# HG changeset patch
# User haftmann
# Date 1271327234 -7200
# Node ID b43b22f636659a86518bffe3416fd21a3316f8d6
# Parent  7bfbb247a5dfcf5d1bb87f1fb666eeaac33bf2ab
theory RBT with abstract type of red-black trees backed by implementation RBT_Impl

diff -r 7bfbb247a5df -r b43b22f63665 NEWS
--- a/NEWS	Wed Apr 14 22:18:10 2010 +0200
+++ b/NEWS	Thu Apr 15 12:27:14 2010 +0200
@@ -74,6 +74,8 @@
 
 *** Pure ***
 
+* Code generator: simple concept for abstract datatypes obeying invariants.
+
 * Local theory specifications may depend on extra type variables that
 are not present in the result type -- arguments TYPE('a) :: 'a itself
 are added internally.  For example:
@@ -106,6 +108,10 @@
 
 *** HOL ***
 
+* Library theory 'RBT' renamed to 'RBT_Impl'; new library theory 'RBT'
+provides abstract red-black tree type which is backed by RBT_Impl
+as implementation.  INCOMPATIBILTY.
+
 * Command 'typedef' now works within a local theory context -- without
 introducing dependencies on parameters or assumptions, which is not
 possible in Isabelle/Pure/HOL.  Note that the logical environment may
diff -r 7bfbb247a5df -r b43b22f63665 src/HOL/Imperative_HOL/ex/SatChecker.thy
--- a/src/HOL/Imperative_HOL/ex/SatChecker.thy	Wed Apr 14 22:18:10 2010 +0200
+++ b/src/HOL/Imperative_HOL/ex/SatChecker.thy	Thu Apr 15 12:27:14 2010 +0200
@@ -5,7 +5,7 @@
 header {* An efficient checker for proofs from a SAT solver *}
 
 theory SatChecker
-imports RBT Sorted_List "~~/src/HOL/Imperative_HOL/Imperative_HOL" 
+imports RBT_Impl Sorted_List "~~/src/HOL/Imperative_HOL/Imperative_HOL" 
 begin
 
 section{* General settings and functions for our representation of clauses *}
@@ -635,24 +635,24 @@
 
 section {* Functional version with RedBlackTrees *}
 
-fun tres_thm :: "(ClauseId, Clause) rbt \<Rightarrow> Lit \<times> ClauseId \<Rightarrow> Clause \<Rightarrow> Clause Heap"
+fun tres_thm :: "(ClauseId, Clause) RBT_Impl.rbt \<Rightarrow> Lit \<times> ClauseId \<Rightarrow> Clause \<Rightarrow> Clause Heap"
 where
   "tres_thm t (l, j) cli =
-  (case (RBT.lookup t j) of 
+  (case (RBT_Impl.lookup t j) of 
      None \<Rightarrow> raise (''MiniSatChecked.res_thm: No resolvant clause in thms array for Conflict step.'')
    | Some clj \<Rightarrow> res_thm' l cli clj)"
 
-fun tdoProofStep :: " ProofStep \<Rightarrow> ((ClauseId, Clause) rbt * Clause list) \<Rightarrow> ((ClauseId, Clause) rbt * Clause list) Heap"
+fun tdoProofStep :: " ProofStep \<Rightarrow> ((ClauseId, Clause) RBT_Impl.rbt * Clause list) \<Rightarrow> ((ClauseId, Clause) RBT_Impl.rbt * Clause list) Heap"
 where
   "tdoProofStep (Conflict saveTo (i, rs)) (t, rcl) =
-     (case (RBT.lookup t i) of
+     (case (RBT_Impl.lookup t i) of
        None \<Rightarrow> raise (''MiniSatChecked.doProofStep: No starting clause in thms array for Conflict step.'')
      | Some cli \<Rightarrow> (do
                       result \<leftarrow> foldM (tres_thm t) rs cli;
-                      return ((RBT.insert saveTo result t), rcl)
+                      return ((RBT_Impl.insert saveTo result t), rcl)
                     done))"
-| "tdoProofStep (Delete cid) (t, rcl) = return ((RBT.delete cid t), rcl)"
-| "tdoProofStep (Root cid clause) (t, rcl) = return (RBT.insert cid (sort clause) t, (remdups(sort clause)) # rcl)"
+| "tdoProofStep (Delete cid) (t, rcl) = return ((RBT_Impl.delete cid t), rcl)"
+| "tdoProofStep (Root cid clause) (t, rcl) = return (RBT_Impl.insert cid (sort clause) t, (remdups(sort clause)) # rcl)"
 | "tdoProofStep (Xstep cid1 cid2) (t, rcl) = raise ''MiniSatChecked.doProofStep: Xstep constructor found.''"
 | "tdoProofStep (ProofDone b) (t, rcl) = raise ''MiniSatChecked.doProofStep: ProofDone constructor found.''"
 
@@ -660,8 +660,8 @@
 where
   "tchecker n p i =
   (do 
-     rcs \<leftarrow> foldM (tdoProofStep) p (RBT.Empty, []);
-     (if (RBT.lookup (fst rcs) i) = Some [] then return (snd rcs) 
+     rcs \<leftarrow> foldM (tdoProofStep) p (RBT_Impl.Empty, []);
+     (if (RBT_Impl.lookup (fst rcs) i) = Some [] then return (snd rcs) 
                 else raise(''No empty clause''))
    done)"
 
diff -r 7bfbb247a5df -r b43b22f63665 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Wed Apr 14 22:18:10 2010 +0200
+++ b/src/HOL/IsaMakefile	Thu Apr 15 12:27:14 2010 +0200
@@ -1,3 +1,4 @@
+
 #
 # IsaMakefile for HOL
 #
@@ -406,14 +407,15 @@
   Library/Library/ROOT.ML Library/Library/document/root.tex		\
   Library/Library/document/root.bib					\
   Library/Transitive_Closure_Table.thy Library/While_Combinator.thy	\
-  Library/Product_ord.thy Library/Char_nat.thy Library/Table.thy	\
+  Library/Product_ord.thy Library/Char_nat.thy				\
   Library/Sublist_Order.thy Library/List_lexord.thy			\
   Library/AssocList.thy Library/Formal_Power_Series.thy			\
   Library/Binomial.thy Library/Eval_Witness.thy Library/Code_Char.thy	\
   Library/Code_Char_chr.thy Library/Code_Integer.thy			\
   Library/Mapping.thy Library/Numeral_Type.thy Library/Reflection.thy	\
   Library/Boolean_Algebra.thy Library/Countable.thy			\
-  Library/Diagonalize.thy Library/RBT.thy Library/Univ_Poly.thy		\
+  Library/Diagonalize.thy Library/RBT.thy Library/RBT_Impl.thy		\
+  Library/Univ_Poly.thy							\
   Library/Poly_Deriv.thy Library/Polynomial.thy Library/Preorder.thy	\
   Library/Product_plus.thy Library/Product_Vector.thy 			\
   Library/Enum.thy Library/Float.thy Library/Quotient_List.thy		\
diff -r 7bfbb247a5df -r b43b22f63665 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Wed Apr 14 22:18:10 2010 +0200
+++ b/src/HOL/Library/Library.thy	Thu Apr 15 12:27:14 2010 +0200
@@ -57,7 +57,6 @@
   SML_Quickcheck
   State_Monad
   Sum_Of_Squares
-  Table
   Transitive_Closure_Table
   Univ_Poly
   While_Combinator
diff -r 7bfbb247a5df -r b43b22f63665 src/HOL/Library/RBT.thy
--- a/src/HOL/Library/RBT.thy	Wed Apr 14 22:18:10 2010 +0200
+++ b/src/HOL/Library/RBT.thy	Thu Apr 15 12:27:14 2010 +0200
@@ -1,1100 +1,253 @@
-(*  Title:      RBT.thy
-    Author:     Markus Reiter, TU Muenchen
-    Author:     Alexander Krauss, TU Muenchen
-*)
+(* Author: Florian Haftmann, TU Muenchen *)
 
-header {* Red-Black Trees *}
+header {* Abstract type of Red-Black Trees *}
 
 (*<*)
 theory RBT
-imports Main
+imports Main RBT_Impl Mapping
 begin
 
-subsection {* Datatype of RB trees *}
-
-datatype color = R | B
-datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"
-
-lemma rbt_cases:
-  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
-
-subsection {* Tree properties *}
-
-subsubsection {* Content of a tree *}
-
-primrec entries :: "('a, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
-where 
-  "entries Empty = []"
-| "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"
-
-abbreviation (input) entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
-where
-  "entry_in_tree k v t \<equiv> (k, v) \<in> set (entries t)"
-
-definition keys :: "('a, 'b) rbt \<Rightarrow> 'a list" where
-  "keys t = map fst (entries t)"
-
-lemma keys_simps [simp, code]:
-  "keys Empty = []"
-  "keys (Branch c l k v r) = keys l @ k # keys r"
-  by (simp_all add: keys_def)
-
-lemma entry_in_tree_keys:
-  assumes "(k, v) \<in> set (entries t)"
-  shows "k \<in> set (keys t)"
-proof -
-  from assms have "fst (k, v) \<in> fst ` set (entries t)" by (rule imageI)
-  then show ?thesis by (simp add: keys_def)
-qed
-
-lemma keys_entries:
-  "k \<in> set (keys t) \<longleftrightarrow> (\<exists>v. (k, v) \<in> set (entries t))"
-  by (auto intro: entry_in_tree_keys) (auto simp add: keys_def)
-
-
-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)"
-
-abbreviation tree_less_symbol (infix "|\<guillemotleft>" 50)
-where "t |\<guillemotleft> x \<equiv> tree_less x t"
-
-definition tree_greater :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
-where
-  tree_greater_prop: "tree_greater k t = (\<forall>x\<in>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 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 tree_ord_props = tree_less_prop tree_greater_prop
-
-lemmas tree_greater_nit = tree_greater_prop entry_in_tree_keys
-lemmas tree_less_nit = tree_less_prop entry_in_tree_keys
-
-lemma tree_less_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"
-where
-  "sorted Empty = True"
-| "sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> sorted l \<and> sorted r)"
-
-lemma sorted_entries:
-  "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
-by (induct t) 
-  (force simp: sorted_append sorted_Cons tree_ord_props 
-      dest!: entry_in_tree_keys)+
-
-lemma distinct_entries:
-  "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
-by (induct t) 
-  (force simp: sorted_append sorted_Cons tree_ord_props 
-      dest!: entry_in_tree_keys)+
-
-
-subsubsection {* Tree lookup *}
-
-primrec lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
-where
-  "lookup Empty k = None"
-| "lookup (Branch _ l x y r) k = (if k < x then lookup l k else if x < k then lookup r k else Some y)"
-
-lemma lookup_keys: "sorted t \<Longrightarrow> dom (lookup t) = set (keys t)"
-  by (induct t) (auto simp: dom_def tree_greater_prop tree_less_prop)
+subsection {* Type definition *}
 
