diff --git a/src/HOL/Library/RBT_Impl.thy b/src/HOL/Library/RBT_Impl.thy
--- a/src/HOL/Library/RBT_Impl.thy
+++ b/src/HOL/Library/RBT_Impl.thy
@@ -1,3087 +1,3087 @@
 (*  Title:      HOL/Library/RBT_Impl.thy
     Author:     Markus Reiter, TU Muenchen
     Author:     Alexander Krauss, TU Muenchen
 *)
 
 section \<open>Implementation of Red-Black Trees\<close>
 
 theory RBT_Impl
 imports Main
 begin
 
 text \<open>
   For applications, you should use theory \<open>RBT\<close> which defines
   an abstract type of red-black tree obeying the invariant.
 \<close>
 
 subsection \<open>Datatype of RB trees\<close>
 
 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 \<open>Tree properties\<close>
 
 subsubsection \<open>Content of a tree\<close>
 
 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)
 
 lemma non_empty_rbt_keys: 
   "t \<noteq> rbt.Empty \<Longrightarrow> keys t \<noteq> []"
   by (cases t) simp_all
 
 subsubsection \<open>Search tree properties\<close>
 
 context ord begin
 
 definition rbt_less :: "'a \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
 where
   rbt_less_prop: "rbt_less k t \<longleftrightarrow> (\<forall>x\<in>set (keys t). x < k)"
 
 abbreviation rbt_less_symbol (infix "|\<guillemotleft>" 50)
 where "t |\<guillemotleft> x \<equiv> rbt_less x t"
 
 definition rbt_greater :: "'a \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
 where
   rbt_greater_prop: "rbt_greater k t = (\<forall>x\<in>set (keys t). k < x)"
 
 lemma rbt_less_simps [simp]:
   "Empty |\<guillemotleft> k = True"
   "Branch c lt kt v rt |\<guillemotleft> k \<longleftrightarrow> kt < k \<and> lt |\<guillemotleft> k \<and> rt |\<guillemotleft> k"
   by (auto simp add: rbt_less_prop)
 
 lemma rbt_greater_simps [simp]:
   "k \<guillemotleft>| Empty = True"
   "k \<guillemotleft>| (Branch c lt kt v rt) \<longleftrightarrow> k < kt \<and> k \<guillemotleft>| lt \<and> k \<guillemotleft>| rt"
   by (auto simp add: rbt_greater_prop)
 
 lemmas rbt_ord_props = rbt_less_prop rbt_greater_prop
 
 lemmas rbt_greater_nit = rbt_greater_prop entry_in_tree_keys
 lemmas rbt_less_nit = rbt_less_prop entry_in_tree_keys
 
 lemma (in order)
   shows rbt_less_eq_trans: "l |\<guillemotleft> u \<Longrightarrow> u \<le> v \<Longrightarrow> l |\<guillemotleft> v"
   and rbt_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
   and rbt_greater_eq_trans: "u \<le> v \<Longrightarrow> v \<guillemotleft>| r \<Longrightarrow> u \<guillemotleft>| r"
   and rbt_greater_trans: "x < y \<Longrightarrow> y \<guillemotleft>| t \<Longrightarrow> x \<guillemotleft>| t"
   by (auto simp: rbt_ord_props)
 
 primrec rbt_sorted :: "('a, 'b) rbt \<Rightarrow> bool"
 where
   "rbt_sorted Empty = True"
 | "rbt_sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> rbt_sorted l \<and> rbt_sorted r)"
 
 end
 
 context linorder begin
 
 lemma rbt_sorted_entries:
   "rbt_sorted t \<Longrightarrow> List.sorted (map fst (entries t))"
 by (induct t)  (force simp: sorted_append rbt_ord_props dest!: entry_in_tree_keys)+
 
 lemma distinct_entries:
   "rbt_sorted t \<Longrightarrow> distinct (map fst (entries t))"
 by (induct t) (force simp: sorted_append rbt_ord_props dest!: entry_in_tree_keys)+
 
 lemma distinct_keys:
   "rbt_sorted t \<Longrightarrow> distinct (keys t)"
   by (simp add: distinct_entries keys_def)
 
 
 subsubsection \<open>Tree lookup\<close>
 
 primrec (in ord) rbt_lookup :: "('a, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
 where
   "rbt_lookup Empty k = None"
 | "rbt_lookup (Branch _ l x y r) k = 
    (if k < x then rbt_lookup l k else if x < k then rbt_lookup r k else Some y)"
 
 lemma rbt_lookup_keys: "rbt_sorted t \<Longrightarrow> dom (rbt_lookup t) = set (keys t)"
   by (induct t) (auto simp: dom_def rbt_greater_prop rbt_less_prop)
 
 lemma dom_rbt_lookup_Branch: 
   "rbt_sorted (Branch c t1 k v t2) \<Longrightarrow> 
     dom (rbt_lookup (Branch c t1 k v t2)) 
     = Set.insert k (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2))"
 proof -
   assume "rbt_sorted (Branch c t1 k v t2)"
   then show ?thesis by (simp add: rbt_lookup_keys)
 qed
 
