbt_lookup t1)"
+      ultimately show "k>k'" using rbt_lookup_keys RBT_SORTED by auto
     qed
     
-    have DOM_T2: "!!k'. k'\<in>dom (lookup t2) \<Longrightarrow> k<k'"
+    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>set (keys t2). k<k'" by auto
-      moreover assume "k'\<in>dom (lookup t2)"
-      ultimately show "k<k'" using lookup_keys SORTED by auto
+      from RBT_SORTED have "k \<guillemotleft>| t2" by simp
+      with rbt_greater_prop have "\<forall>k'\<in>set (keys t2). k<k'" by auto
+      moreover assume "k'\<in>dom (rbt_lookup t2)"
+      ultimately show "k<k'" using rbt_lookup_keys RBT_SORTED by auto
     qed
     
     {
       assume C: "x<k"
-      hence "lookup (Branch c t1 k v t2) x = lookup 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 (lookup t2)" proof
-        assume "x\<in>dom (lookup 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 "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)"
+      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 "lookup (Branch c t1 k v t2) x = lookup 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 (lookup t1)" proof
-        assume "x\<in>dom (lookup 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
       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
+  also from Branch 
+  have "rbt_lookup t2 ++ [k \<mapsto> v] ++ rbt_lookup t1 = map_of (entries (Branch c t1 k v t2))" by simp
   finally show ?case by simp
 qed
 
-lemma lookup_in_tree: "sorted t \<Longrightarrow> lookup t k = Some v \<longleftrightarrow> (k, v) \<in> set (entries t)"
+lemma rbt_lookup_in_tree: "rbt_sorted t \<Longrightarrow> rbt_lookup t k = Some v \<longleftrightarrow> (k, v) \<in> set (entries t)"
   by (simp add: map_of_entries [symmetric] distinct_entries)
 
 lemma set_entries_inject:
-  assumes sorted: "sorted t1" "sorted t2" 
+  assumes rbt_sorted: "rbt_sorted t1" "rbt_sorted t2" 
   shows "set (entries t1) = set (entries t2) \<longleftrightarrow> entries t1 = entries t2"
 proof -
-  from sorted have "distinct (map fst (entries t1))"
+  from rbt_sorted have "distinct (map fst (entries t1))"
     "distinct (map fst (entries t2))"
     by (auto intro: distinct_entries)
-  with sorted show ?thesis
-    by (auto intro: map_sorted_distinct_set_unique sorted_entries simp add: distinct_map)
+  with rbt_sorted show ?thesis
+    by (auto intro: map_sorted_distinct_set_unique rbt_sorted_entries simp add: distinct_map)
 qed
 
 lemma entries_eqI:
-  assumes sorted: "sorted t1" "sorted t2" 
-  assumes lookup: "lookup t1 = lookup t2"
+  assumes rbt_sorted: "rbt_sorted t1" "rbt_sorted t2" 
+  assumes rbt_lookup: "rbt_lookup t1 = rbt_lookup t2"
   shows "entries t1 = entries t2"
 proof -
-  from sorted lookup have "map_of (entries t1) = map_of (entries t2)"
+  from rbt_sorted rbt_lookup have "map_of (entries t1) = map_of (entries t2)"
     by (simp add: map_of_entries)
-  with sorted have "set (entries t1) = set (entries t2)"
+  with rbt_sorted have "set (entries t1) = set (entries t2)"
     by (simp add: map_of_inject_set distinct_entries)
-  with sorted show ?thesis by (simp add: set_entries_inject)
+  with rbt_sorted show ?thesis by (simp add: set_entries_inject)
 qed
 
-lemma entries_lookup:
-  assumes "sorted t1" "sorted t2" 
-  shows "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
+lemma entries_rbt_lookup:
+  assumes "rbt_sorted t1" "rbt_sorted t2" 
+  shows "entries t1 = entries t2 \<longleftrightarrow> rbt_lookup t1 = rbt_lookup t2"
   using assms by (auto intro: entries_eqI simp add: map_of_entries [symmetric])
 
-lemma lookup_from_in_tree: 
-  assumes "sorted t1" "sorted t2" 
-  and "\<And>v. (k\<Colon>'a\<Colon>linorder, v) \<in> set (entries t1) \<longleftrightarrow> (k, v) \<in> set (entries t2)" 
-  shows "lookup t1 k = lookup t2 k"
+lemma rbt_lookup_from_in_tree: 
+  assumes "rbt_sorted t1" "rbt_sorted t2" 
+  and "\<And>v. (k, v) \<in> set (entries t1) \<longleftrightarrow> (k, v) \<in> set (entries t2)" 
+  shows "rbt_lookup t1 k = rbt_lookup t2 k"
 proof -
-  from assms have "k \<in> dom (lookup t1) \<longleftrightarrow> k \<in> dom (lookup t2)"
-    by (simp add: keys_entries lookup_keys)
-  with assms show ?thesis by (auto simp add: lookup_in_tree [symmetric])
+  from assms have "k \<in> dom (rbt_lookup t1) \<longleftrightarrow> k \<in> dom (rbt_lookup t2)"
+    by (simp add: keys_entries rbt_lookup_keys)
+  with assms show ?thesis by (auto simp add: rbt_lookup_in_tree [symmetric])
 qed
 
+end
 
 subsubsection {* Red-black properties *}
 
@@ -290,15 +309,18 @@
   "inv2 Empty = True"
 | "inv2 (Branch c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bheight lt = bheight rt)"
 
-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"
+context ord begin
 
-lemma is_rbt_sorted [simp]:
-  "is_rbt t \<Longrightarrow> sorted t" by (simp add: is_rbt_def)
+definition is_rbt :: "('a, 'b) rbt \<Rightarrow> bool" where
+  "is_rbt t \<longleftrightarrow> inv1 t \<and> inv2 t \<and> color_of t = B \<and> rbt_sorted t"
+
+lemma is_rbt_rbt_sorted [simp]:
+  "is_rbt t \<Longrightarrow> rbt_sorted t" by (simp add: is_rbt_def)
 
 theorem Empty_is_rbt [simp]:
   "is_rbt Empty" by (simp add: is_rbt_def)
 