-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))"
+typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}"
+  morphisms impl_of RBT
 proof -
-  assume "sorted (Branch c t1 k v t2)"
-  moreover from this have "sorted t1" "sorted t2" by simp_all
-  ultimately show ?thesis by (simp add: lookup_keys)
-qed
-
-lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
-proof (induct t)
-  case Empty then show ?case by simp
-next
-  case (Branch color t1 a b t2)
-  let ?A = "Set.insert a (dom (lookup t1) \<union> dom (lookup t2))"
-  have "dom (lookup (Branch color t1 a b t2)) \<subseteq> ?A" by (auto split: split_if_asm)
-  moreover from Branch have "finite (insert a (dom (lookup t1) \<union> dom (lookup t2)))" by simp
-  ultimately show ?case by (rule finite_subset)
-qed 
-
-lemma lookup_tree_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> lookup t k = None" 
-by (induct t) auto
-
-lemma lookup_tree_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> lookup t k = None"
-by (induct t) auto
-
-lemma lookup_Empty: "lookup Empty = empty"
-by (rule ext) simp
-
-lemma map_of_entries:
-  "sorted t \<Longrightarrow> map_of (entries t) = lookup t"
-proof (induct t)
-  case Empty thus ?case by (simp add: lookup_Empty)
-next
-  case (Branch c t1 k v t2)
-  have "lookup (Branch c t1 k v t2) = lookup t2 ++ [k\<mapsto>v] ++ lookup t1"
-  proof (rule ext)
-    fix x
-    from Branch have SORTED: "sorted (Branch c t1 k v t2)" by simp
-    let ?thesis = "lookup (Branch c t1 k v t2) x = (lookup t2 ++ [k \<mapsto> v] ++ lookup t1) x"
-
-    have DOM_T1: "!!k'. k'\<in>dom (lookup t1) \<Longrightarrow> k>k'"
-    proof -
-      fix k'
-      from SORTED have "t1 |\<guillemotleft> k" by simp
-      with tree_less_prop have "\<forall>k'\<in>set (keys t1). k>k'" by auto
-      moreover assume "k'\<in>dom (lookup t1)"
-      ultimately show "k>k'" using lookup_keys SORTED by auto
-    qed
-    
-    have DOM_T2: "!!k'. k'\<in>dom (lookup t2) \<Longrightarrow> k<k'"
-    proof -
-      fix k'
-      from SORTED have "k \<guillemotleft>| t2" by simp
-      with tree_greater_prop have "\<forall>k'\<in>set (keys t2). k<k'" by auto
-      moreover assume "k'\<in>dom (lookup t2)"
-      ultimately show "k<k'" using lookup_keys SORTED by auto
-    qed
-    
-    {
-      assume C: "x<k"
-      hence "lookup (Branch c t1 k v t2) x = lookup t1 x" by simp
-      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
-      moreover have "x\<notin>dom (lookup t2)" proof
-        assume "x\<in>dom (lookup t2)"
-        with DOM_T2 have "k<x" by blast
-        with C show False by simp
-      qed
-      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
-    } moreover {
-      assume [simp]: "x=k"
-      hence "lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
-      moreover have "x\<notin>dom (lookup t1)" proof
-        assume "x\<in>dom (lookup t1)"
-        with DOM_T1 have "k>x" by blast
-        thus False by simp
-      qed
-      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
-    } moreover {
-      assume C: "x>k"
-      hence "lookup (Branch c t1 k v t2) x = lookup t2 x" by (simp add: less_not_sym[of k x])
-      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
-      moreover have "x\<notin>dom (lookup t1)" proof
-        assume "x\<in>dom (lookup t1)"
-        with DOM_T1 have "k>x" by simp
-        with C show False by simp
-      qed
-      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
-    } ultimately show ?thesis using less_linear by blast
-  qed
-  also from Branch have "lookup t2 ++ [k \<mapsto> v] ++ lookup t1 = map_of (entries (Branch c t1 k v t2))" by simp
-  finally show ?case by simp
-qed
-
-lemma lookup_in_tree: "sorted t \<Longrightarrow> 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" 
-  shows "set (entries t1) = set (entries t2) \<longleftrightarrow> entries t1 = entries t2"
-proof -
-  from 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)
-qed
-
-lemma entries_eqI:
-  assumes sorted: "sorted t1" "sorted t2" 
-  assumes lookup: "lookup t1 = lookup t2"
-  shows "entries t1 = entries t2"
-proof -
-  from sorted 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)"
-    by (simp add: map_of_inject_set distinct_entries)
-  with sorted show ?thesis by (simp add: set_entries_inject)
+  have "RBT_Impl.Empty \<in> ?rbt" by simp
+  then show ?thesis ..
 qed
 
-lemma entries_lookup:
-  assumes "sorted t1" "sorted t2" 
-  shows "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = 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"
-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])
-qed
-
-
-subsubsection {* Red-black properties *}
-
-primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"
-where
-  "color_of Empty = B"
-| "color_of (Branch c _ _ _ _) = c"
+lemma is_rbt_impl_of [simp, intro]:
+  "is_rbt (impl_of t)"
+  using impl_of [of t] by simp
 
-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 (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)"
+lemma rbt_eq:
+  "t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"
+  by (simp add: impl_of_inject)
 
-primrec inv1l :: "('a, 'b) rbt \<Rightarrow> bool" -- {* Weaker version *}
-where
-  "inv1l Empty = True"
-| "inv1l (Branch c l k v r) = (inv1 l \<and> inv1 r)"
-lemma [simp]: "inv1 t \<Longrightarrow> inv1l t" by (cases t) simp+
-
-primrec inv2 :: "('a, 'b) rbt \<Rightarrow> bool"
-where
-  "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"
-
-lemma is_rbt_sorted [simp]:
-  "is_rbt t \<Longrightarrow> sorted t" by (simp add: is_rbt_def)
-
-theorem Empty_is_rbt [simp]:
-  "is_rbt Empty" by (simp add: is_rbt_def)
+lemma [code abstype]:
+  "RBT (impl_of t) = t"
+  by (simp add: impl_of_inverse)
 
 
-subsection {* Insertion *}
-
-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 (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_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" "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_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_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
+subsection {* Primitive operations *}
 
-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)"
-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)
-  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)" 
-    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> 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> 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> 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> 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> 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> 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+
-
-lemma entries_balance [simp]:
-  "entries (balance l k v r) = entries l @ (k, v) # entries r"
-  by (induct l k v r rule: balance.induct) auto
-
-lemma keys_balance [simp]: 
-  "keys (balance l k v r) = keys l @ k # keys r"
-  by (simp add: keys_def)
+definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" where
+  [code]: "lookup t = RBT_Impl.lookup (impl_of t)"
 
-lemma balance_in_tree:  
-  "entry_in_tree k x (balance l v y r) \<longleftrightarrow> entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r"
-  by (auto simp add: keys_def)
-
-lemma lookup_balance[simp]: 
-fixes k :: "'a::linorder"
-assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-shows "lookup (balance l k v r) x = lookup (Branch B l k v r) x"
-by (rule lookup_from_in_tree) (auto simp:assms balance_in_tree balance_sorted)
-
-primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "paint c Empty = Empty"
-| "paint c (Branch _ l k v r) = Branch c l k v r"
-
-lemma paint_inv1l[simp]: "inv1l t \<Longrightarrow> inv1l (paint c t)" by (cases t) auto
-lemma paint_inv1[simp]: "inv1l t \<Longrightarrow> inv1 (paint B t)" by (cases t) auto
-lemma paint_inv2[simp]: "inv2 t \<Longrightarrow> inv2 (paint c t)" by (cases t) auto
-lemma paint_color_of[simp]: "color_of (paint B t) = B" by (cases t) auto
-lemma paint_sorted[simp]: "sorted t \<Longrightarrow> sorted (paint c t)" by (cases t) auto
-lemma paint_in_tree[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
-lemma paint_lookup[simp]: "lookup (paint c t) = lookup t" by (rule ext) (cases t, auto)
-lemma paint_tree_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
-lemma paint_tree_less[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
-
-fun
-  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 = 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)"
-
-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)"
-  using assms
-  by (induct f k x t rule: ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
+definition empty :: "('a\<Colon>linorder, 'b) rbt" where
+  "empty = RBT RBT_Impl.Empty"
 
-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 keys_ins: "set (keys (ins f k v t)) = { k } \<union> set (keys t)"
-  by (induct f k v t rule: ins.induct) auto
-
-lemma lookup_ins: 
-  fixes k :: "'a::linorder"
-  assumes "sorted t"
-  shows "lookup (ins f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v 
-                                                       | Some w \<Rightarrow> f k w v)) x"
-using assms by (induct f k v t rule: ins.induct) auto
-
-definition
-  insert_with_key :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "insert_with_key f k v t = paint B (ins f k v t)"
-
-lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insert_with_key f k x t)"
-  by (auto simp: insert_with_key_def)
-
-theorem insertwk_is_rbt: 
-  assumes inv: "is_rbt t" 
-  shows "is_rbt (insert_with_key f k x t)"
-using assms
-unfolding 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 
-                                                       | Some w \<Rightarrow> f k w v)) x"
-unfolding insert_with_key_def using assms
-by (simp add:lookup_ins)
-
-definition
-  insertw_def: "insert_with f = insert_with_key (\<lambda>_. f)"
-
-lemma insertw_sorted: "sorted t \<Longrightarrow> sorted (insert_with f k v t)" by (simp add: insertwk_sorted insertw_def)
-theorem insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insert_with f k v t)" by (simp add: insertwk_is_rbt insertw_def)
-
-lemma lookup_insertw:
-  assumes "is_rbt t"
-  shows "lookup (insert_with f k v t) = (lookup t)(k \<mapsto> (if k:dom (lookup t) then f (the (lookup t k)) v else v))"
-using assms
-unfolding insertw_def
-by (rule_tac ext) (cases "lookup t k", auto simp:lookup_insertwk dom_def)
+lemma impl_of_empty [code abstract]:
+  "impl_of empty = RBT_Impl.Empty"
+  by (simp add: empty_def RBT_inverse)
 
 definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "insert = insert_with_key (\<lambda>_ _ nv. nv)"
+  "insert k v t = RBT (RBT_Impl.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)"
+  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))"
 
-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 impl_of_delete [code abstract]:
+  "impl_of (delete k t) = RBT_Impl.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
+  [code]: "entries t = RBT_Impl.entries (impl_of t)"
+
+definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" where
+  [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)"
 