 lemma finite_dom_rbt_lookup [simp, intro!]: "finite (dom (rbt_lookup t))"
 proof (induct t)
   case Empty then show ?case by simp
 next
   case (Branch color t1 a b t2)
   let ?A = "Set.insert a (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2))"
   have "dom (rbt_lookup (Branch color t1 a b t2)) \<subseteq> ?A" by (auto split: if_split_asm)
   moreover from Branch have "finite (insert a (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2)))" by simp
   ultimately show ?case by (rule finite_subset)
 qed 
 
 end
 
 context ord begin
 
 lemma rbt_lookup_rbt_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> rbt_lookup t k = None" 
 by (induct t) auto
 
 lemma rbt_lookup_rbt_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> rbt_lookup t k = None"
 by (induct t) auto
 
 lemma rbt_lookup_Empty: "rbt_lookup Empty = Map.empty"
 by (rule ext) simp
 
 end
 
 context linorder begin
 
 lemma map_of_entries:
   "rbt_sorted t \<Longrightarrow> map_of (entries t) = rbt_lookup t"
 proof (induct t)
   case Empty thus ?case by (simp add: rbt_lookup_Empty)
 next
   case (Branch c t1 k v t2)
   have "rbt_lookup (Branch c t1 k v t2) = rbt_lookup t2 ++ [k\<mapsto>v] ++ rbt_lookup t1"
   proof (rule ext)
     fix x
     from Branch have RBT_SORTED: "rbt_sorted (Branch c t1 k v t2)" by simp
     let ?thesis = "rbt_lookup (Branch c t1 k v t2) x = (rbt_lookup t2 ++ [k \<mapsto> v] ++ rbt_lookup t1) x"
 
     have DOM_T1: "!!k'. k'\<in>dom (rbt_lookup t1) \<Longrightarrow> k>k'"
     proof -
       fix k'
       from RBT_SORTED have "t1 |\<guillemotleft> k" by simp
       with rbt_less_prop have "\<forall>k'\<in>set (keys t1). k>k'" by auto
       moreover assume "k'\<in>dom (rbt_lookup t1)"
       ultimately show "k>k'" using rbt_lookup_keys RBT_SORTED by auto
     qed
     
     have DOM_T2: "!!k'. k'\<in>dom (rbt_lookup t2) \<Longrightarrow> k<k'"
     proof -
       fix k'
       from RBT_SORTED have "k \<guillemotleft>| t2" by simp
       with rbt_greater_prop have "\<forall>k'\<in>set (keys t2). k<k'" by auto
       moreover assume "k'\<in>dom (rbt_lookup t2)"
       ultimately show "k<k'" using rbt_lookup_keys RBT_SORTED by auto
     qed
     
     {
       assume C: "x<k"
       hence "rbt_lookup (Branch c t1 k v t2) x = rbt_lookup t1 x" by simp
       moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
       moreover have "x \<notin> dom (rbt_lookup t2)"
       proof
         assume "x \<in> dom (rbt_lookup t2)"
         with DOM_T2 have "k<x" by blast
         with C show False by simp
       qed
       ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
     } moreover {
       assume [simp]: "x=k"
       hence "rbt_lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
       moreover have "x \<notin> dom (rbt_lookup t1)" 
       proof
         assume "x \<in> dom (rbt_lookup t1)"
         with DOM_T1 have "k>x" by blast
         thus False by simp
       qed
       ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
     } moreover {
       assume C: "x>k"
       hence "rbt_lookup (Branch c t1 k v t2) x = rbt_lookup t2 x" by (simp add: less_not_sym[of k x])
       moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
       moreover have "x\<notin>dom (rbt_lookup t1)" proof
         assume "x\<in>dom (rbt_lookup t1)"
         with DOM_T1 have "k>x" by simp
         with C show False by simp
       qed
       ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
     } ultimately show ?thesis using less_linear by blast
   qed
   also from Branch 
   have "rbt_lookup t2 ++ [k \<mapsto> v] ++ rbt_lookup t1 = map_of (entries (Branch c t1 k v t2))" by simp
   finally show ?case by simp
 qed
 
 lemma rbt_lookup_in_tree: "rbt_sorted t \<Longrightarrow> rbt_lookup t k = Some v \<longleftrightarrow> (k, v) \<in> set (entries t)"
   by (simp add: map_of_entries [symmetric] distinct_entries)
 
 lemma set_entries_inject:
   assumes rbt_sorted: "rbt_sorted t1" "rbt_sorted t2" 
   shows "set (entries t1) = set (entries t2) \<longleftrightarrow> entries t1 = entries t2"
 proof -
   from rbt_sorted have "distinct (map fst (entries t1))"
     "distinct (map fst (entries t2))"
     by (auto intro: distinct_entries)
   with rbt_sorted show ?thesis
     by (auto intro: map_sorted_distinct_set_unique rbt_sorted_entries simp add: distinct_map)
 qed
 