+end
 
 subsection {* Insertion *}
 
@@ -324,61 +346,65 @@
   using assms
   by (induct l k v r rule: balance.induct) auto
 
-lemma balance_tree_greater[simp]: "(v \<guillemotleft>| balance a k x b) = (v \<guillemotleft>| a \<and> v \<guillemotleft>| b \<and> v < k)" 
+context ord begin
+
+lemma balance_rbt_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_tree_less[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"
+lemma balance_rbt_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_sorted: 
-  fixes k :: "'a::linorder"
-  assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-  shows "sorted (balance l k v r)"
+end
+
+lemma (in linorder) balance_rbt_sorted: 
+  fixes k :: "'a"
+  assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+  shows "rbt_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>| 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)
+    by (auto simp add: rbt_ord_props)
+  hence "y \<guillemotleft>| (Branch B va vb vd vc)" by (blast dest: rbt_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> tree_less x (Branch B va vb vd vc)" 
+  from "3_2" have "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" 
     by simp
-  hence "tree_less y (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
+  hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_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> 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)
+  from "3_3" have "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" by simp
+  hence "y \<guillemotleft>| Branch B va vb vd vc" by (blast dest: rbt_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> 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)
+  hence "x < y \<and> Branch B vd ve vg vf |\<guillemotleft> x" by simp
+  hence 1: "Branch B vd ve vg vf |\<guillemotleft> y" by (blast dest: rbt_less_trans)
+  from "3_4" have "y < z \<and> z \<guillemotleft>| Branch B va vb vii vc" by simp
+  hence "y \<guillemotleft>| Branch B va vb vii vc" by (blast dest: rbt_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> 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)
+  hence "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" by simp
+  hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_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> 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)
+  hence "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" by simp
+  hence "y \<guillemotleft>| Branch B va vb vd vc" by (blast dest: rbt_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> 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)
+  hence "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" by simp
+  hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_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> 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)
+  hence "x < y \<and> Branch B va vb vg vc |\<guillemotleft> x" by simp
+  hence 1: "Branch B va vb vg vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
+  from "5_4" have "y < z \<and> z \<guillemotleft>| Branch B vd ve vii vf" by simp
+  hence "y \<guillemotleft>| Branch B vd ve vii vf" by (blast dest: rbt_greater_trans)
   with 1 "5_4" show ?case by simp
 qed simp+
 
@@ -394,11 +420,11 @@
   "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_in_tree) (auto simp:assms balance_in_tree balance_sorted)
+lemma (in linorder) rbt_lookup_balance[simp]: 
+fixes k :: "'a"
+assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+shows "rbt_lookup (balance l k v r) x = rbt_lookup (Branch B l k v r) x"
+by (rule rbt_lookup_from_in_tree) (auto simp:assms balance_in_tree balance_rbt_sorted)
 
 primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
@@ -409,95 +435,112 @@
 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_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
+
+context ord begin
+
+lemma paint_rbt_sorted[simp]: "rbt_sorted t \<Longrightarrow> rbt_sorted (paint c t)" by (cases t) auto
+lemma paint_rbt_lookup[simp]: "rbt_lookup (paint c t) = rbt_lookup t" by (rule ext) (cases t, auto)
+lemma paint_rbt_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
+lemma paint_rbt_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"
+  rbt_ins :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
-  "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 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)"
+  "rbt_ins f k v Empty = Branch R Empty k v Empty" |
+  "rbt_ins f k v (Branch B l x y r) = (if k < x then balance (rbt_ins f k v l) x y r
+                                       else if k > x then balance l x y (rbt_ins f k v r)
+                                       else Branch B l x (f k y v) r)" |
+  "rbt_ins f k v (Branch R l x y r) = (if k < x then Branch R (rbt_ins f k v l) x y r
+                                       else if k > x then Branch R l x y (rbt_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)" "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)"
+  shows "inv2 (rbt_ins f k x t)" "bheight (rbt_ins f k x t) = bheight t" 
+  "color_of t = B \<Longrightarrow> inv1 (rbt_ins f k x t)" "inv1l (rbt_ins f k x t)"
   using assms
-  by (induct f k x t rule: ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
+  by (induct f k x t rule: rbt_ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
+
+end
+
+context linorder begin
 
-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_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 ins_rbt_greater[simp]: "(v \<guillemotleft>| rbt_ins f (k :: 'a) x t) = (v \<guillemotleft>| t \<and> k > v)"
+  by (induct f k x t rule: rbt_ins.induct) auto
+lemma ins_rbt_less[simp]: "(rbt_ins f k x t |\<guillemotleft> v) = (t |\<guillemotleft> v \<and> k < v)"
+  by (induct f k x t rule: rbt_ins.induct) auto
+lemma ins_rbt_sorted[simp]: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_ins f k x t)"
+  by (induct f k x t rule: rbt_ins.induct) (auto simp: balance_rbt_sorted)
 
-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 keys_ins: "set (keys (rbt_ins f k v t)) = { k } \<union> set (keys t)"
+  by (induct f k v t rule: rbt_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
+lemma rbt_lookup_ins: 
+  fixes k :: "'a"
+  assumes "rbt_sorted t"
+  shows "rbt_lookup (rbt_ins f k v t) x = ((rbt_lookup t)(k |-> case rbt_lookup t k of None \<Rightarrow> v 
+                                                                | Some w \<Rightarrow> f k w v)) x"
+using assms by (induct f k v t rule: rbt_ins.induct) auto
+
+end
+
+context ord begin
+
+definition rbt_insert_with_key :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where "rbt_insert_with_key f k v t = paint B (rbt_ins f k v t)"
+
+definition rbt_insertw_def: "rbt_insert_with f = rbt_insert_with_key (\<lambda>_. f)"
 
-definition
-  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)"
+definition rbt_insert :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
+  "rbt_insert = rbt_insert_with_key (\<lambda>_ _ nv. nv)"
+
+end
+
+context linorder begin
 
-lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insert_with_key f k x t)"
-  by (auto simp: insert_with_key_def)
+lemma rbt_insertwk_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert_with_key f (k :: 'a) x t)"
+  by (auto simp: rbt_insert_with_key_def)
 
-theorem insertwk_is_rbt: 
+theorem rbt_insertwk_is_rbt: 
   assumes inv: "is_rbt t" 
-  shows "is_rbt (insert_with_key f k x t)"
+  shows "is_rbt (rbt_insert_with_key f k x t)"
 using assms
-unfolding insert_with_key_def is_rbt_def
+unfolding rbt_insert_with_key_def is_rbt_def
 by (auto simp: ins_inv1_inv2)
 
-lemma lookup_insertwk: 
-  assumes "sorted t"
-  shows "lookup (insert_with_key f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v 
+lemma rbt_lookup_rbt_insertwk: 
+  assumes "rbt_sorted t"
+  shows "rbt_lookup (rbt_insert_with_key f k v t) x = ((rbt_lookup t)(k |-> case rbt_lookup t k of None \<Rightarrow> v 
                                                        | Some w \<Rightarrow> f k w v)) x"
-unfolding insert_with_key_def using assms
-by (simp add:lookup_ins)
+unfolding rbt_insert_with_key_def using assms
+by (simp add:rbt_lookup_ins)
 
-definition
-  insertw_def: "insert_with f = insert_with_key (\<lambda>_. f)"
+lemma rbt_insertw_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert_with f k v t)" 
+  by (simp add: rbt_insertwk_rbt_sorted rbt_insertw_def)
+theorem rbt_insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (rbt_insert_with f k v t)"
+  by (simp add: rbt_insertwk_is_rbt rbt_insertw_def)
 
-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:
+lemma rbt_lookup_rbt_insertw:
   assumes "is_rbt t"
-  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))"
+  shows "rbt_lookup (rbt_insert_with f k v t) = (rbt_lookup t)(k \<mapsto> (if k:dom (rbt_lookup t) then f (the (rbt_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)
+unfolding rbt_insertw_def
+by (rule_tac ext) (cases "rbt_lookup t k", auto simp:rbt_lookup_rbt_insertwk dom_def)
 
-definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "insert = insert_with_key (\<lambda>_ _ nv. nv)"
+lemma rbt_insert_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert k v t)"
+  by (simp add: rbt_insertwk_rbt_sorted rbt_insert_def)
+theorem rbt_insert_is_rbt [simp]: "is_rbt t \<Longrightarrow> is_rbt (rbt_insert k v t)"
+  by (simp add: rbt_insertwk_is_rbt rbt_insert_def)
 
-lemma insert_sorted: "sorted t \<Longrightarrow> sorted (insert k v t)" by (simp add: insertwk_sorted 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: 
+lemma rbt_lookup_rbt_insert: 
   assumes "is_rbt t"
-  shows "lookup (insert k v t) = (lookup t)(k\<mapsto>v)"
-unfolding insert_def
+  shows "rbt_lookup (rbt_insert k v t) = (rbt_lookup t)(k\<mapsto>v)"
+unfolding rbt_insert_def
 using assms
-by (rule_tac ext) (simp add: lookup_insertwk split:option.split)
+by (rule_tac ext) (simp add: rbt_lookup_rbt_insertwk split:option.split)
 
+end
 
 subsection {* Deletion *}
 
@@ -532,26 +575,31 @@
 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)"
+lemma (in linorder) balance_left_rbt_sorted: 
+  "\<lbrakk> rbt_sorted l; rbt_sorted r; rbt_less k l; k \<guillemotleft>| r \<rbrakk> \<Longrightarrow> rbt_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)
+apply (auto simp: balance_rbt_sorted)
+apply (unfold rbt_greater_prop rbt_less_prop)
 by force+
 
-lemma balance_left_tree_greater: 
-  fixes k :: "'a::order"
+context order begin
+
+lemma balance_left_rbt_greater: 
+  fixes k :: "'a"
   assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
   shows "k \<guillemotleft>| balance_left a x t b"
 using assms 
 by (induct a x t b rule: balance_left.induct) auto
 
-lemma balance_left_tree_less: 
-  fixes k :: "'a::order"
+lemma balance_left_rbt_less: 
+  fixes k :: "'a"
   assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
   shows "balance_left a x t b |\<guillemotleft> k"
 using assms
 by (induct a x t b rule: balance_left.induct) auto
 
+end
+
 lemma balance_left_in_tree: 
   assumes "inv1l l" "inv1 r" "bheight l + 1 = bheight 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)"
@@ -578,24 +626,29 @@
 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)"
+lemma (in linorder) balance_right_rbt_sorted:
+  "\<lbrakk> rbt_sorted l; rbt_sorted r; rbt_less k l; k \<guillemotleft>| r \<rbrakk> \<Longrightarrow> rbt_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)
+apply (auto simp:balance_rbt_sorted)
+apply (unfold rbt_less_prop rbt_greater_prop)
 by force+
 
-lemma balance_right_tree_greater: 
-  fixes k :: "'a::order"
+context order begin
+
+lemma balance_right_rbt_greater: 
+  fixes k :: "'a"
   assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
   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"
+lemma balance_right_rbt_less: 
+  fixes k :: "'a"
   assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
   shows "balance_right a x t b |\<guillemotleft> k"
 using assms by (induct a x t b rule: balance_right.induct) auto
 
+end
+
 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 (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)"
@@ -607,11 +660,11 @@
   "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))" 
+                                    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))" 
 | "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> balance_left a k x (Branch B bc s y d))" 
+                                    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> 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)" 
 
@@ -630,26 +683,28 @@
 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"
+context linorder begin
+
+lemma combine_rbt_greater[simp]: 
+  fixes k :: "'a"
   assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r" 
   shows "k \<guillemotleft>| combine l r"
 using assms 
 by (induct l r rule: combine.induct)
-   (auto simp: balance_left_tree_greater split:rbt.splits color.splits)
+   (auto simp: balance_left_rbt_greater split:rbt.splits color.splits)
 