-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: lookup_insertwk split:option.split)
+lemma impl_of_bulkload [code abstract]:
+  "impl_of (bulkload xs) = RBT_Impl.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))"
+
+lemma impl_of_map_entry [code abstract]:
+  "impl_of (map_entry k f t) = RBT_Impl.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
+  "map f t = RBT (RBT_Impl.map f (impl_of t))"
+
+lemma impl_of_map [code abstract]:
+  "impl_of (map f t) = RBT_Impl.map f (impl_of t)"
+  by (simp add: map_def RBT_inverse)
+
+definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
+  [code]: "fold f t = RBT_Impl.fold f (impl_of t)"
+
+
+subsection {* Derived operations *}
+
+definition is_empty :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where
+  [code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
 
 
-subsection {* Deletion *}
-
-lemma bheight_paintR'[simp]: "color_of t = B \<Longrightarrow> bheight (paint R t) = bheight t - 1"
-by (cases t rule: rbt_cases) auto
-
-fun
-  balance_left :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "balance_left (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
-  "balance_left bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
-  "balance_left bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
-  "balance_left t k x s = Empty"
+subsection {* Abstract lookup properties *}
 
-lemma balance_left_inv2_with_inv1:
-  assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"
-  shows "bheight (balance_left lt k v rt) = bheight lt + 1"
-  and   "inv2 (balance_left lt k v rt)"
-using assms 
-by (induct lt k v rt rule: balance_left.induct) (auto simp: balance_inv2 balance_bheight)
-
-lemma balance_left_inv2_app: 
-  assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"
-  shows "inv2 (balance_left lt k v rt)" 
-        "bheight (balance_left lt k v rt) = bheight rt"
-using assms 
-by (induct lt k v rt rule: balance_left.induct) (auto simp add: balance_inv2 balance_bheight)+ 
-
-lemma balance_left_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balance_left a k x b)"
-  by (induct a k x b rule: balance_left.induct) (simp add: balance_inv1)+
-
-lemma balance_left_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balance_left lt k x rt)"
-by (induct lt k x rt rule: balance_left.induct) (auto simp: balance_inv1)
+lemma lookup_RBT:
+  "is_rbt t \<Longrightarrow> lookup (RBT t) = RBT_Impl.lookup t"
+  by (simp add: lookup_def RBT_inverse)
 
-lemma balance_left_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_left l k v r)"
-apply (induct l k v r rule: balance_left.induct)
-apply (auto simp: balance_sorted)
-apply (unfold tree_greater_prop tree_less_prop)
-by force+
-
-lemma balance_left_tree_greater: 
-  fixes k :: "'a::order"
-  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"
-  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
+lemma lookup_impl_of:
+  "RBT_Impl.lookup (impl_of t) = lookup t"
+  by (simp add: lookup_def)
 
-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)"
-using assms 
-by (induct l k v r rule: balance_left.induct) (auto simp: balance_in_tree)
-
-fun
-  balance_right :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "balance_right a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
-  "balance_right (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
-  "balance_right (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
-  "balance_right t k x s = Empty"
-
-lemma balance_right_inv2_with_inv1:
-  assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt + 1" "inv1 lt"
-  shows "inv2 (balance_right lt k v rt) \<and> bheight (balance_right lt k v rt) = bheight lt"
-using assms
-by (induct lt k v rt rule: balance_right.induct) (auto simp: balance_inv2 balance_bheight)
+lemma entries_impl_of:
+  "RBT_Impl.entries (impl_of t) = entries t"
+  by (simp add: entries_def)
 
-lemma balance_right_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balance_right a k x b)"
-by (induct a k x b rule: balance_right.induct) (simp add: balance_inv1)+
-
-lemma balance_right_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balance_right lt k x rt)"
-by (induct lt k x rt rule: balance_right.induct) (auto simp: balance_inv1)
-
-lemma balance_right_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_right l k v r)"
-apply (induct l k v r rule: balance_right.induct)
-apply (auto simp:balance_sorted)
-apply (unfold tree_less_prop tree_greater_prop)
-by force+
+lemma keys_impl_of:
+  "RBT_Impl.keys (impl_of t) = keys t"
+  by (simp add: keys_def)
 
-lemma balance_right_tree_greater: 
-  fixes k :: "'a::order"
-  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"
-  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
-
-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)"
-using assms by (induct l k v r rule: balance_right.induct) (auto simp: balance_in_tree)
+lemma lookup_empty [simp]:
+  "lookup empty = Map.empty"
+  by (simp add: empty_def lookup_RBT expand_fun_eq)
 
-fun
-  combine :: "('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
-where
-  "combine Empty x = x" 
-| "combine x Empty = x" 
-| "combine (Branch R a k x b) (Branch R c s y d) = (case (combine b c) of
-                                      Branch R b2 t z c2 \<Rightarrow> (Branch R (Branch R a k x b2) t z (Branch R c2 s y d)) |
-                                      bc \<Rightarrow> Branch R a k x (Branch R bc s y d))" 
-| "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))" 
-| "combine a (Branch R b k x c) = Branch R (combine a b) k x c" 
-| "combine (Branch R a k x b) c = Branch R a k x (combine b c)" 
-
-lemma combine_inv2:
-  assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
-  shows "bheight (combine lt rt) = bheight lt" "inv2 (combine lt rt)"
-using assms 
-by (induct lt rt rule: combine.induct) 
-   (auto simp: balance_left_inv2_app split: rbt.splits color.splits)
+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)
 
-lemma combine_inv1: 
-  assumes "inv1 lt" "inv1 rt"
-  shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (combine lt rt)"
-         "inv1l (combine lt rt)"
-using assms 
-by (induct lt rt rule: combine.induct)
-   (auto simp: balance_left_inv1 split: rbt.splits color.splits)
+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)
 
-lemma combine_tree_greater[simp]: 
-  fixes k :: "'a::linorder"
-  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)
-
-lemma combine_tree_less[simp]: 
-  fixes k :: "'a::linorder"
-  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)
+lemma map_of_entries [simp]:
+  "map_of (entries t) = lookup t"
+  by (simp add: entries_def map_of_entries lookup_impl_of)
 
-lemma combine_sorted: 
-  fixes k :: "'a::linorder"
-  assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-  shows "sorted (combine l r)"
-using assms proof (induct l r rule: combine.induct)
-  case (3 a x v b c y w d)
-  hence ineqs: "a |\<guillemotleft> x" "x \<guillemotleft>| b" "b |\<guillemotleft> k" "k \<guillemotleft>| c" "c |\<guillemotleft> y" "y \<guillemotleft>| d"
-    by auto
-  with 3
-  show ?case
-    by (cases "combine b c" rule: rbt_cases)
-      (auto, (metis combine_tree_greater combine_tree_less ineqs ineqs tree_less_simps(2) tree_greater_simps(2) tree_greater_trans tree_less_trans)+)
-next
-  case (4 a x v b c y w d)
-  hence "x < k \<and> tree_greater k c" by simp
-  hence "tree_greater x c" by (blast dest: tree_greater_trans)
-  with 4 have 2: "tree_greater x (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)
-  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)
-  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 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)
-  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)
-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)
-qed simp+
+lemma entries_lookup:
+  "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
+  by (simp add: entries_def lookup_def entries_lookup)
+
+lemma lookup_bulkload [simp]:
+  "lookup (bulkload xs) = map_of xs"
+  by (simp add: bulkload_def lookup_RBT RBT_Impl.lookup_bulkload)
 
-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)"
-using assms 
-proof (induct l r rule: combine.induct)
-  case (4 _ _ _ b c)
-  hence a: "bheight (combine b c) = bheight b" by (simp add: combine_inv2)
-  from 4 have b: "inv1l (combine b c)" by (simp add: combine_inv1)
+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 lookup_map_entry lookup_impl_of)
 
-  show ?case
-  proof (cases "combine b c" rule: rbt_cases)
-    case Empty
-    with 4 a show ?thesis by (auto simp: balance_left_in_tree)
-  next
-    case (Red lta ka va rta)
-    with 4 show ?thesis by auto
-  next
-    case (Black lta ka va rta)
-    with a b 4  show ?thesis by (auto simp: balance_left_in_tree)
-  qed 
-qed (auto split: rbt.splits color.splits)
+lemma lookup_map [simp]:
+  "lookup (map f t) k = Option.map (f k) (lookup t k)"
+  by (simp add: map_def lookup_RBT lookup_map lookup_impl_of)
 
-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"
-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)"
+lemma fold_fold:
+  "fold f t = (\<lambda>s. foldl (\<lambda>s (k, v). f k v s) s (entries t))"
+  by (simp add: fold_def expand_fun_eq RBT_Impl.fold_def entries_impl_of)
 
-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))"
-  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))"
-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)
-case (2 y c _ y')
-  have "y = y' \<or> y < y' \<or> y > y'" by auto
-  thus ?case proof (elim disjE)
-    assume "y = y'"
-    with 2 show ?thesis by (cases c) (simp add: combine_inv2 combine_inv1)+
-  next
-    assume "y < y'"
-    with 2 show ?thesis by (cases c) auto
-  next
-    assume "y' < y"
-    with 2 show ?thesis by (cases c) auto
-  qed
-next
-  case (3 y lt z v rta y' ss bb) 
-  thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balance_left_inv2_with_inv1 balance_left_inv1 balance_left_inv1l)+
-next
-  case (5 y a y' ss lt z v rta)
-  thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balance_right_inv2_with_inv1 balance_right_inv1 balance_right_inv1l)+
-next
-  case ("6_1" y a y' ss) thus ?case by (cases "color_of a = B \<and> color_of Empty = B") simp+
-qed auto
+lemma is_empty_empty [simp]:
+  "is_empty t \<longleftrightarrow> t = empty"
+  by (simp add: rbt_eq is_empty_def impl_of_empty split: rbt.split)
 
-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)
-
-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 "\<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)
-  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)
-next
-  case ("4_2" x vaa vbb vdd vc yy ss bb)
-  hence "tree_less yy (Branch R vaa vbb vdd vc)" by simp
-  hence "tree_less yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_less)
-  with "4_2" show ?case by simp
-next
-  case (5 x aa yy ss lta zz v rta) 
-  hence "tree_greater yy (Branch B lta zz v rta)" by simp
-  hence "tree_greater yy (del x (Branch B lta zz v rta))" by (rule del_tree_greater)
-  with 5 show ?case by (simp add: balance_right_sorted)
-next
-  case ("6_2" x aa yy ss vaa vbb vdd vc)
-  hence "tree_greater yy (Branch R vaa vbb vdd vc)" by simp
-  hence "tree_greater yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_greater)
-  with "6_2" show ?case by simp
-qed (auto simp: combine_sorted)
+lemma RBT_lookup_empty [simp]: (*FIXME*)
+  "RBT_Impl.lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
+  by (cases t) (auto simp add: expand_fun_eq)
 