 lemma entries_eqI:
   assumes rbt_sorted: "rbt_sorted t1" "rbt_sorted t2" 
   assumes rbt_lookup: "rbt_lookup t1 = rbt_lookup t2"
   shows "entries t1 = entries t2"
 proof -
   from rbt_sorted rbt_lookup have "map_of (entries t1) = map_of (entries t2)"
     by (simp add: map_of_entries)
   with rbt_sorted have "set (entries t1) = set (entries t2)"
     by (simp add: map_of_inject_set distinct_entries)
   with rbt_sorted show ?thesis by (simp add: set_entries_inject)
 qed
 
 lemma entries_rbt_lookup:
   assumes "rbt_sorted t1" "rbt_sorted t2" 
   shows "entries t1 = entries t2 \<longleftrightarrow> rbt_lookup t1 = rbt_lookup t2"
   using assms by (auto intro: entries_eqI simp add: map_of_entries [symmetric])
 
 lemma rbt_lookup_from_in_tree: 
   assumes "rbt_sorted t1" "rbt_sorted t2" 
   and "\<And>v. (k, v) \<in> set (entries t1) \<longleftrightarrow> (k, v) \<in> set (entries t2)" 
   shows "rbt_lookup t1 k = rbt_lookup t2 k"
 proof -
   from assms have "k \<in> dom (rbt_lookup t1) \<longleftrightarrow> k \<in> dom (rbt_lookup t2)"
     by (simp add: keys_entries rbt_lookup_keys)
   with assms show ?thesis by (auto simp add: rbt_lookup_in_tree [symmetric])
 qed
 
 end
 
 subsubsection \<open>Red-black properties\<close>
 
 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" \<comment> \<open>Weaker version\<close>
 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)"
 
 context ord begin
 
 definition is_rbt :: "('a, 'b) rbt \<Rightarrow> bool" where
   "is_rbt t \<longleftrightarrow> inv1 t \<and> inv2 t \<and> color_of t = B \<and> rbt_sorted t"
 
 lemma is_rbt_rbt_sorted [simp]:
   "is_rbt t \<Longrightarrow> rbt_sorted t" by (simp add: is_rbt_def)
 
 theorem Empty_is_rbt [simp]:
   "is_rbt Empty" by (simp add: is_rbt_def)
 
 end
 
 subsection \<open>Insertion\<close>
 
 text \<open>The function definitions are based on the book by Okasaki.\<close>
 
 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
 
 context ord begin
 
 lemma balance_rbt_greater[simp]: "(v \<guillemotleft>| balance a k x b) = (v \<guillemotleft>| a \<and> v \<guillemotleft>| b \<and> v < k)" 
   by (induct a k x b rule: balance.induct) auto
 
 lemma balance_rbt_less[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"
   by (induct a k x b rule: balance.induct) auto
 
 end
 
 lemma (in linorder) balance_rbt_sorted: 
   fixes k :: "'a"
   assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
   shows "rbt_sorted (balance l k v r)"
 using assms proof (induct l k v r rule: balance.induct)
   case ("2_2" a x w b y t c z s va vb vd vc)
   hence "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" 
     by (auto simp add: rbt_ord_props)
   hence "y \<guillemotleft>| (Branch B va vb vd vc)" by (blast dest: rbt_greater_trans)
   with "2_2" show ?case by simp
 next
   case ("3_2" va vb vd vc x w b y s c z)
   from "3_2" have "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" 
     by simp
   hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
   with "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> z \<guillemotleft>| Branch B va vb vd vc" by simp
   hence "y \<guillemotleft>| Branch B va vb vd vc" by (blast dest: rbt_greater_trans)
   with "3_3" show ?case by simp
 next
   case ("3_4" vd ve vg vf x w b y s c z t va vb vii vc)
   hence "x < y \<and> Branch B vd ve vg vf |\<guillemotleft> x" by simp
   hence 1: "Branch B vd ve vg vf |\<guillemotleft> y" by (blast dest: rbt_less_trans)
   from "3_4" have "y < z \<and> z \<guillemotleft>| Branch B va vb vii vc" by simp
   hence "y \<guillemotleft>| Branch B va vb vii vc" by (blast dest: rbt_greater_trans)
   with 1 "3_4" show ?case by simp
 next
   case ("4_2" va vb vd vc x w b y s c z t dd)
   hence "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" by simp
   hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
   with "4_2" show ?case by simp
 next
   case ("5_2" x w b y s c z t va vb vd vc)
   hence "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" by simp
   hence "y \<guillemotleft>| Branch B va vb vd vc" by (blast dest: rbt_greater_trans)
   with "5_2" show ?case by simp
 next
   case ("5_3" va vb vd vc x w b y s c z t)
   hence "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" by simp
   hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
   with "5_3" show ?case by simp
 next
   case ("5_4" va vb vg vc x w b y s c z t vd ve vii vf)
   hence "x < y \<and> Branch B va vb vg vc |\<guillemotleft> x" by simp
   hence 1: "Branch B va vb vg vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
   from "5_4" have "y < z \<and> z \<guillemotleft>| Branch B vd ve vii vf" by simp
   hence "y \<guillemotleft>| Branch B vd ve vii vf" by (blast dest: rbt_greater_trans)
   with 1 "5_4" show ?case by simp
 qed simp+
 