-lemma combine_tree_less[simp]: 
-  fixes k :: "'a::linorder"
+lemma combine_rbt_less[simp]: 
+  fixes k :: "'a"
   assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k" 
   shows "combine l r |\<guillemotleft> k"
 using assms 
 by (induct l r rule: combine.induct)
-   (auto simp: balance_left_tree_less split:rbt.splits color.splits)
+   (auto simp: balance_left_rbt_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 (combine l r)"
+lemma combine_rbt_sorted: 
+  fixes k :: "'a"
+  assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+  shows "rbt_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"
@@ -657,48 +712,52 @@
   with 3
   show ?case
     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)+)
+      (auto, (metis combine_rbt_greater combine_rbt_less ineqs ineqs rbt_less_simps(2) rbt_greater_simps(2) rbt_greater_trans rbt_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 (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 (combine b c)" by (simp add: combine_tree_less)
+  hence "x < k \<and> rbt_greater k c" by simp
+  hence "rbt_greater x c" by (blast dest: rbt_greater_trans)
+  with 4 have 2: "rbt_greater x (combine b c)" by (simp add: combine_rbt_greater)
+  from 4 have "k < y \<and> rbt_less k b" by simp
+  hence "rbt_less y b" by (blast dest: rbt_less_trans)
+  with 4 have 3: "rbt_less y (combine b c)" by (simp add: combine_rbt_less)
   show ?case
   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: balance_left_sorted)
+    from 4 have "x < y \<and> rbt_greater y d" by auto
+    hence "rbt_greater x d" by (blast dest: rbt_greater_trans)
+    with 4 Empty have "rbt_sorted a" and "rbt_sorted (Branch B Empty y w d)"
+      and "rbt_less x a" and "rbt_greater x (Branch B Empty y w d)" by auto
+    with Empty show ?thesis by (simp add: balance_left_rbt_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
-    hence "tree_greater va d" by (blast dest: tree_greater_trans)
+    with 2 4 have "x < va \<and> rbt_less x a" by simp
+    hence 5: "rbt_less va a" by (blast dest: rbt_less_trans)
+    from Red 3 4 have "va < y \<and> rbt_greater y d" by simp
+    hence "rbt_greater va d" by (blast dest: rbt_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> 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 (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)
+    from 4 have "x < y \<and> rbt_greater y d" by auto
+    hence "rbt_greater x d" by (blast dest: rbt_greater_trans)
+    with Black 2 3 4 have "rbt_sorted a" and "rbt_sorted (Branch B (combine b c) y w d)" 
+      and "rbt_less x a" and "rbt_greater x (Branch B (combine b c) y w d)" by auto
+    with Black show ?thesis by (simp add: balance_left_rbt_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: combine_tree_less)
+  hence "k < x \<and> rbt_less k (Branch B va vb vd vc)" by simp
+  hence "rbt_less x (Branch B va vb vd vc)" by (blast dest: rbt_less_trans)
+  with 5 show ?case by (simp add: combine_rbt_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: combine_tree_greater)
+  hence "x < k \<and> rbt_greater k (Branch B va vb vd vc)" by simp
+  hence "rbt_greater x (Branch B va vb vd vc)" by (blast dest: rbt_greater_trans)
+  with 6 show ?case by (simp add: combine_rbt_greater)
 qed simp+
 
+end
+
 lemma combine_in_tree: 
   assumes "inv2 l" "inv2 r" "bheight l = bheight r" "inv1 l" "inv1 r"
   shows "entry_in_tree k v (combine l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
@@ -721,29 +780,43 @@
   qed 
 qed (auto split: rbt.splits color.splits)
 
+context ord begin
+
 fun
-  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"
+  rbt_del_from_left :: "'a \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
+  rbt_del_from_right :: "'a \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
+  rbt_del :: "'a\<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 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)"
+  "rbt_del x Empty = Empty" |
+  "rbt_del x (Branch c a y s b) = 
+   (if x < y then rbt_del_from_left x a y s b 
+    else (if x > y then rbt_del_from_right x a y s b else combine a b))" |
+  "rbt_del_from_left x (Branch B lt z v rt) y s b = balance_left (rbt_del x (Branch B lt z v rt)) y s b" |
+  "rbt_del_from_left x a y s b = Branch R (rbt_del x a) y s b" |
+  "rbt_del_from_right x a y s (Branch B lt z v rt) = balance_right a y s (rbt_del x (Branch B lt z v rt))" | 
+  "rbt_del_from_right x a y s b = Branch R a y s (rbt_del x b)"
+
+end
+
+context linorder begin
 
 lemma 
   assumes "inv2 lt" "inv1 lt"
   shows
   "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
-  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))"
+   inv2 (rbt_del_from_left x lt k v rt) \<and> 
+   bheight (rbt_del_from_left x lt k v rt) = bheight lt \<and> 
+   (color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_left x lt k v rt) \<or> 
+    (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_left x lt k v rt))"
   and "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
-  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))"
+  inv2 (rbt_del_from_right x lt k v rt) \<and> 
+  bheight (rbt_del_from_right x lt k v rt) = bheight lt \<and> 
+  (color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_right x lt k v rt) \<or> 
+   (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_right x lt k v rt))"
+  and rbt_del_inv1_inv2: "inv2 (rbt_del x lt) \<and> (color_of lt = R \<and> bheight (rbt_del x lt) = bheight lt \<and> inv1 (rbt_del x lt) 
+  \<or> color_of lt = B \<and> bheight (rbt_del x lt) = bheight lt - 1 \<and> inv1l (rbt_del x lt))"
 using assms
-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)
+proof (induct x lt k v rt and x lt k v rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
 case (2 y c _ y')
   have "y = y' \<or> y < y' \<or> y > y'" by auto
   thus ?case proof (elim disjE)
@@ -767,55 +840,55 @@
 qed auto
 