-lemma "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_left x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
-  and "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_right x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
-  and del_in_tree: "\<lbrakk>sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
-proof (induct x lt kt y rt and x lt kt y rt and x t rule: del_from_left_del_from_right_del.induct)
-  case (2 xx c aa yy ss bb)
-  have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
-  from this 2 show ?case proof (elim disjE)
-    assume "xx = yy"
-    with 2 show ?thesis proof (cases "xx = k")
-      case True
-      from 2 `xx = yy` `xx = k` have "sorted (Branch c aa yy ss bb) \<and> k = yy" by simp
-      hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: tree_less_nit tree_greater_prop)
-      with `xx = yy` 2 `xx = k` show ?thesis by (simp add: combine_in_tree)
-    qed (simp add: combine_in_tree)
-  qed simp+
-next    
-  case (3 xx lta zz vv rta yy ss bb)
-  def mt[simp]: mt == "Branch B lta zz vv rta"
-  from 3 have "inv2 mt \<and> inv1 mt" by simp
-  hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
-  with 3 have 4: "entry_in_tree k v (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)
-  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" `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)
-    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)
-  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)
-  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)
-  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)
-  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)
-  qed auto
-qed simp
+lemma lookup_empty_empty [simp]:
+  "lookup t = Map.empty \<longleftrightarrow> t = empty"
+  by (cases t) (simp add: empty_def lookup_def RBT_inject RBT_inverse)
+
+lemma sorted_keys [iff]:
+  "sorted (keys t)"
+  by (simp add: keys_def RBT_Impl.keys_def sorted_entries)
+
+lemma distinct_keys [iff]:
+  "distinct (keys t)"
+  by (simp add: keys_def RBT_Impl.keys_def distinct_entries)
 
 
-definition delete where
-  delete_def: "delete k t = paint B (del k t)"
+subsection {* Implementation of mappings *}
 
-theorem delete_is_rbt [simp]: assumes "is_rbt t" shows "is_rbt (delete k t)"
-proof -
-  from assms have "inv2 t" and "inv1 t" unfolding is_rbt_def by auto 
-  hence "inv2 (del k t) \<and> (color_of t = R \<and> bheight (del k t) = bheight t \<and> inv1 (del k t) \<or> color_of t = B \<and> bheight (del k t) = bheight t - 1 \<and> inv1l (del k t))" by (rule del_inv1_inv2)
-  hence "inv2 (del k t) \<and> inv1l (del k t)" by (cases "color_of t") auto
-  with assms show ?thesis
-    unfolding is_rbt_def delete_def
-    by (auto intro: paint_sorted del_sorted)
-qed
-
-lemma delete_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)
+definition Mapping :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) mapping" where
+  "Mapping t = Mapping.Mapping (lookup t)"
 
-lemma lookup_delete:
-  assumes is_rbt: "is_rbt t"
-  shows "lookup (delete k t) = (lookup t)|`(-{k})"
-proof
-  fix x
-  show "lookup (delete k t) x = (lookup t |` (-{k})) x" 
-  proof (cases "x = k")
-    assume "x = k" 
-    with is_rbt show ?thesis
-      by (cases "lookup (delete k t) k") (auto simp: lookup_in_tree 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)
-  qed
-qed
+code_datatype Mapping
 
-
-subsection {* Union *}
+lemma lookup_Mapping [simp, code]:
+  "Mapping.lookup (Mapping t) = lookup t"
+  by (simp add: Mapping_def)
 
-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"
-where
-  "union_with_key f t Empty = t"
-| "union_with_key f t (Branch c lt k v rt) = union_with_key f (union_with_key f (insert_with_key f k v t) lt) rt"
-
-lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (union_with_key f lt rt)" 
-  by (induct rt arbitrary: lt) (auto simp: insertwk_sorted)
-theorem unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (union_with_key f lt rt)" 
-  by (induct rt arbitrary: lt) (simp add: insertwk_is_rbt)+
+lemma empty_Mapping [code]:
+  "Mapping.empty = Mapping empty"
+  by (rule mapping_eqI) simp
 
-definition
-  union_with where
-  "union_with f = union_with_key (\<lambda>_. f)"
-
-theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union_with f lt rt)" unfolding union_with_def by simp
-
-definition union where
-  "union = union_with_key (%_ _ rv. rv)"
-
-theorem union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union lt rt)" unfolding union_def by simp
+lemma is_empty_Mapping [code]:
+  "Mapping.is_empty (Mapping t) \<longleftrightarrow> is_empty t"
+  by (simp add: rbt_eq Mapping.is_empty_empty Mapping_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 insert_Mapping [code]:
+  "Mapping.update k v (Mapping t) = Mapping (insert k v t)"
+  by (rule mapping_eqI) simp
 
-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 (Branch c l k v r s)
-  then have "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
-
-  have meq: "lookup s(k \<mapsto> v) ++ lookup l ++ lookup r =
-    lookup s ++
-    (\<lambda>a. if a < k then lookup l a
-    else if k < a then lookup r a else Some v)" (is "?m1 = ?m2")
-  proof (rule ext)
-    fix a
+lemma delete_Mapping [code]:
+  "Mapping.delete k (Mapping xs) = Mapping (delete k xs)"
+  by (rule mapping_eqI) simp
 
-   have "k < a \<or> k = a \<or> k > a" by auto
-    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 `k < a` show ?thesis
-        by (auto simp: map_add_def split: option.splits)
-    next
-      assume "k = a"
-      with `l |\<guillemotleft> k` `k \<guillemotleft>| r` 
-      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)
-      with `a < k` show ?thesis
-        by (auto simp: map_add_def split: option.splits)
-    qed
-  qed
+lemma keys_Mapping [code]:
+  "Mapping.keys (Mapping t) = set (keys t)"
+  by (simp add: keys_def Mapping_def Mapping.keys_def lookup_def lookup_keys)
 
-  from Branch have is_rbt: "is_rbt (RBT.union (RBT.insert k v s) l)"
-    by (auto intro: union_is_rbt insert_is_rbt)
-  with Branch have IHs:
-    "lookup (union (union (insert k v s) l) r) = lookup (union (insert k v s) l) ++ lookup r"
-    "lookup (union (insert k v s) l) = lookup (insert k v s) ++ lookup l"
-    by auto
-  
-  with meq show ?case
-    by (auto simp: lookup_insert[OF Branch(3)])
+lemma ordered_keys_Mapping [code]:
+  "Mapping.ordered_keys (Mapping t) = keys t"
+  by (rule sorted_distinct_set_unique) (simp_all add: ordered_keys_def keys_Mapping)
 
-qed
-
-
-subsection {* Modifying existing entries *}
-
-primrec
-  map_entry :: "'a\<Colon>linorder \<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))
-    else Branch c lt x (f v) rt)"
+lemma Mapping_size_card_keys: (*FIXME*)
+  "Mapping.size m = card (Mapping.keys m)"
+  by (simp add: Mapping.size_def Mapping.keys_def)
 
-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)
-
-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 lookup_map_entry:
-  "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
-  by (induct t) (auto split: option.splits simp add: expand_fun_eq)
-
+lemma size_Mapping [code]:
+  "Mapping.size (Mapping t) = length (keys t)"
+  by (simp add: Mapping_size_card_keys keys_Mapping distinct_card)
 
-subsection {* Mapping all entries *}
-
-primrec
-  map :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'c) rbt"
-where
-  "map f Empty = Empty"
-| "map f (Branch c lt k v rt) = Branch c (map f lt) k (f k v) (map f rt)"
+lemma tabulate_Mapping [code]:
+  "Mapping.tabulate ks f = Mapping (bulkload (List.map (\<lambda>k. (k, f k)) ks))"
+  by (rule mapping_eqI) (simp add: map_of_map_restrict)
 
-lemma map_entries [simp]: "entries (map f t) = List.map (\<lambda>(k, v). (k, f k v)) (entries t)"
-  by (induct t) auto
-lemma map_keys [simp]: "keys (map f t) = keys t" by (simp add: keys_def split_def)
-lemma map_tree_greater: "tree_greater k (map f t) = tree_greater k t" by (induct t) simp+
-lemma map_tree_less: "tree_less k (map f t) = tree_less k t" by (induct t) simp+
-lemma map_sorted: "sorted (map f t) = sorted t"  by (induct t) (simp add: map_tree_less map_tree_greater)+
-lemma map_color_of: "color_of (map f t) = color_of t" by (induct t) simp+
-lemma map_inv1: "inv1 (map f t) = inv1 t" by (induct t) (simp add: map_color_of)+
-lemma map_inv2: "inv2 (map f t) = inv2 t" "bheight (map f t) = bheight t" by (induct t) simp+
-theorem map_is_rbt [simp]: "is_rbt (map f t) = is_rbt t" 
-unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_sorted map_color_of)
-
-theorem lookup_map: "lookup (map f t) x = Option.map (f x) (lookup t x)"
-  by (induct t) auto
-
-
-subsection {* Folding over entries *}
-
-definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
-  "fold f t s = foldl (\<lambda>s (k, v). f k v s) s (entries t)"
+lemma bulkload_Mapping [code]:
+  "Mapping.bulkload vs = Mapping (bulkload (List.map (\<lambda>n. (n, vs ! n)) [0..<length vs]))"
+  by (rule mapping_eqI) (simp add: map_of_map_restrict expand_fun_eq)
 
-lemma fold_simps [simp, code]:
-  "fold f Empty = id"
-  "fold f (Branch c lt k v rt) = fold f rt \<circ> f k v \<circ> fold f lt"
-  by (simp_all add: fold_def expand_fun_eq)
-
-
-subsection {* Bulkloading a tree *}
-
-definition bulkload :: "('a \<times> 'b) list \<Rightarrow> ('a\<Colon>linorder, 'b) rbt" where
-  "bulkload xs = foldr (\<lambda>(k, v). RBT.insert k v) xs RBT.Empty"
-
-lemma bulkload_is_rbt [simp, intro]:
-  "is_rbt (bulkload xs)"
-  unfolding bulkload_def by (induct xs) auto
+lemma [code, code del]: "HOL.eq (x :: (_, _) mapping) y \<longleftrightarrow> x = y" by (fact eq_equals) (*FIXME*)
 