 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 (in linorder) rbt_lookup_balance[simp]: 
 fixes k :: "'a"
 assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
 shows "rbt_lookup (balance l k v r) x = rbt_lookup (Branch B l k v r) x"
 by (rule rbt_lookup_from_in_tree) (auto simp:assms balance_in_tree balance_rbt_sorted)
 
 primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
   "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_in_tree[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
 
 context ord begin
 
 lemma paint_rbt_sorted[simp]: "rbt_sorted t \<Longrightarrow> rbt_sorted (paint c t)" by (cases t) auto
 lemma paint_rbt_lookup[simp]: "rbt_lookup (paint c t) = rbt_lookup t" by (rule ext) (cases t, auto)
 lemma paint_rbt_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
 lemma paint_rbt_less[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
 
 fun
   rbt_ins :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
   "rbt_ins f k v Empty = Branch R Empty k v Empty" |
   "rbt_ins f k v (Branch B l x y r) = (if k < x then balance (rbt_ins f k v l) x y r
                                        else if k > x then balance l x y (rbt_ins f k v r)
                                        else Branch B l x (f k y v) r)" |
   "rbt_ins f k v (Branch R l x y r) = (if k < x then Branch R (rbt_ins f k v l) x y r
                                        else if k > x then Branch R l x y (rbt_ins f k v r)
                                        else Branch R l x (f k y v) r)"
 
 lemma ins_inv1_inv2: 
   assumes "inv1 t" "inv2 t"
   shows "inv2 (rbt_ins f k x t)" "bheight (rbt_ins f k x t) = bheight t" 
   "color_of t = B \<Longrightarrow> inv1 (rbt_ins f k x t)" "inv1l (rbt_ins f k x t)"
   using assms
   by (induct f k x t rule: rbt_ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
 
 end
 
 context linorder begin
 
 lemma ins_rbt_greater[simp]: "(v \<guillemotleft>| rbt_ins f (k :: 'a) x t) = (v \<guillemotleft>| t \<and> k > v)"
   by (induct f k x t rule: rbt_ins.induct) auto
 lemma ins_rbt_less[simp]: "(rbt_ins f k x t |\<guillemotleft> v) = (t |\<guillemotleft> v \<and> k < v)"
   by (induct f k x t rule: rbt_ins.induct) auto
 lemma ins_rbt_sorted[simp]: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_ins f k x t)"
   by (induct f k x t rule: rbt_ins.induct) (auto simp: balance_rbt_sorted)
 
 lemma keys_ins: "set (keys (rbt_ins f k v t)) = { k } \<union> set (keys t)"
   by (induct f k v t rule: rbt_ins.induct) auto
 
 lemma rbt_lookup_ins: 
   fixes k :: "'a"
   assumes "rbt_sorted t"
   shows "rbt_lookup (rbt_ins f k v t) x = ((rbt_lookup t)(k |-> case rbt_lookup t k of None \<Rightarrow> v 
                                                                 | Some w \<Rightarrow> f k w v)) x"
 using assms by (induct f k v t rule: rbt_ins.induct) auto
 
 end
 
 context ord begin
 
 definition rbt_insert_with_key :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where "rbt_insert_with_key f k v t = paint B (rbt_ins f k v t)"
 
 definition rbt_insertw_def: "rbt_insert_with f = rbt_insert_with_key (\<lambda>_. f)"
 
 definition rbt_insert :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_insert = rbt_insert_with_key (\<lambda>_ _ nv. nv)"
 
 end
 
 context linorder begin
 
 lemma rbt_insertwk_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert_with_key f (k :: 'a) x t)"
   by (auto simp: rbt_insert_with_key_def)
 
 theorem rbt_insertwk_is_rbt: 
   assumes inv: "is_rbt t" 
   shows "is_rbt (rbt_insert_with_key f k x t)"
 using assms
 unfolding rbt_insert_with_key_def is_rbt_def
 by (auto simp: ins_inv1_inv2)
 
 lemma rbt_lookup_rbt_insertwk: 
   assumes "rbt_sorted t"
   shows "rbt_lookup (rbt_insert_with_key f k v t) x = ((rbt_lookup t)(k |-> case rbt_lookup t k of None \<Rightarrow> v 
                                                        | Some w \<Rightarrow> f k w v)) x"
 unfolding rbt_insert_with_key_def using assms
 by (simp add:rbt_lookup_ins)
 