 lemma 
-  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: del_from_left_del_from_right_del.induct) 
-   (auto simp: balance_left_tree_less balance_right_tree_less)
+  rbt_del_from_left_rbt_less: "\<lbrakk> lt |\<guillemotleft> v; rt |\<guillemotleft> v; k < v\<rbrakk> \<Longrightarrow> rbt_del_from_left x lt k y rt |\<guillemotleft> v"
+  and rbt_del_from_right_rbt_less: "\<lbrakk>lt |\<guillemotleft> v; rt |\<guillemotleft> v; k < v\<rbrakk> \<Longrightarrow> rbt_del_from_right x lt k y rt |\<guillemotleft> v"
+  and rbt_del_rbt_less: "lt |\<guillemotleft> v \<Longrightarrow> rbt_del x lt |\<guillemotleft> v"
+by (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct) 
+   (auto simp: balance_left_rbt_less balance_right_rbt_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: del_from_left_del_from_right_del.induct)
-   (auto simp: balance_left_tree_greater balance_right_tree_greater)
+lemma rbt_del_from_left_rbt_greater: "\<lbrakk>v \<guillemotleft>| lt; v \<guillemotleft>| rt; k > v\<rbrakk> \<Longrightarrow> v \<guillemotleft>| rbt_del_from_left x lt k y rt"
+  and rbt_del_from_right_rbt_greater: "\<lbrakk>v \<guillemotleft>| lt; v \<guillemotleft>| rt; k > v\<rbrakk> \<Longrightarrow> v \<guillemotleft>| rbt_del_from_right x lt k y rt"
+  and rbt_del_rbt_greater: "v \<guillemotleft>| lt \<Longrightarrow> v \<guillemotleft>| rbt_del x lt"
+by (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
+   (auto simp: balance_left_rbt_greater balance_right_rbt_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: del_from_left_del_from_right_del.induct)
+lemma "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> k; k \<guillemotleft>| rt\<rbrakk> \<Longrightarrow> rbt_sorted (rbt_del_from_left x lt k y rt)"
+  and "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> k; k \<guillemotleft>| rt\<rbrakk> \<Longrightarrow> rbt_sorted (rbt_del_from_right x lt k y rt)"
+  and rbt_del_rbt_sorted: "rbt_sorted lt \<Longrightarrow> rbt_sorted (rbt_del x lt)"
+proof (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_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: balance_left_sorted)
+  from 3 have "Branch B lta zz v rta |\<guillemotleft> yy" by simp
+  hence "rbt_del x (Branch B lta zz v rta) |\<guillemotleft> yy" by (rule rbt_del_rbt_less)
+  with 3 show ?case by (simp add: balance_left_rbt_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)
+  hence "Branch R vaa vbb vdd vc |\<guillemotleft> yy" by simp
+  hence "rbt_del x (Branch R vaa vbb vdd vc) |\<guillemotleft> yy" by (rule rbt_del_rbt_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: balance_right_sorted)
+  hence "yy \<guillemotleft>| Branch B lta zz v rta" by simp
+  hence "yy \<guillemotleft>| rbt_del x (Branch B lta zz v rta)" by (rule rbt_del_rbt_greater)
+  with 5 show ?case by (simp add: balance_right_rbt_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)
+  hence "yy \<guillemotleft>| Branch R vaa vbb vdd vc" by simp
+  hence "yy \<guillemotleft>| rbt_del x (Branch R vaa vbb vdd vc)" by (rule rbt_del_rbt_greater)
   with "6_2" show ?case by simp
-qed (auto simp: combine_sorted)
+qed (auto simp: combine_rbt_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)
+lemma "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> kt; kt \<guillemotleft>| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_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>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> kt; kt \<guillemotleft>| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_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 rbt_del_in_tree: "\<lbrakk>rbt_sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_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: rbt_del_from_left_rbt_del_from_right_rbt_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)
+      from 2 `xx = yy` `xx = k` have "rbt_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: rbt_less_nit rbt_greater_prop)
       with `xx = yy` 2 `xx = k` show ?thesis by (simp add: combine_in_tree)
     qed (simp add: combine_in_tree)
   qed simp+
@@ -823,143 +896,147 @@
   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 (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)
+  hence "inv2 (rbt_del xx mt) \<and> (color_of mt = R \<and> bheight (rbt_del xx mt) = bheight mt \<and> inv1 (rbt_del xx mt) \<or> color_of mt = B \<and> bheight (rbt_del xx mt) = bheight mt - 1 \<and> inv1l (rbt_del xx mt))" by (blast dest: rbt_del_inv1_inv2)
+  with 3 have 4: "entry_in_tree k v (rbt_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)
+    from 3 True have "yy \<guillemotleft>| bb \<and> yy > k" by simp
+    hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
+    with 3 4 True show ?thesis by (auto simp: rbt_greater_nit)
   qed auto
 next
   case ("4_1" xx yy ss bb)
   show ?case proof (cases "xx = k")
     case True
-    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" have "yy \<guillemotleft>| bb \<and> k < yy" by simp
+    hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
     with "4_1" `xx = k` 
-   have "entry_in_tree k v (Branch R Empty yy ss bb) = entry_in_tree k v Empty" by (auto simp: tree_greater_nit)
+   have "entry_in_tree k v (Branch R Empty yy ss bb) = entry_in_tree k v Empty" by (auto simp: rbt_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> 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)
+    with "4_2" have "k < yy \<and> yy \<guillemotleft>| bb" by simp
+    hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
+    with True "4_2" show ?thesis by (auto simp: rbt_greater_nit)
   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 (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)
+  hence "inv2 (rbt_del xx mt) \<and> (color_of mt = R \<and> bheight (rbt_del xx mt) = bheight mt \<and> inv1 (rbt_del xx mt) \<or> color_of mt = B \<and> bheight (rbt_del xx mt) = bheight mt - 1 \<and> inv1l (rbt_del xx mt))" by (blast dest: rbt_del_inv1_inv2)
+  with 5 have 3: "entry_in_tree k v (rbt_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)
+    from 5 True have "aa |\<guillemotleft> yy \<and> yy < k" by simp
+    hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
+    with 3 5 True show ?thesis by (auto simp: rbt_less_nit)
   qed auto
 next
   case ("6_1" xx aa yy ss)
   show ?case proof (cases "xx = k")
     case True
-    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)
+    with "6_1" have "aa |\<guillemotleft> yy \<and> k > yy" by simp
+    hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
+    with "6_1" `xx = k` show ?thesis by (auto simp: rbt_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> 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)
+    with "6_2" have "k > yy \<and> aa |\<guillemotleft> yy" by simp
+    hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
+    with True "6_2" show ?thesis by (auto simp: rbt_less_nit)
   qed auto
 qed simp
 