-lemma lookup_bulkload:
-  "RBT.lookup (bulkload xs) = map_of xs"
-proof -
-  obtain ys where "ys = rev xs" by simp
-  have "\<And>t. is_rbt t \<Longrightarrow>
-    RBT.lookup (foldl (\<lambda>t (k, v). RBT.insert k v t) t ys) = RBT.lookup t ++ map_of (rev ys)"
-      by (induct ys) (simp_all add: bulkload_def split_def RBT.lookup_insert)
-  from this Empty_is_rbt have
-    "RBT.lookup (foldl (\<lambda>t (k, v). RBT.insert k v t) RBT.Empty (rev xs)) = RBT.lookup RBT.Empty ++ map_of xs"
-     by (simp add: `ys = rev xs`)
-  then show ?thesis by (simp add: bulkload_def foldl_foldr lookup_Empty split_def)
-qed
+lemma eq_Mapping [code]:
+  "HOL.eq (Mapping t1) (Mapping t2) \<longleftrightarrow> entries t1 = entries t2"
+  by (simp add: eq Mapping_def entries_lookup)
 
-hide (open) const Empty insert delete entries bulkload lookup map_entry map fold union sorted
+hide (open) const impl_of lookup empty insert delete
+  entries keys bulkload map_entry map fold
 (*>*)
 
 text {* 
-  This theory defines purely functional red-black trees which can be
-  used as an efficient representation of finite maps.
+  This theory defines abstract red-black trees as an efficient
+  representation of finite maps, backed by the implementation
+  in @{theory RBT_Impl}.
 *}
 
-
 subsection {* Data type and invariant *}
 
 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 musorted belong to the @{text "linorder"} class.
+  The type @{typ "('k, 'v) RBT_Impl.rbt"} denotes red-black trees with
+  keys of type @{typ "'k"} and values of type @{typ "'v"}. To function
+  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 "is_rbt t"}.
-  This theory provides lemmas to prove that the invariant is
-  satisfied throughout the computation.
+  satisfies the invariant @{text "is_rbt t"}.  The abstract type @{typ
+  "('k, 'v) rbt"} always obeys this invariant, and for this reason you
+  should only use this in our application.  Going back to @{typ "('k,
+  'v) RBT_Impl.rbt"} may be necessary in proofs if not yet proven
+  properties about the operations must be established.
 
   The interpretation function @{const "RBT.lookup"} returns the partial
   map represented by a red-black tree:
@@ -1106,15 +259,12 @@
   $O(\log n)$.  
 *}
 
-
 subsection {* Operations *}
 
-print_antiquotations
-
 text {*
   Currently, the following operations are supported:
 
-  @{term_type [display] "RBT.Empty"}
+  @{term_type [display] "RBT.empty"}
   Returns the empty tree. $O(1)$
 
   @{term_type [display] "RBT.insert"}
@@ -1137,9 +287,6 @@
 
   @{term_type [display] "RBT.fold"}
   Folds over all entries in a tree. $O(n)$
-
-  @{term_type [display] "RBT.union"}
-  Forms the union of two trees, preferring entries from the first one.
 *}
 
 
@@ -1173,8 +320,8 @@
 
 text {*
   \noindent
-  \underline{@{text "lookup_Empty"}}
-  @{thm [display] lookup_Empty}
+  \underline{@{text "lookup_empty"}}
+  @{thm [display] lookup_empty}
   \vspace{1ex}
 
   \noindent
@@ -1196,11 +343,6 @@
   \underline{@{text "lookup_map"}}
   @{thm [display] lookup_map}
   \vspace{1ex}
-
-  \noindent
-  \underline{@{text "lookup_union"}}
-  @{thm [display] lookup_union}
-  \vspace{1ex}
 *}
 