 lemma rbt_insertw_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert_with f k v t)" 
   by (simp add: rbt_insertwk_rbt_sorted rbt_insertw_def)
 theorem rbt_insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (rbt_insert_with f k v t)"
   by (simp add: rbt_insertwk_is_rbt rbt_insertw_def)
 
 lemma rbt_lookup_rbt_insertw:
   "is_rbt t \<Longrightarrow>
     rbt_lookup (rbt_insert_with f k v t) =
       (rbt_lookup t)(k \<mapsto> (if k \<in> dom (rbt_lookup t) then f (the (rbt_lookup t k)) v else v))"
   by (rule ext, cases "rbt_lookup t k") (auto simp: rbt_lookup_rbt_insertwk dom_def rbt_insertw_def)
 
 lemma rbt_insert_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert k v t)"
   by (simp add: rbt_insertwk_rbt_sorted rbt_insert_def)
 theorem rbt_insert_is_rbt [simp]: "is_rbt t \<Longrightarrow> is_rbt (rbt_insert k v t)"
   by (simp add: rbt_insertwk_is_rbt rbt_insert_def)
 
 lemma rbt_lookup_rbt_insert: "is_rbt t \<Longrightarrow> rbt_lookup (rbt_insert k v t) = (rbt_lookup t)(k\<mapsto>v)"
   by (rule ext) (simp add: rbt_insert_def rbt_lookup_rbt_insertwk split: option.split)
 
 end
 
 subsection \<open>Deletion\<close>
 
 lemma bheight_paintR'[simp]: "color_of t = B \<Longrightarrow> bheight (paint R t) = bheight t - 1"
 by (cases t rule: rbt_cases) auto
 
 text \<open>
   The function definitions are based on the Haskell code by Stefan Kahrs
   at \<^url>\<open>http://www.cs.ukc.ac.uk/people/staff/smk/redblack/rb.html\<close>.
 \<close>
 
 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 (in linorder) balance_left_rbt_sorted: 
   "\<lbrakk> rbt_sorted l; rbt_sorted r; rbt_less k l; k \<guillemotleft>| r \<rbrakk> \<Longrightarrow> rbt_sorted (balance_left l k v r)"
 apply (induct l k v r rule: balance_left.induct)
 apply (auto simp: balance_rbt_sorted)
 apply (unfold rbt_greater_prop rbt_less_prop)
 by force+
 
 context order begin
 
 lemma balance_left_rbt_greater: 
   fixes k :: "'a"
   assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
   shows "k \<guillemotleft>| balance_left a x t b"
 using assms 
 by (induct a x t b rule: balance_left.induct) auto
 
 lemma balance_left_rbt_less: 
   fixes k :: "'a"
   assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
   shows "balance_left a x t b |\<guillemotleft> k"
 using assms
 by (induct a x t b rule: balance_left.induct) auto
 
 end
 
 lemma balance_left_in_tree: 
   assumes "inv1l l" "inv1 r" "bheight l + 1 = bheight r"
   shows "entry_in_tree k v (balance_left l a b r) = (entry_in_tree k v l \<or> k = a \<and> v = b \<or> entry_in_tree k v r)"
 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 (in linorder) balance_right_rbt_sorted:
   "\<lbrakk> rbt_sorted l; rbt_sorted r; rbt_less k l; k \<guillemotleft>| r \<rbrakk> \<Longrightarrow> rbt_sorted (balance_right l k v r)"
 apply (induct l k v r rule: balance_right.induct)
 apply (auto simp:balance_rbt_sorted)
 apply (unfold rbt_less_prop rbt_greater_prop)
 by force+
 
 context order begin
 
 lemma balance_right_rbt_greater: 
   fixes k :: "'a"
   assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
   shows "k \<guillemotleft>| balance_right a x t b"
 using assms by (induct a x t b rule: balance_right.induct) auto
 
 lemma balance_right_rbt_less: 
   fixes k :: "'a"
   assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
   shows "balance_right a x t b |\<guillemotleft> k"
 using assms by (induct a x t b rule: balance_right.induct) auto
 
 end
 
 lemma balance_right_in_tree:
   assumes "inv1 l" "inv1l r" "bheight l = bheight r + 1" "inv2 l" "inv2 r"
   shows "entry_in_tree x y (balance_right l k v r) = (entry_in_tree x y l \<or> x = k \<and> y = v \<or> entry_in_tree x y r)"
 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)
 
 context linorder begin
 
 lemma combine_rbt_greater[simp]: 
   fixes k :: "'a"
   assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r" 
   shows "k \<guillemotleft>| combine l r"
 using assms 
 by (induct l r rule: combine.induct)
    (auto simp: balance_left_rbt_greater split:rbt.splits color.splits)
 
 lemma combine_rbt_less[simp]: 
   fixes k :: "'a"
   assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k" 
   shows "combine l r |\<guillemotleft> k"
 using assms 
 by (induct l r rule: combine.induct)
    (auto simp: balance_left_rbt_less split:rbt.splits color.splits)
 