+definition (in ord) rbt_delete where
+  "rbt_delete k t = paint B (rbt_del k t)"
 
-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 rbt_delete_is_rbt [simp]: assumes "is_rbt t" shows "is_rbt (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
+  hence "inv2 (rbt_del k t) \<and> (color_of t = R \<and> bheight (rbt_del k t) = bheight t \<and> inv1 (rbt_del k t) \<or> color_of t = B \<and> bheight (rbt_del k t) = bheight t - 1 \<and> inv1l (rbt_del k t))" by (rule rbt_del_inv1_inv2)
+  hence "inv2 (rbt_del k t) \<and> inv1l (rbt_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)
+    unfolding is_rbt_def rbt_delete_def
+    by (auto intro: paint_rbt_sorted rbt_del_rbt_sorted)
 qed
 
-lemma delete_in_tree: 
+lemma rbt_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_in_tree)
+  shows "entry_in_tree k v (rbt_delete x t) = (x \<noteq> k \<and> entry_in_tree k v t)"
+  using assms unfolding is_rbt_def rbt_delete_def
+  by (auto simp: rbt_del_in_tree)
 
-lemma lookup_delete:
+lemma rbt_lookup_rbt_delete:
   assumes is_rbt: "is_rbt t"
-  shows "lookup (delete k t) = (lookup t)|`(-{k})"
+  shows "rbt_lookup (rbt_delete k t) = (rbt_lookup t)|`(-{k})"
 proof
   fix x
-  show "lookup (delete k t) x = (lookup t |` (-{k})) x" 
+  show "rbt_lookup (rbt_delete k t) x = (rbt_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_in_tree delete_in_tree)
+      by (cases "rbt_lookup (rbt_delete k t) k") (auto simp: rbt_lookup_in_tree rbt_delete_in_tree)
   next
     assume "x \<noteq> k"
     thus ?thesis
-      by auto (metis is_rbt delete_is_rbt delete_in_tree is_rbt_sorted lookup_from_in_tree)
+      by auto (metis is_rbt rbt_delete_is_rbt rbt_delete_in_tree is_rbt_rbt_sorted rbt_lookup_from_in_tree)
   qed
 qed
 
+end
 
 subsection {* Union *}
 
-primrec
-  union_with_key :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+context ord begin
+
+primrec rbt_union_with_key :: "('a \<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"
+  "rbt_union_with_key f t Empty = t"
+| "rbt_union_with_key f t (Branch c lt k v rt) = rbt_union_with_key f (rbt_union_with_key f (rbt_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 (union_with_key f lt rt)" 
-  by (induct rt arbitrary: lt) (simp add: insertwk_is_rbt)+
+definition rbt_union_with where
+  "rbt_union_with f = rbt_union_with_key (\<lambda>_. f)"
+
+definition rbt_union where
+  "rbt_union = rbt_union_with_key (%_ _ rv. rv)"
+
+end
 
-definition
-  union_with where
-  "union_with f = union_with_key (\<lambda>_. f)"
+context linorder begin
 
-theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union_with f lt rt)" unfolding union_with_def by simp
+lemma rbt_unionwk_rbt_sorted: "rbt_sorted lt \<Longrightarrow> rbt_sorted (rbt_union_with_key f lt rt)" 
+  by (induct rt arbitrary: lt) (auto simp: rbt_insertwk_rbt_sorted)
+theorem rbt_unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (rbt_union_with_key f lt rt)" 
+  by (induct rt arbitrary: lt) (simp add: rbt_insertwk_is_rbt)+
 
-definition union where
-  "union = union_with_key (%_ _ rv. rv)"
+theorem rbt_unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (rbt_union_with f lt rt)" unfolding rbt_union_with_def by simp
+
+theorem rbt_union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (rbt_union lt rt)" unfolding rbt_union_def by simp
 
-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"
-  unfolding union_def insert_def
+lemma (in ord) rbt_union_Branch[simp]:
+  "rbt_union t (Branch c lt k v rt) = rbt_union (rbt_union (rbt_insert k v t) lt) rt"
+  unfolding rbt_union_def rbt_insert_def
   by simp
 
-lemma lookup_union:
-  assumes "is_rbt s" "sorted t"
-  shows "lookup (union s t) = lookup s ++ lookup t"
+lemma rbt_lookup_rbt_union:
+  assumes "is_rbt s" "rbt_sorted t"
+  shows "rbt_lookup (rbt_union s t) = rbt_lookup s ++ rbt_lookup t"
 using assms
 proof (induct t arbitrary: s)
-  case Empty thus ?case by (auto simp: union_def)
+  case Empty thus ?case by (auto simp: rbt_union_def)
 next
   case (Branch c l k v r s)
-  then have "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
+  then have "rbt_sorted r" "rbt_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")
+  have meq: "rbt_lookup s(k \<mapsto> v) ++ rbt_lookup l ++ rbt_lookup r =
+    rbt_lookup s ++
+    (\<lambda>a. if a < k then rbt_lookup l a
+    else if k < a then rbt_lookup r a else Some v)" (is "?m1 = ?m2")
   proof (rule ext)
     fix a
 
@@ -967,7 +1044,7 @@
     thus "?m1 a = ?m2 a"
     proof (elim disjE)
       assume "k < a"
-      with `l |\<guillemotleft> k` have "l |\<guillemotleft> a" by (rule tree_less_trans)
+      with `l |\<guillemotleft> k` have "l |\<guillemotleft> a" by (rule rbt_less_trans)
       with `k < a` show ?thesis
         by (auto simp: map_add_def split: option.splits)
     next
@@ -976,52 +1053,57 @@
       show ?thesis by (auto simp: map_add_def)
     next
       assume "a < k"
-      from this `k \<guillemotleft>| r` have "a \<guillemotleft>| r" by (rule tree_greater_trans)
+      from this `k \<guillemotleft>| r` have "a \<guillemotleft>| r" by (rule rbt_greater_trans)
       with `a < k` show ?thesis
         by (auto simp: map_add_def split: option.splits)
     qed
   qed
 