 end
diff -r 7bfbb247a5df -r b43b22f63665 src/HOL/Library/RBT_Impl.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/RBT_Impl.thy	Thu Apr 15 12:27:14 2010 +0200
@@ -0,0 +1,1084 @@
+(*  Title:      RBT_Impl.thy
+    Author:     Markus Reiter, TU Muenchen
+    Author:     Alexander Krauss, TU Muenchen
+*)
+
+header {* Implementation of Red-Black Trees *}
+
+theory RBT_Impl
+imports Main
+begin
+
+text {*
+  For applications, you should use theory @{text RBT} which defines
+  an abstract type of red-black tree obeying the invariant.
+*}
+
+subsection {* Datatype of RB trees *}
+
+datatype color = R | B
+datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"
+
+lemma rbt_cases:
+  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
+
+subsection {* Tree properties *}
+
+subsubsection {* Content of a tree *}
+
+primrec entries :: "('a, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
+where 
+  "entries Empty = []"
+| "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"
+
+abbreviation (input) entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
+where
+  "entry_in_tree k v t \<equiv> (k, v) \<in> set (entries t)"
+
+definition keys :: "('a, 'b) rbt \<Rightarrow> 'a list" where
+  "keys t = map fst (entries t)"
+
+lemma keys_simps [simp, code]:
+  "keys Empty = []"
+  "keys (Branch c l k v r) = keys l @ k # keys r"
+  by (simp_all add: keys_def)
+
+lemma entry_in_tree_keys:
+  assumes "(k, v) \<in> set (entries t)"
+  shows "k \<in> set (keys t)"
+proof -
+  from assms have "fst (k, v) \<in> fst ` set (entries t)" by (rule imageI)
+  then show ?thesis by (simp add: keys_def)
+qed
+
+lemma keys_entries:
+  "k \<in> set (keys t) \<longleftrightarrow> (\<exists>v. (k, v) \<in> set (entries t))"
+  by (auto intro: entry_in_tree_keys) (auto simp add: keys_def)
+
+
+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)"
+
+abbreviation tree_less_symbol (infix "|\<guillemotleft>" 50)
+where "t |\<guillemotleft> x \<equiv> tree_less x t"
+
+definition tree_greater :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
+where
+  tree_greater_prop: "tree_greater k t = (\<forall>x\<in>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 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 tree_ord_props = tree_less_prop tree_greater_prop
+
+lemmas tree_greater_nit = tree_greater_prop entry_in_tree_keys
+lemmas tree_less_nit = tree_less_prop entry_in_tree_keys
+
+lemma tree_less_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"
+where
+  "sorted Empty = True"
+| "sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> sorted l \<and> sorted r)"
+
+lemma sorted_entries:
+  "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
+by (induct t) 
+  (force simp: sorted_append sorted_Cons tree_ord_props 
+      dest!: entry_in_tree_keys)+
+
+lemma distinct_entries:
+  "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
+by (induct t) 
+  (force simp: sorted_append sorted_Cons tree_ord_props 
+      dest!: entry_in_tree_keys)+
+
+
+subsubsection {* Tree lookup *}
+
+primrec lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
+where
+  "lookup Empty k = None"
+| "lookup (Branch _ l x y r) k = (if k < x then lookup l k else if x < k then lookup r k else Some y)"
+
+lemma lookup_keys: "sorted t \<Longrightarrow> dom (lookup t) = set (keys t)"
+  by (induct t) (auto simp: dom_def tree_greater_prop tree_less_prop)
+
+lemma dom_lookup_Branch: 
+  "sorted (Branch c t1 k v t2) \<Longrightarrow> 
+    dom (lookup (Branch c t1 k v t2)) 
+    = Set.insert k (dom (lookup t1) \<union> dom (lookup t2))"
+proof -
+  assume "sorted (Branch c t1 k v t2)"
+  moreover from this have "sorted t1" "sorted t2" by simp_all
+  ultimately show ?thesis by (simp add: lookup_keys)
+qed
+
+lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
+proof (induct t)
+  case Empty then show ?case by simp
+next
+  case (Branch color t1 a b t2)
+  let ?A = "Set.insert a (dom (lookup t1) \<union> dom (lookup t2))"
+  have "dom (lookup (Branch color t1 a b t2)) \<subseteq> ?A" by (auto split: split_if_asm)
+  moreover from Branch have "finite (insert a (dom (lookup t1) \<union> dom (lookup t2)))" by simp
+  ultimately show ?case by (rule finite_subset)
+qed 
+
+lemma lookup_tree_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> lookup t k = None" 
+by (induct t) auto
+
+lemma lookup_tree_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> lookup t k = None"
+by (induct t) auto
+
+lemma lookup_Empty: "lookup Empty = empty"
+by (rule ext) simp
+
+lemma map_of_entries:
+  "sorted t \<Longrightarrow> map_of (entries t) = lookup t"
+proof (induct t)
+  case Empty thus ?case by (simp add: lookup_Empty)
+next
+  case (Branch c t1 k v t2)
+  have "lookup (Branch c t1 k v t2) = lookup t2 ++ [k\<mapsto>v] ++ lookup t1"
+  proof (rule ext)
+    fix x
+    from Branch have SORTED: "sorted (Branch c t1 k v t2)" by simp
+    let ?thesis = "lookup (Branch c t1 k v t2) x = (lookup t2 ++ [k \<mapsto> v] ++ lookup t1) x"
+
+    have DOM_T1: "!!k'. k'\<in>dom (lookup t1) \<Longrightarrow> k>k'"
+    proof -
+      fix k'
+      from SORTED have "t1 |\<guillemotleft> k" by simp
+      with tree_less_prop have "\<forall>k'\<in>set (keys t1). k>k'" by auto
+      moreover assume "k'\<in>dom (lookup t1)"
+      ultimately show "k>k'" using lookup_keys SORTED by auto
+    qed
+    
+    have DOM_T2: "!!k'. k'\<in>dom (lookup t2) \<Longrightarrow> k<k'"
+    proof -
+      fix k'
+      from SORTED have "k \<guillemotleft>| t2" by simp
+      with tree_greater_prop have "\<forall>k'\<in>set (keys t2). k<k'" by auto
+      moreover assume "k'\<in>dom (lookup t2)"
+      ultimately show "k<k'" using lookup_keys SORTED by auto
+    qed
+    
+    {
+      assume C: "x<k"
+      hence "lookup (Branch c t1 k v t2) x = lookup t1 x" by simp
+      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
+      moreover have "x\<notin>dom (lookup t2)" proof
+        assume "x\<in>dom (lookup t2)"
+        with DOM_T2 have "k<x" by blast
+        with C show False by simp
+      qed
+      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+    } moreover {
+      assume [simp]: "x=k"
+      hence "lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
+      moreover have "x\<notin>dom (lookup t1)" proof
+        assume "x\<in>dom (lookup t1)"
+        with DOM_T1 have "k>x" by blast
+        thus False by simp
+      qed
+      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+    } moreover {
+      assume C: "x>k"
+      hence "lookup (Branch c t1 k v t2) x = lookup t2 x" by (simp add: less_not_sym[of k x])
+      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
+      moreover have "x\<notin>dom (lookup t1)" proof
+        assume "x\<in>dom (lookup t1)"
+        with DOM_T1 have "k>x" by simp
+        with C show False by simp
+      qed
+      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+    } ultimately show ?thesis using less_linear by blast
+  qed
+  also from Branch have "lookup t2 ++ [k \<mapsto> v] ++ lookup t1 = map_of (entries (Branch c t1 k v t2))" by simp
+  finally show ?case by simp
+qed
+
+lemma lookup_in_tree: "sorted t \<Longrightarrow> 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" 
+  shows "set (entries t1) = set (entries t2) \<longleftrightarrow> entries t1 = entries t2"
+proof -
+  from 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)
+qed
+
+lemma entries_eqI:
+  assumes sorted: "sorted t1" "sorted t2" 
+  assumes lookup: "lookup t1 = lookup t2"
+  shows "entries t1 = entries t2"
+proof -
+  from sorted 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)"
+    by (simp add: map_of_inject_set distinct_entries)
+  with 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"
+  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"
+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])
+qed
+
+
+subsubsection {* Red-black properties *}
+
+primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"
+where
+  "color_of Empty = B"
+| "color_of (Branch c _ _ _ _) = c"
+
+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 (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)"
+
+primrec inv1l :: "('a, 'b) rbt \<Rightarrow> bool" -- {* Weaker version *}
+where
+  "inv1l Empty = True"
+| "inv1l (Branch c l k v r) = (inv1 l \<and> inv1 r)"
+lemma [simp]: "inv1 t \<Longrightarrow> inv1l t" by (cases t) simp+
+
+primrec inv2 :: "('a, 'b) rbt \<Rightarrow> bool"
+where
+  "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"
+
+lemma is_rbt_sorted [simp]:
+  "is_rbt t \<Longrightarrow> sorted t" by (simp add: is_rbt_def)
+
+theorem Empty_is_rbt [simp]:
+  "is_rbt Empty" by (simp add: is_rbt_def)
+
+
+subsection {* Insertion *}
+
+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 (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_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" "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_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_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_sorted: 
+  fixes k :: "'a::linorder"
+  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>| 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> 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> 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> 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> 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> 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> 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> 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+
+
+lemma entries_balance [simp]:
+  "entries (balance l k v r) = entries l @ (k, v) # entries r"
+  by (induct l k v r rule: balance.induct) auto
+
+lemma keys_balance [simp]: 
+  "keys (balance l k v r) = keys l @ k # keys r"
+  by (simp add: keys_def)
+
+lemma balance_in_tree:  
+  "entry_in_tree k x (balance l v y r) \<longleftrightarrow> entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r"
+  by (auto simp add: keys_def)
+
+lemma lookup_balance[simp]: 
+fixes k :: "'a::linorder"
+assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+shows "lookup (balance l k v r) x = lookup (Branch B l k v r) x"
+by (rule lookup_from_in_tree) (auto simp:assms balance_in_tree balance_sorted)
+
+primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "paint c Empty = Empty"
+| "paint c (Branch _ l k v r) = Branch c l k v r"
+
+lemma paint_inv1l[simp]: "inv1l t \<Longrightarrow> inv1l (paint c t)" by (cases t) auto
+lemma paint_inv1[simp]: "inv1l t \<Longrightarrow> inv1 (paint B t)" by (cases t) auto
+lemma paint_inv2[simp]: "inv2 t \<Longrightarrow> inv2 (paint c t)" by (cases t) auto
+lemma paint_color_of[simp]: "color_of (paint B t) = B" by (cases t) auto
+lemma paint_sorted[simp]: "sorted t \<Longrightarrow> sorted (paint c t)" by (cases t) auto
+lemma paint_in_tree[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
+lemma paint_lookup[simp]: "lookup (paint c t) = lookup t" by (rule ext) (cases t, auto)
+lemma paint_tree_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
+lemma paint_tree_less[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
+
+fun
+  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 = 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)"
+
+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)"
+  using assms
+  by (induct f k x t rule: ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
+
+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 keys_ins: "set (keys (ins f k v t)) = { k } \<union> set (keys t)"
+  by (induct f k v t rule: ins.induct) auto
+
+lemma lookup_ins: 
+  fixes k :: "'a::linorder"
+  assumes "sorted t"
+  shows "lookup (ins f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v 
+                                                       | Some w \<Rightarrow> f k w v)) x"
+using assms by (induct f k v t rule: ins.induct) auto
+
+definition
+  insert_with_key :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "insert_with_key f k v t = paint B (ins f k v t)"
+
+lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insert_with_key f k x t)"
+  by (auto simp: insert_with_key_def)
+
+theorem insertwk_is_rbt: 
+  assumes inv: "is_rbt t" 
+  shows "is_rbt (insert_with_key f k x t)"
+using assms
+unfolding 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 
+                                                       | Some w \<Rightarrow> f k w v)) x"
+unfolding insert_with_key_def using assms
+by (simp add:lookup_ins)
+
+definition
+  insertw_def: "insert_with f = insert_with_key (\<lambda>_. f)"
+
+lemma insertw_sorted: "sorted t \<Longrightarrow> sorted (insert_with f k v t)" by (simp add: insertwk_sorted insertw_def)
+theorem insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insert_with f k v t)" by (simp add: insertwk_is_rbt insertw_def)
+
+lemma lookup_insertw:
+  assumes "is_rbt t"
+  shows "lookup (insert_with f k v t) = (lookup t)(k \<mapsto> (if k:dom (lookup t) then f (the (lookup t k)) v else v))"