 lemma combine_rbt_sorted: 
   fixes k :: "'a"
   assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
   shows "rbt_sorted (combine l r)"
 using assms proof (induct l r rule: combine.induct)
   case (3 a x v b c y w d)
   hence ineqs: "a |\<guillemotleft> x" "x \<guillemotleft>| b" "b |\<guillemotleft> k" "k \<guillemotleft>| c" "c |\<guillemotleft> y" "y \<guillemotleft>| d"
     by auto
   with 3
   show ?case
     by (cases "combine b c" rule: rbt_cases)
       (auto, (metis combine_rbt_greater combine_rbt_less ineqs ineqs rbt_less_simps(2) rbt_greater_simps(2) rbt_greater_trans rbt_less_trans)+)
 next
   case (4 a x v b c y w d)
   hence "x < k \<and> rbt_greater k c" by simp
   hence "rbt_greater x c" by (blast dest: rbt_greater_trans)
   with 4 have 2: "rbt_greater x (combine b c)" by (simp add: combine_rbt_greater)
   from 4 have "k < y \<and> rbt_less k b" by simp
   hence "rbt_less y b" by (blast dest: rbt_less_trans)
   with 4 have 3: "rbt_less y (combine b c)" by (simp add: combine_rbt_less)
   show ?case
   proof (cases "combine b c" rule: rbt_cases)
     case Empty
     from 4 have "x < y \<and> rbt_greater y d" by auto
     hence "rbt_greater x d" by (blast dest: rbt_greater_trans)
     with 4 Empty have "rbt_sorted a" and "rbt_sorted (Branch B Empty y w d)"
       and "rbt_less x a" and "rbt_greater x (Branch B Empty y w d)" by auto
     with Empty show ?thesis by (simp add: balance_left_rbt_sorted)
   next
     case (Red lta va ka rta)
     with 2 4 have "x < va \<and> rbt_less x a" by simp
     hence 5: "rbt_less va a" by (blast dest: rbt_less_trans)
     from Red 3 4 have "va < y \<and> rbt_greater y d" by simp
     hence "rbt_greater va d" by (blast dest: rbt_greater_trans)
     with Red 2 3 4 5 show ?thesis by simp
   next
     case (Black lta va ka rta)
     from 4 have "x < y \<and> rbt_greater y d" by auto
     hence "rbt_greater x d" by (blast dest: rbt_greater_trans)
     with Black 2 3 4 have "rbt_sorted a" and "rbt_sorted (Branch B (combine b c) y w d)" 
       and "rbt_less x a" and "rbt_greater x (Branch B (combine b c) y w d)" by auto
     with Black show ?thesis by (simp add: balance_left_rbt_sorted)
   qed
 next
   case (5 va vb vd vc b x w c)
   hence "k < x \<and> rbt_less k (Branch B va vb vd vc)" by simp
   hence "rbt_less x (Branch B va vb vd vc)" by (blast dest: rbt_less_trans)
   with 5 show ?case by (simp add: combine_rbt_less)
 next
   case (6 a x v b va vb vd vc)
   hence "x < k \<and> rbt_greater k (Branch B va vb vd vc)" by simp
   hence "rbt_greater x (Branch B va vb vd vc)" by (blast dest: rbt_greater_trans)
   with 6 show ?case by (simp add: combine_rbt_greater)
 qed simp+
 
 end
 
 lemma combine_in_tree: 
   assumes "inv2 l" "inv2 r" "bheight l = bheight r" "inv1 l" "inv1 r"
   shows "entry_in_tree k v (combine l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
 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)
 
 context ord begin
 
 fun
   rbt_del_from_left :: "'a \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
   rbt_del_from_right :: "'a \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
   rbt_del :: "'a\<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
 where
   "rbt_del x Empty = Empty" |
   "rbt_del x (Branch c a y s b) = 
    (if x < y then rbt_del_from_left x a y s b 
     else (if x > y then rbt_del_from_right x a y s b else combine a b))" |
   "rbt_del_from_left x (Branch B lt z v rt) y s b = balance_left (rbt_del x (Branch B lt z v rt)) y s b" |
   "rbt_del_from_left x a y s b = Branch R (rbt_del x a) y s b" |
   "rbt_del_from_right x a y s (Branch B lt z v rt) = balance_right a y s (rbt_del x (Branch B lt z v rt))" | 
   "rbt_del_from_right x a y s b = Branch R a y s (rbt_del x b)"
 