-  from Branch have is_rbt: "is_rbt (RBT_Impl.union (RBT_Impl.insert k v s) l)"
-    by (auto intro: union_is_rbt insert_is_rbt)
+  from Branch have is_rbt: "is_rbt (RBT_Impl.rbt_union (RBT_Impl.rbt_insert k v s) l)"
+    by (auto intro: rbt_union_is_rbt 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"
+    "rbt_lookup (rbt_union (rbt_union (rbt_insert k v s) l) r) = rbt_lookup (rbt_union (rbt_insert k v s) l) ++ rbt_lookup r"
+    "rbt_lookup (rbt_union (rbt_insert k v s) l) = rbt_lookup (rbt_insert k v s) ++ rbt_lookup l"
     by auto
   
   with meq show ?case
-    by (auto simp: lookup_insert[OF Branch(3)])
+    by (auto simp: rbt_lookup_rbt_insert[OF Branch(3)])
 
 qed
 
+end
 
 subsection {* Modifying existing entries *}
 
+context ord begin
+
 primrec
-  map_entry :: "'a\<Colon>linorder \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
+  rbt_map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
 where
-  "map_entry k f Empty = Empty"
-| "map_entry k f (Branch c lt x v rt) =
-    (if k < x then Branch c (map_entry k f lt) x v rt
-    else if k > x then (Branch c lt x v (map_entry k f rt))
+  "rbt_map_entry k f Empty = Empty"
+| "rbt_map_entry k f (Branch c lt x v rt) =
+    (if k < x then Branch c (rbt_map_entry k f lt) x v rt
+    else if k > x then (Branch c lt x v (rbt_map_entry k f rt))
     else Branch c lt x (f v) rt)"
 
-lemma map_entry_color_of: "color_of (map_entry k f t) = color_of t" by (induct t) simp+
-lemma map_entry_inv1: "inv1 (map_entry k f t) = inv1 t" by (induct t) (simp add: map_entry_color_of)+
-lemma map_entry_inv2: "inv2 (map_entry k f t) = inv2 t" "bheight (map_entry k f t) = bheight t" by (induct t) simp+
-lemma map_entry_tree_greater: "tree_greater a (map_entry k f t) = tree_greater a t" by (induct t) simp+
-lemma map_entry_tree_less: "tree_less a (map_entry k f t) = tree_less a t" by (induct t) simp+
-lemma map_entry_sorted: "sorted (map_entry k f t) = sorted t"
-  by (induct t) (simp_all add: map_entry_tree_less map_entry_tree_greater)
+
+lemma rbt_map_entry_color_of: "color_of (rbt_map_entry k f t) = color_of t" by (induct t) simp+
+lemma rbt_map_entry_inv1: "inv1 (rbt_map_entry k f t) = inv1 t" by (induct t) (simp add: rbt_map_entry_color_of)+
+lemma rbt_map_entry_inv2: "inv2 (rbt_map_entry k f t) = inv2 t" "bheight (rbt_map_entry k f t) = bheight t" by (induct t) simp+
+lemma rbt_map_entry_rbt_greater: "rbt_greater a (rbt_map_entry k f t) = rbt_greater a t" by (induct t) simp+
+lemma rbt_map_entry_rbt_less: "rbt_less a (rbt_map_entry k f t) = rbt_less a t" by (induct t) simp+
+lemma rbt_map_entry_rbt_sorted: "rbt_sorted (rbt_map_entry k f t) = rbt_sorted t"
+  by (induct t) (simp_all add: rbt_map_entry_rbt_less rbt_map_entry_rbt_greater)
 
-theorem map_entry_is_rbt [simp]: "is_rbt (map_entry k f t) = is_rbt t" 
-unfolding is_rbt_def by (simp add: map_entry_inv2 map_entry_color_of map_entry_sorted map_entry_inv1 )
+theorem rbt_map_entry_is_rbt [simp]: "is_rbt (rbt_map_entry k f t) = is_rbt t" 
+unfolding is_rbt_def by (simp add: rbt_map_entry_inv2 rbt_map_entry_color_of rbt_map_entry_rbt_sorted rbt_map_entry_inv1 )
 
-theorem lookup_map_entry:
-  "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
+end
+
+theorem (in linorder) rbt_lookup_rbt_map_entry:
+  "rbt_lookup (rbt_map_entry k f t) = (rbt_lookup t)(k := Option.map f (rbt_lookup t k))"
   by (induct t) (auto split: option.splits simp add: fun_eq_iff)
 
-
 subsection {* Mapping all entries *}
 
 primrec
@@ -1033,18 +1115,28 @@
 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+
+
+context ord begin
+
+lemma map_rbt_greater: "rbt_greater k (map f t) = rbt_greater k t" by (induct t) simp+
+lemma map_rbt_less: "rbt_less k (map f t) = rbt_less k t" by (induct t) simp+
+lemma map_rbt_sorted: "rbt_sorted (map f t) = rbt_sorted t"  by (induct t) (simp add: map_rbt_less map_rbt_greater)+
 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)
+unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_rbt_sorted map_color_of)
 
-theorem lookup_map: "lookup (map f t) x = Option.map (f x) (lookup t x)"
-  by (induct t) auto
+end
 
+theorem (in linorder) rbt_lookup_map: "rbt_lookup (map f t) x = Option.map (f x) (rbt_lookup t x)"
+  apply(induct t)
+  apply auto
+  apply(subgoal_tac "x = a")
+  apply auto
+  done
+ (* FIXME: simproc "antisym less" does not work for linorder context, only for linorder type class
+    by (induct t) auto *)
 
 subsection {* Folding over entries *}
 
@@ -1059,26 +1151,73 @@
 
 subsection {* Bulkloading a tree *}
 
-definition bulkload :: "('a \<times> 'b) list \<Rightarrow> ('a\<Colon>linorder, 'b) rbt" where
-  "bulkload xs = foldr (\<lambda>(k, v). insert k v) xs Empty"
+definition (in ord) rbt_bulkload :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where
+  "rbt_bulkload xs = foldr (\<lambda>(k, v). rbt_insert k v) xs Empty"
+
+context linorder begin
 