+using assms
+unfolding insertw_def
+by (rule_tac ext) (cases "lookup t k", auto simp:lookup_insertwk dom_def)
+
+definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
+  "insert = insert_with_key (\<lambda>_ _ nv. nv)"
+
+lemma insert_sorted: "sorted t \<Longrightarrow> sorted (insert k v t)" by (simp add: insertwk_sorted insert_def)
+theorem insert_is_rbt [simp]: "is_rbt t \<Longrightarrow> is_rbt (insert k v t)" by (simp add: insertwk_is_rbt insert_def)
+
+lemma lookup_insert: 
+  assumes "is_rbt t"
+  shows "lookup (insert k v t) = (lookup t)(k\<mapsto>v)"
+unfolding insert_def
+using assms
+by (rule_tac ext) (simp add: lookup_insertwk split:option.split)
+
+
+subsection {* Deletion *}
+
+lemma bheight_paintR'[simp]: "color_of t = B \<Longrightarrow> bheight (paint R t) = bheight t - 1"
+by (cases t rule: rbt_cases) auto
+
+fun
+  balance_left :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "balance_left (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
+  "balance_left bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
+  "balance_left bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
+  "balance_left t k x s = Empty"
+
+lemma balance_left_inv2_with_inv1:
+  assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"
+  shows "bheight (balance_left lt k v rt) = bheight lt + 1"
+  and   "inv2 (balance_left lt k v rt)"
+using assms 
+by (induct lt k v rt rule: balance_left.induct) (auto simp: balance_inv2 balance_bheight)
+
+lemma balance_left_inv2_app: 
+  assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"
+  shows "inv2 (balance_left lt k v rt)" 
+        "bheight (balance_left lt k v rt) = bheight rt"
+using assms 
+by (induct lt k v rt rule: balance_left.induct) (auto simp add: balance_inv2 balance_bheight)+ 
+
+lemma balance_left_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balance_left a k x b)"
+  by (induct a k x b rule: balance_left.induct) (simp add: balance_inv1)+
+
+lemma balance_left_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balance_left lt k x rt)"
+by (induct lt k x rt rule: balance_left.induct) (auto simp: balance_inv1)
+
+lemma balance_left_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_left l k v r)"
+apply (induct l k v r rule: balance_left.induct)
+apply (auto simp: balance_sorted)
+apply (unfold tree_greater_prop tree_less_prop)
+by force+
+
+lemma balance_left_tree_greater: 
+  fixes k :: "'a::order"
+  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"
+  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
+
+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)"
+using assms 
+by (induct l k v r rule: balance_left.induct) (auto simp: balance_in_tree)
+
+fun
+  balance_right :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "balance_right a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
+  "balance_right (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
+  "balance_right (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
+  "balance_right t k x s = Empty"
+
+lemma balance_right_inv2_with_inv1:
+  assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt + 1" "inv1 lt"
+  shows "inv2 (balance_right lt k v rt) \<and> bheight (balance_right lt k v rt) = bheight lt"
+using assms
+by (induct lt k v rt rule: balance_right.induct) (auto simp: balance_inv2 balance_bheight)
+
+lemma balance_right_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balance_right a k x b)"
+by (induct a k x b rule: balance_right.induct) (simp add: balance_inv1)+
+
+lemma balance_right_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balance_right lt k x rt)"
+by (induct lt k x rt rule: balance_right.induct) (auto simp: balance_inv1)
+
+lemma balance_right_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_right l k v r)"
+apply (induct l k v r rule: balance_right.induct)
+apply (auto simp:balance_sorted)
+apply (unfold tree_less_prop tree_greater_prop)
+by force+
+
+lemma balance_right_tree_greater: 
+  fixes k :: "'a::order"
+  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"
+  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
+
+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)"
+using assms by (induct l k v r rule: balance_right.induct) (auto simp: balance_in_tree)
+
+fun
+  combine :: "('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+where
+  "combine Empty x = x" 
+| "combine x Empty = x" 
+| "combine (Branch R a k x b) (Branch R c s y d) = (case (combine b c) of
+                                      Branch R b2 t z c2 \<Rightarrow> (Branch R (Branch R a k x b2) t z (Branch R c2 s y d)) |
+                                      bc \<Rightarrow> Branch R a k x (Branch R bc s y d))" 
+| "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))" 
+| "combine a (Branch R b k x c) = Branch R (combine a b) k x c" 
+| "combine (Branch R a k x b) c = Branch R a k x (combine b c)" 
+
+lemma combine_inv2:
+  assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
+  shows "bheight (combine lt rt) = bheight lt" "inv2 (combine lt rt)"
+using assms 
+by (induct lt rt rule: combine.induct) 
+   (auto simp: balance_left_inv2_app split: rbt.splits color.splits)
+
+lemma combine_inv1: 
+  assumes "inv1 lt" "inv1 rt"
+  shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (combine lt rt)"
+         "inv1l (combine lt rt)"
+using assms 
+by (induct lt rt rule: combine.induct)
+   (auto simp: balance_left_inv1 split: rbt.splits color.splits)
+
+lemma combine_tree_greater[simp]: 
+  fixes k :: "'a::linorder"
+  assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r" 
+  shows "k \<guillemotleft>| combine l r"
+using assms 
+by (induct l r rule: combine.induct)
+   (auto simp: balance_left_tree_greater split:rbt.splits color.splits)
+
+lemma combine_tree_less[simp]: 
+  fixes k :: "'a::linorder"
+  assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k" 
+  shows "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)
+
+lemma combine_sorted: 
+  fixes k :: "'a::linorder"
+  assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+  shows "sorted (combine l r)"
+using assms proof (induct l r rule: combine.induct)
+  case (3 a x v b c y w d)
+  hence ineqs: "a |\<guillemotleft> x" "x \<guillemotleft>| b" "b |\<guillemotleft> k" "k \<guillemotleft>| c" "c |\<guillemotleft> y" "y \<guillemotleft>| d"
+    by auto
+  with 3
+  show ?case
+    by (cases "combine b c" rule: rbt_cases)
+      (auto, (metis combine_tree_greater combine_tree_less ineqs ineqs tree_less_simps(2) tree_greater_simps(2) tree_greater_trans tree_less_trans)+)
+next
+  case (4 a x v b c y w d)
+  hence "x < k \<and> tree_greater k c" by simp
+  hence "tree_greater x c" by (blast dest: tree_greater_trans)
+  with 4 have 2: "tree_greater x (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)
+  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)
+  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 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)
+  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)
+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)
+qed simp+
+
+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)"
+using assms 
+proof (induct l r rule: combine.induct)
+  case (4 _ _ _ b c)
+  hence a: "bheight (combine b c) = bheight b" by (simp add: combine_inv2)
+  from 4 have b: "inv1l (combine b c)" by (simp add: combine_inv1)
+
+  show ?case
+  proof (cases "combine b c" rule: rbt_cases)
+    case Empty
+    with 4 a show ?thesis by (auto simp: balance_left_in_tree)
+  next
+    case (Red lta ka va rta)
+    with 4 show ?thesis by auto
+  next
+    case (Black lta ka va rta)
+    with a b 4  show ?thesis by (auto simp: balance_left_in_tree)
+  qed 
+qed (auto split: rbt.splits color.splits)
+
+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"
+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)"
+
+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))"
+  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))"
+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)
+case (2 y c _ y')
+  have "y = y' \<or> y < y' \<or> y > y'" by auto
+  thus ?case proof (elim disjE)
+    assume "y = y'"
+    with 2 show ?thesis by (cases c) (simp add: combine_inv2 combine_inv1)+
+  next
+    assume "y < y'"
+    with 2 show ?thesis by (cases c) auto
+  next
+    assume "y' < y"
+    with 2 show ?thesis by (cases c) auto
+  qed
+next
+  case (3 y lt z v rta y' ss bb) 
+  thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balance_left_inv2_with_inv1 balance_left_inv1 balance_left_inv1l)+
+next
+  case (5 y a y' ss lt z v rta)
+  thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balance_right_inv2_with_inv1 balance_right_inv1 balance_right_inv1l)+
+next
+  case ("6_1" y a y' ss) thus ?case by (cases "color_of a = B \<and> color_of Empty = B") simp+
+qed auto
+
+lemma 
+  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)
+
+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 "\<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)
+  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)
+next
+  case ("4_2" x vaa vbb vdd vc yy ss bb)
+  hence "tree_less yy (Branch R vaa vbb vdd vc)" by simp
+  hence "tree_less yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_less)
+  with "4_2" show ?case by simp
+next
+  case (5 x aa yy ss lta zz v rta) 
+  hence "tree_greater yy (Branch B lta zz v rta)" by simp
+  hence "tree_greater yy (del x (Branch B lta zz v rta))" by (rule del_tree_greater)
+  with 5 show ?case by (simp add: balance_right_sorted)
+next
+  case ("6_2" x aa yy ss vaa vbb vdd vc)
+  hence "tree_greater yy (Branch R vaa vbb vdd vc)" by simp
+  hence "tree_greater yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_greater)
+  with "6_2" show ?case by simp
+qed (auto simp: combine_sorted)
+
+lemma "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_left x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+  and "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_right x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+  and del_in_tree: "\<lbrakk>sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
+proof (induct x lt kt y rt and x lt kt y rt and x t rule: del_from_left_del_from_right_del.induct)
+  case (2 xx c aa yy ss bb)
+  have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
+  from this 2 show ?case proof (elim disjE)
+    assume "xx = yy"
+    with 2 show ?thesis proof (cases "xx = k")
+      case True
+      from 2 `xx = yy` `xx = k` have "sorted (Branch c aa yy ss bb) \<and> k = yy" by simp
+      hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: tree_less_nit tree_greater_prop)
+      with `xx = yy` 2 `xx = k` show ?thesis by (simp add: combine_in_tree)
+    qed (simp add: combine_in_tree)
+  qed simp+
+next    
+  case (3 xx lta zz vv rta yy ss bb)
+  def mt[simp]: mt == "Branch B lta zz vv rta"
+  from 3 have "inv2 mt \<and> inv1 mt" by simp
+  hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
+  with 3 have 4: "entry_in_tree k v (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)
+  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" `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)
+    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)
+  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)
+  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)
+  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)
+  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)
+  qed auto
+qed simp
+
+
+definition delete where
+  delete_def: "delete k t = paint B (del k t)"
+
+theorem delete_is_rbt [simp]: assumes "is_rbt t" shows "is_rbt (delete k t)"
+proof -
+  from assms have "inv2 t" and "inv1 t" unfolding is_rbt_def by auto 
+  hence "inv2 (del k t) \<and> (color_of t = R \<and> bheight (del k t) = bheight t \<and> inv1 (del k t) \<or> color_of t = B \<and> bheight (del k t) = bheight t - 1 \<and> inv1l (del k t))" by (rule del_inv1_inv2)
+  hence "inv2 (del k t) \<and> inv1l (del k t)" by (cases "color_of t") auto
+  with assms show ?thesis
+    unfolding is_rbt_def delete_def
+    by (auto intro: paint_sorted del_sorted)
+qed
+
+lemma delete_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)
+
+lemma lookup_delete:
+  assumes is_rbt: "is_rbt t"
+  shows "lookup (delete k t) = (lookup t)|`(-{k})"
+proof
+  fix x
+  show "lookup (delete k t) x = (lookup t |` (-{k})) x" 
+  proof (cases "x = k")
+    assume "x = k" 
+    with is_rbt show ?thesis
+      by (cases "lookup (delete k t) k") (auto simp: lookup_in_tree 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)
+  qed
+qed
+
+
+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"
+where
+  "union_with_key f t Empty = t"
+| "union_with_key f t (Branch c lt k v rt) = union_with_key f (union_with_key f (insert_with_key f k v t) lt) rt"
+
+lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (union_with_key f lt rt)" 
+  by (induct rt arbitrary: lt) (auto simp: insertwk_sorted)