 end
 
 context linorder begin
 
 lemma 
   assumes "inv2 lt" "inv1 lt"
   shows
   "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
    inv2 (rbt_del_from_left x lt k v rt) \<and> 
    bheight (rbt_del_from_left x lt k v rt) = bheight lt \<and> 
    (color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_left x lt k v rt) \<or> 
     (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_left x lt k v rt))"
   and "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
   inv2 (rbt_del_from_right x lt k v rt) \<and> 
   bheight (rbt_del_from_right x lt k v rt) = bheight lt \<and> 
   (color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_right x lt k v rt) \<or> 
    (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_right x lt k v rt))"
   and rbt_del_inv1_inv2: "inv2 (rbt_del x lt) \<and> (color_of lt = R \<and> bheight (rbt_del x lt) = bheight lt \<and> inv1 (rbt_del x lt) 
   \<or> color_of lt = B \<and> bheight (rbt_del x lt) = bheight lt - 1 \<and> inv1l (rbt_del x lt))"
 using assms
 proof (induct x lt k v rt and x lt k v rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
 case (2 y c _ y')
   have "y = y' \<or> y < y' \<or> y > y'" by auto
   thus ?case proof (elim disjE)
     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 
   rbt_del_from_left_rbt_less: "\<lbrakk> lt |\<guillemotleft> v; rt |\<guillemotleft> v; k < v\<rbrakk> \<Longrightarrow> rbt_del_from_left x lt k y rt |\<guillemotleft> v"
   and rbt_del_from_right_rbt_less: "\<lbrakk>lt |\<guillemotleft> v; rt |\<guillemotleft> v; k < v\<rbrakk> \<Longrightarrow> rbt_del_from_right x lt k y rt |\<guillemotleft> v"
   and rbt_del_rbt_less: "lt |\<guillemotleft> v \<Longrightarrow> rbt_del x lt |\<guillemotleft> v"
 by (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct) 
    (auto simp: balance_left_rbt_less balance_right_rbt_less)
 
 lemma rbt_del_from_left_rbt_greater: "\<lbrakk>v \<guillemotleft>| lt; v \<guillemotleft>| rt; k > v\<rbrakk> \<Longrightarrow> v \<guillemotleft>| rbt_del_from_left x lt k y rt"
   and rbt_del_from_right_rbt_greater: "\<lbrakk>v \<guillemotleft>| lt; v \<guillemotleft>| rt; k > v\<rbrakk> \<Longrightarrow> v \<guillemotleft>| rbt_del_from_right x lt k y rt"
   and rbt_del_rbt_greater: "v \<guillemotleft>| lt \<Longrightarrow> v \<guillemotleft>| rbt_del x lt"
 by (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
    (auto simp: balance_left_rbt_greater balance_right_rbt_greater)
 
 lemma "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> k; k \<guillemotleft>| rt\<rbrakk> \<Longrightarrow> rbt_sorted (rbt_del_from_left x lt k y rt)"
   and "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> k; k \<guillemotleft>| rt\<rbrakk> \<Longrightarrow> rbt_sorted (rbt_del_from_right x lt k y rt)"
   and rbt_del_rbt_sorted: "rbt_sorted lt \<Longrightarrow> rbt_sorted (rbt_del x lt)"
 proof (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
   case (3 x lta zz v rta yy ss bb)
   from 3 have "Branch B lta zz v rta |\<guillemotleft> yy" by simp
   hence "rbt_del x (Branch B lta zz v rta) |\<guillemotleft> yy" by (rule rbt_del_rbt_less)
   with 3 show ?case by (simp add: balance_left_rbt_sorted)
 next
   case ("4_2" x vaa vbb vdd vc yy ss bb)
   hence "Branch R vaa vbb vdd vc |\<guillemotleft> yy" by simp
   hence "rbt_del x (Branch R vaa vbb vdd vc) |\<guillemotleft> yy" by (rule rbt_del_rbt_less)
   with "4_2" show ?case by simp
 next
   case (5 x aa yy ss lta zz v rta) 
   hence "yy \<guillemotleft>| Branch B lta zz v rta" by simp
   hence "yy \<guillemotleft>| rbt_del x (Branch B lta zz v rta)" by (rule rbt_del_rbt_greater)
   with 5 show ?case by (simp add: balance_right_rbt_sorted)
 next
   case ("6_2" x aa yy ss vaa vbb vdd vc)
   hence "yy \<guillemotleft>| Branch R vaa vbb vdd vc" by simp
   hence "yy \<guillemotleft>| rbt_del x (Branch R vaa vbb vdd vc)" by (rule rbt_del_rbt_greater)
   with "6_2" show ?case by simp
 qed (auto simp: combine_rbt_sorted)
 