-lemma bulkload_is_rbt [simp, intro]:
-  "is_rbt (bulkload xs)"
-  unfolding bulkload_def by (induct xs) auto
+lemma rbt_bulkload_is_rbt [simp, intro]:
+  "is_rbt (rbt_bulkload xs)"
+  unfolding rbt_bulkload_def by (induct xs) auto
 
-lemma lookup_bulkload:
-  "lookup (bulkload xs) = map_of xs"
+lemma rbt_lookup_rbt_bulkload:
+  "rbt_lookup (rbt_bulkload xs) = map_of xs"
 proof -
   obtain ys where "ys = rev xs" by simp
   have "\<And>t. is_rbt t \<Longrightarrow>
-    lookup (List.fold (prod_case insert) ys t) = lookup t ++ map_of (rev ys)"
-      by (induct ys) (simp_all add: bulkload_def lookup_insert prod_case_beta)
+    rbt_lookup (List.fold (prod_case rbt_insert) ys t) = rbt_lookup t ++ map_of (rev ys)"
+      by (induct ys) (simp_all add: rbt_bulkload_def rbt_lookup_rbt_insert prod_case_beta)
   from this Empty_is_rbt have
-    "lookup (List.fold (prod_case insert) (rev xs) Empty) = lookup Empty ++ map_of xs"
+    "rbt_lookup (List.fold (prod_case rbt_insert) (rev xs) Empty) = rbt_lookup Empty ++ map_of xs"
      by (simp add: `ys = rev xs`)
-  then show ?thesis by (simp add: bulkload_def lookup_Empty foldr_conv_fold)
+  then show ?thesis by (simp add: rbt_bulkload_def rbt_lookup_Empty foldr_conv_fold)
 qed
 
-hide_const (open) R B Empty insert delete entries keys bulkload lookup map_entry map fold union sorted
+end
+
+lemmas [code] =
+  ord.rbt_less_prop
+  ord.rbt_greater_prop
+  ord.rbt_sorted.simps
+  ord.rbt_lookup.simps
+  ord.is_rbt_def
+  ord.rbt_ins.simps
+  ord.rbt_insert_with_key_def
+  ord.rbt_insertw_def
+  ord.rbt_insert_def
+  ord.rbt_del_from_left.simps
+  ord.rbt_del_from_right.simps
+  ord.rbt_del.simps
+  ord.rbt_delete_def
+  ord.rbt_union_with_key.simps
+  ord.rbt_union_with_def
+  ord.rbt_union_def
+  ord.rbt_map_entry.simps
+  ord.rbt_bulkload_def
+
+text {* Restore original type constraints for constants *}
+setup {*
+  fold Sign.add_const_constraint
+    [(@{const_name rbt_less}, SOME @{typ "('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"}),
+     (@{const_name rbt_greater}, SOME @{typ "('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"}),
+     (@{const_name rbt_sorted}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> bool"}),
+     (@{const_name rbt_lookup}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"}),
+     (@{const_name is_rbt}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> bool"}),
+     (@{const_name rbt_ins}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_insert_with_key}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_insert_with}, SOME @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_insert}, SOME @{typ "('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_del_from_left}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_del_from_right}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_del}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_delete}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_union_with_key}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_union_with}, SOME @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_union}, SOME @{typ "('a\<Colon>linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_map_entry}, SOME @{typ "'a\<Colon>linorder \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
+     (@{const_name rbt_bulkload}, SOME @{typ "('a \<times> 'b) list \<Rightarrow> ('a\<Colon>linorder,'b) rbt"})]
+*}
+
+hide_const (open) R B Empty entries keys map fold
 
 end
diff -r 11a0aa6cc677 -r 2ada2be850cb src/HOL/Library/RBT_Mapping.thy
--- a/src/HOL/Library/RBT_Mapping.thy	Thu Apr 12 13:47:21 2012 +0200
+++ b/src/HOL/Library/RBT_Mapping.thy	Fri Apr 13 11:45:30 2012 +0200
@@ -40,7 +40,7 @@
 
 lemma keys_Mapping [code]:
   "Mapping.keys (Mapping t) = set (keys t)"
-  by (simp add: RBT.keys_def Mapping_def Mapping.keys_def lookup_def lookup_keys)
+  by (simp add: RBT.keys_def Mapping_def Mapping.keys_def lookup_def rbt_lookup_keys)
 
 lemma ordered_keys_Mapping [code]:
   "Mapping.ordered_keys (Mapping t) = keys t"
@@ -144,22 +144,22 @@
   @{thm Empty_is_rbt}\hfill(@{text "Empty_is_rbt"})
 
   \noindent
-  @{thm insert_is_rbt}\hfill(@{text "insert_is_rbt"})
+  @{thm rbt_insert_is_rbt}\hfill(@{text "rbt_insert_is_rbt"})
 
   \noindent
-  @{thm delete_is_rbt}\hfill(@{text "delete_is_rbt"})
+  @{thm rbt_delete_is_rbt}\hfill(@{text "delete_is_rbt"})
 
   \noindent
-  @{thm bulkload_is_rbt}\hfill(@{text "bulkload_is_rbt"})
+  @{thm rbt_bulkload_is_rbt}\hfill(@{text "bulkload_is_rbt"})
 
   \noindent
-  @{thm map_entry_is_rbt}\hfill(@{text "map_entry_is_rbt"})
+  @{thm rbt_map_entry_is_rbt}\hfill(@{text "map_entry_is_rbt"})
 
   \noindent
   @{thm map_is_rbt}\hfill(@{text "map_is_rbt"})
 
   \noindent
-  @{thm union_is_rbt}\hfill(@{text "union_is_rbt"})
+  @{thm rbt_union_is_rbt}\hfill(@{text "union_is_rbt"})
 *}