+theorem unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (union_with_key f lt rt)" 
+  by (induct rt arbitrary: lt) (simp add: insertwk_is_rbt)+
+
+definition
+  union_with where
+  "union_with f = union_with_key (\<lambda>_. f)"
+
+theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union_with f lt rt)" unfolding union_with_def by simp
+
+definition union where
+  "union = union_with_key (%_ _ rv. rv)"
+
+theorem union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union lt rt)" unfolding union_def by simp
+
+lemma union_Branch[simp]:
+  "union t (Branch c lt k v rt) = union (union (insert k v t) lt) rt"
+  unfolding union_def insert_def
+  by simp
+
+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 (Branch c l k v r s)
+  then have "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
+
+  have meq: "lookup s(k \<mapsto> v) ++ lookup l ++ lookup r =
+    lookup s ++
+    (\<lambda>a. if a < k then lookup l a
+    else if k < a then lookup r a else Some v)" (is "?m1 = ?m2")
+  proof (rule ext)
+    fix a
+
+   have "k < a \<or> k = a \<or> k > a" by auto
+    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 `k < a` show ?thesis
+        by (auto simp: map_add_def split: option.splits)
+    next
+      assume "k = a"
+      with `l |\<guillemotleft> k` `k \<guillemotleft>| r` 
+      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)
+      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)
+  with Branch have IHs:
+    "lookup (union (union (insert k v s) l) r) = lookup (union (insert k v s) l) ++ lookup r"
+    "lookup (union (insert k v s) l) = lookup (insert k v s) ++ lookup l"
+    by auto
+  
+  with meq show ?case
+    by (auto simp: lookup_insert[OF Branch(3)])
+
+qed
+
+
+subsection {* Modifying existing entries *}
+
+primrec
+  map_entry :: "'a\<Colon>linorder \<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))
+    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)
+
+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 lookup_map_entry:
+  "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
+  by (induct t) (auto split: option.splits simp add: expand_fun_eq)
+
+
+subsection {* Mapping all entries *}
+
+primrec
+  map :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'c) rbt"
+where
+  "map f Empty = Empty"
+| "map f (Branch c lt k v rt) = Branch c (map f lt) k (f k v) (map f rt)"
+
+lemma map_entries [simp]: "entries (map f t) = List.map (\<lambda>(k, v). (k, f k v)) (entries t)"
+  by (induct t) auto
+lemma map_keys [simp]: "keys (map f t) = keys t" by (simp add: keys_def split_def)
+lemma map_tree_greater: "tree_greater k (map f t) = tree_greater k t" by (induct t) simp+
+lemma map_tree_less: "tree_less k (map f t) = tree_less k t" by (induct t) simp+
+lemma map_sorted: "sorted (map f t) = sorted t"  by (induct t) (simp add: map_tree_less map_tree_greater)+
+lemma map_color_of: "color_of (map f t) = color_of t" by (induct t) simp+
+lemma map_inv1: "inv1 (map f t) = inv1 t" by (induct t) (simp add: map_color_of)+
+lemma map_inv2: "inv2 (map f t) = inv2 t" "bheight (map f t) = bheight t" by (induct t) simp+
+theorem map_is_rbt [simp]: "is_rbt (map f t) = is_rbt t" 
+unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_sorted map_color_of)
+
+theorem lookup_map: "lookup (map f t) x = Option.map (f x) (lookup t x)"
+  by (induct t) auto
+
+
+subsection {* Folding over entries *}
+
+definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
+  "fold f t s = foldl (\<lambda>s (k, v). f k v s) s (entries t)"
+
+lemma fold_simps [simp, code]:
+  "fold f Empty = id"
+  "fold f (Branch c lt k v rt) = fold f rt \<circ> f k v \<circ> fold f lt"
+  by (simp_all add: fold_def expand_fun_eq)
+
+
+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"
+
+lemma bulkload_is_rbt [simp, intro]:
+  "is_rbt (bulkload xs)"
+  unfolding bulkload_def by (induct xs) auto
+
+lemma lookup_bulkload:
+  "lookup (bulkload xs) = map_of xs"
+proof -
+  obtain ys where "ys = rev xs" by simp
+  have "\<And>t. is_rbt t \<Longrightarrow>
+    lookup (foldl (\<lambda>t (k, v). insert k v t) t ys) = lookup t ++ map_of (rev ys)"
+      by (induct ys) (simp_all add: bulkload_def split_def lookup_insert)
+  from this Empty_is_rbt have
+    "lookup (foldl (\<lambda>t (k, v). insert k v t) Empty (rev xs)) = lookup Empty ++ map_of xs"
+     by (simp add: `ys = rev xs`)
+  then show ?thesis by (simp add: bulkload_def foldl_foldr lookup_Empty split_def)
+qed
+
+hide (open) const Empty insert delete entries keys bulkload lookup map_entry map fold union sorted
+
+end
diff -r 7bfbb247a5df -r b43b22f63665 src/HOL/Library/Table.thy
--- a/src/HOL/Library/Table.thy	Wed Apr 14 22:18:10 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,229 +0,0 @@
-(* Author: Florian Haftmann, TU Muenchen *)
-
-header {* Tables: finite mappings implemented by red-black trees *}
-
-theory Table
-imports Main RBT Mapping
-begin
-
-subsection {* Type definition *}
-
-typedef (open) ('a, 'b) table = "{t :: ('a\<Colon>linorder, 'b) rbt. is_rbt t}"
-  morphisms tree_of Table
-proof -
-  have "RBT.Empty \<in> ?table" by simp
-  then show ?thesis ..
-qed
-
-lemma is_rbt_tree_of [simp, intro]:
-  "is_rbt (tree_of t)"
-  using tree_of [of t] by simp
-
-lemma table_eq:
-  "t1 = t2 \<longleftrightarrow> tree_of t1 = tree_of t2"
-  by (simp add: tree_of_inject)
-
-lemma [code abstype]:
-  "Table (tree_of t) = t"
-  by (simp add: tree_of_inverse)
-
-
-subsection {* Primitive operations *}
-
-definition lookup :: "('a\<Colon>linorder, 'b) table \<Rightarrow> 'a \<rightharpoonup> 'b" where
-  [code]: "lookup t = RBT.lookup (tree_of t)"
-
-definition empty :: "('a\<Colon>linorder, 'b) table" where
-  "empty = Table RBT.Empty"
-
-lemma tree_of_empty [code abstract]:
-  "tree_of empty = RBT.Empty"
-  by (simp add: empty_def Table_inverse)
-
-definition update :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) table \<Rightarrow> ('a, 'b) table" where
-  "update k v t = Table (RBT.insert k v (tree_of t))"
-
-lemma tree_of_update [code abstract]:
-  "tree_of (update k v t) = RBT.insert k v (tree_of t)"
-  by (simp add: update_def Table_inverse)
-
-definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) table \<Rightarrow> ('a, 'b) table" where
-  "delete k t = Table (RBT.delete k (tree_of t))"
-
-lemma tree_of_delete [code abstract]:
-  "tree_of (delete k t) = RBT.delete k (tree_of t)"
-  by (simp add: delete_def Table_inverse)
-
-definition entries :: "('a\<Colon>linorder, 'b) table \<Rightarrow> ('a \<times> 'b) list" where
-  [code]: "entries t = RBT.entries (tree_of t)"
-
-definition keys :: "('a\<Colon>linorder, 'b) table \<Rightarrow> 'a list" where
-  [code]: "keys t = RBT.keys (tree_of t)"
-
-definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) table" where
-  "bulkload xs = Table (RBT.bulkload xs)"
-
-lemma tree_of_bulkload [code abstract]:
-  "tree_of (bulkload xs) = RBT.bulkload xs"
-  by (simp add: bulkload_def Table_inverse)
-
-definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) table \<Rightarrow> ('a, 'b) table" where
-  "map_entry k f t = Table (RBT.map_entry k f (tree_of t))"
-
-lemma tree_of_map_entry [code abstract]:
-  "tree_of (map_entry k f t) = RBT.map_entry k f (tree_of t)"
-  by (simp add: map_entry_def Table_inverse)
-
-definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) table \<Rightarrow> ('a, 'b) table" where
-  "map f t = Table (RBT.map f (tree_of t))"
-
-lemma tree_of_map [code abstract]:
-  "tree_of (map f t) = RBT.map f (tree_of t)"
-  by (simp add: map_def Table_inverse)
-
-definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) table \<Rightarrow> 'c \<Rightarrow> 'c" where
-  [code]: "fold f t = RBT.fold f (tree_of t)"
-
-
-subsection {* Derived operations *}
-
-definition is_empty :: "('a\<Colon>linorder, 'b) table \<Rightarrow> bool" where
-  [code]: "is_empty t = (case tree_of t of RBT.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
-
-
-subsection {* Abstract lookup properties *}
-
-lemma lookup_Table:
-  "is_rbt t \<Longrightarrow> lookup (Table t) = RBT.lookup t"
-  by (simp add: lookup_def Table_inverse)
-
-lemma lookup_tree_of:
-  "RBT.lookup (tree_of t) = lookup t"
-  by (simp add: lookup_def)
-
-lemma entries_tree_of:
-  "RBT.entries (tree_of t) = entries t"
-  by (simp add: entries_def)
-
-lemma keys_tree_of:
-  "RBT.keys (tree_of t) = keys t"
-  by (simp add: keys_def)
-
-lemma lookup_empty [simp]:
-  "lookup empty = Map.empty"
-  by (simp add: empty_def lookup_Table expand_fun_eq)
-
-lemma lookup_update [simp]:
-  "lookup (update k v t) = (lookup t)(k \<mapsto> v)"
-  by (simp add: update_def lookup_Table lookup_insert lookup_tree_of)
-
-lemma lookup_delete [simp]:
-  "lookup (delete k t) = (lookup t)(k := None)"
-  by (simp add: delete_def lookup_Table RBT.lookup_delete lookup_tree_of restrict_complement_singleton_eq)
-
-lemma map_of_entries [simp]:
-  "map_of (entries t) = lookup t"
-  by (simp add: entries_def map_of_entries lookup_tree_of)
-
-lemma entries_lookup:
-  "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
-  by (simp add: entries_def lookup_def entries_lookup)
-
-lemma lookup_bulkload [simp]:
-  "lookup (bulkload xs) = map_of xs"
-  by (simp add: bulkload_def lookup_Table RBT.lookup_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_Table lookup_map_entry lookup_tree_of)
-
-lemma lookup_map [simp]:
-  "lookup (map f t) k = Option.map (f k) (lookup t k)"
-  by (simp add: map_def lookup_Table lookup_map lookup_tree_of)
-
-lemma fold_fold:
-  "fold f t = (\<lambda>s. foldl (\<lambda>s (k, v). f k v s) s (entries t))"
-  by (simp add: fold_def expand_fun_eq RBT.fold_def entries_tree_of)
-
-lemma is_empty_empty [simp]:
-  "is_empty t \<longleftrightarrow> t = empty"
-  by (simp add: table_eq is_empty_def tree_of_empty split: rbt.split)
-
-lemma RBT_lookup_empty [simp]: (*FIXME*)
-  "RBT.lookup t = Map.empty \<longleftrightarrow> t = RBT.Empty"
-  by (cases t) (auto simp add: expand_fun_eq)
-
-lemma lookup_empty_empty [simp]:
-  "lookup t = Map.empty \<longleftrightarrow> t = empty"
-  by (cases t) (simp add: empty_def lookup_def Table_inject Table_inverse)
-
-lemma sorted_keys [iff]:
-  "sorted (keys t)"
-  by (simp add: keys_def RBT.keys_def sorted_entries)
-
-lemma distinct_keys [iff]:
-  "distinct (keys t)"
-  by (simp add: keys_def RBT.keys_def distinct_entries)
-
-
-subsection {* Implementation of mappings *}
-
-definition Mapping :: "('a\<Colon>linorder, 'b) table \<Rightarrow> ('a, 'b) mapping" where
-  "Mapping t = Mapping.Mapping (lookup t)"
-
-code_datatype Mapping
-
-lemma lookup_Mapping [simp, code]:
-  "Mapping.lookup (Mapping t) = lookup t"
-  by (simp add: Mapping_def)
-
-lemma empty_Mapping [code]:
-  "Mapping.empty = Mapping empty"
-  by (rule mapping_eqI) simp
-
-lemma is_empty_Mapping [code]:
-  "Mapping.is_empty (Mapping t) \<longleftrightarrow> is_empty t"
-  by (simp add: table_eq Mapping.is_empty_empty Mapping_def)
-
-lemma update_Mapping [code]:
-  "Mapping.update k v (Mapping t) = Mapping (update k v t)"
-  by (rule mapping_eqI) simp
-
-lemma delete_Mapping [code]:
-  "Mapping.delete k (Mapping xs) = Mapping (delete k xs)"
-  by (rule mapping_eqI) simp
-
-lemma keys_Mapping [code]:
-  "Mapping.keys (Mapping t) = set (keys t)"
-  by (simp add: keys_def Mapping_def Mapping.keys_def lookup_def lookup_keys)
-
-lemma ordered_keys_Mapping [code]:
-  "Mapping.ordered_keys (Mapping t) = keys t"
-  by (rule sorted_distinct_set_unique) (simp_all add: ordered_keys_def keys_Mapping)
-
-lemma Mapping_size_card_keys: (*FIXME*)
-  "Mapping.size m = card (Mapping.keys m)"
-  by (simp add: Mapping.size_def Mapping.keys_def)
-
-lemma size_Mapping [code]:
-  "Mapping.size (Mapping t) = length (keys t)"
-  by (simp add: Mapping_size_card_keys keys_Mapping distinct_card)
-
-lemma tabulate_Mapping [code]:
-  "Mapping.tabulate ks f = Mapping (bulkload (List.map (\<lambda>k. (k, f k)) ks))"
-  by (rule mapping_eqI) (simp add: map_of_map_restrict)
-
-lemma bulkload_Mapping [code]:
-  "Mapping.bulkload vs = Mapping (bulkload (List.map (\<lambda>n. (n, vs ! n)) [0..<length vs]))"
-  by (rule mapping_eqI) (simp add: map_of_map_restrict expand_fun_eq)
-
-lemma [code, code del]: "HOL.eq (x :: (_, _) mapping) y \<longleftrightarrow> x = y" by (fact eq_equals) (*FIXME*)
-
-lemma eq_Mapping [code]:
-  "HOL.eq (Mapping t1) (Mapping t2) \<longleftrightarrow> entries t1 = entries t2"
-  by (simp add: eq Mapping_def entries_lookup)
-
-hide (open) const tree_of lookup empty update delete
-  entries keys bulkload map_entry map fold
-
-end
diff -r 7bfbb247a5df -r b43b22f63665 src/HOL/ex/Codegenerator_Candidates.thy
--- a/src/HOL/ex/Codegenerator_Candidates.thy	Wed Apr 14 22:18:10 2010 +0200
+++ b/src/HOL/ex/Codegenerator_Candidates.thy	Thu Apr 15 12:27:14 2010 +0200
@@ -20,8 +20,8 @@
   "~~/src/HOL/Number_Theory/Primes"
   Product_ord
   "~~/src/HOL/ex/Records"
+  RBT
   SetsAndFunctions
-  Table
   While_Combinator
   Word
 begin