 lemma "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> kt; kt \<guillemotleft>| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_del_from_left x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
   and "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> kt; kt \<guillemotleft>| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_del_from_right x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
   and rbt_del_in_tree: "\<lbrakk>rbt_sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
 proof (induct x lt kt y rt and x lt kt y rt and x t rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
   case (2 xx c aa yy ss bb)
   have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
   from this 2 show ?case proof (elim disjE)
     assume "xx = yy"
     with 2 show ?thesis proof (cases "xx = k")
       case True
       from 2 \<open>xx = yy\<close> \<open>xx = k\<close> have "rbt_sorted (Branch c aa yy ss bb) \<and> k = yy" by simp
       hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: rbt_less_nit rbt_greater_prop)
       with \<open>xx = yy\<close> 2 \<open>xx = k\<close> 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)
   define mt where [simp]: "mt = Branch B lta zz vv rta"
   from 3 have "inv2 mt \<and> inv1 mt" by simp
   hence "inv2 (rbt_del xx mt) \<and> (color_of mt = R \<and> bheight (rbt_del xx mt) = bheight mt \<and> inv1 (rbt_del xx mt) \<or> color_of mt = B \<and> bheight (rbt_del xx mt) = bheight mt - 1 \<and> inv1l (rbt_del xx mt))" by (blast dest: rbt_del_inv1_inv2)
   with 3 have 4: "entry_in_tree k v (rbt_del_from_left xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> entry_in_tree k v mt \<or> (k = yy \<and> v = ss) \<or> entry_in_tree k v bb)" by (simp add: balance_left_in_tree)
   thus ?case proof (cases "xx = k")
     case True
     from 3 True have "yy \<guillemotleft>| bb \<and> yy > k" by simp
     hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
     with 3 4 True show ?thesis by (auto simp: rbt_greater_nit)
   qed auto
 next
   case ("4_1" xx yy ss bb)
   show ?case proof (cases "xx = k")
     case True
     with "4_1" have "yy \<guillemotleft>| bb \<and> k < yy" by simp
     hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
     with "4_1" \<open>xx = k\<close> 
    have "entry_in_tree k v (Branch R Empty yy ss bb) = entry_in_tree k v Empty" by (auto simp: rbt_greater_nit)
     thus ?thesis by auto
   qed simp+
 next
   case ("4_2" xx vaa vbb vdd vc yy ss bb)
   thus ?case proof (cases "xx = k")
     case True
     with "4_2" have "k < yy \<and> yy \<guillemotleft>| bb" by simp
     hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
     with True "4_2" show ?thesis by (auto simp: rbt_greater_nit)
   qed auto
 next
   case (5 xx aa yy ss lta zz vv rta)
   define mt where [simp]: "mt = Branch B lta zz vv rta"
   from 5 have "inv2 mt \<and> inv1 mt" by simp
   hence "inv2 (rbt_del xx mt) \<and> (color_of mt = R \<and> bheight (rbt_del xx mt) = bheight mt \<and> inv1 (rbt_del xx mt) \<or> color_of mt = B \<and> bheight (rbt_del xx mt) = bheight mt - 1 \<and> inv1l (rbt_del xx mt))" by (blast dest: rbt_del_inv1_inv2)
   with 5 have 3: "entry_in_tree k v (rbt_del_from_right xx aa yy ss mt) = (entry_in_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> entry_in_tree k v mt)" by (simp add: balance_right_in_tree)
   thus ?case proof (cases "xx = k")
     case True
     from 5 True have "aa |\<guillemotleft> yy \<and> yy < k" by simp
     hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
     with 3 5 True show ?thesis by (auto simp: rbt_less_nit)
   qed auto
 next
   case ("6_1" xx aa yy ss)
   show ?case proof (cases "xx = k")
     case True
     with "6_1" have "aa |\<guillemotleft> yy \<and> k > yy" by simp
     hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
     with "6_1" \<open>xx = k\<close> show ?thesis by (auto simp: rbt_less_nit)
   qed simp
 next
   case ("6_2" xx aa yy ss vaa vbb vdd vc)
   thus ?case proof (cases "xx = k")
     case True
     with "6_2" have "k > yy \<and> aa |\<guillemotleft> yy" by simp
     hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
     with True "6_2" show ?thesis by (auto simp: rbt_less_nit)
   qed auto
 qed simp
 
 definition (in ord) rbt_delete where
   "rbt_delete k t = paint B (rbt_del k t)"
 
 theorem rbt_delete_is_rbt [simp]: assumes "is_rbt t" shows "is_rbt (rbt_delete k t)"
 proof -
   from assms have "inv2 t" and "inv1 t" unfolding is_rbt_def by auto 
   hence "inv2 (rbt_del k t) \<and> (color_of t = R \<and> bheight (rbt_del k t) = bheight t \<and> inv1 (rbt_del k t) \<or> color_of t = B \<and> bheight (rbt_del k t) = bheight t - 1 \<and> inv1l (rbt_del k t))" by (rule rbt_del_inv1_inv2)
   hence "inv2 (rbt_del k t) \<and> inv1l (rbt_del k t)" by (cases "color_of t") auto
   with assms show ?thesis
     unfolding is_rbt_def rbt_delete_def
     by (auto intro: paint_rbt_sorted rbt_del_rbt_sorted)
 qed
 
 lemma rbt_delete_in_tree: 
   assumes "is_rbt t" 
   shows "entry_in_tree k v (rbt_delete x t) = (x \<noteq> k \<and> entry_in_tree k v t)"
   using assms unfolding is_rbt_def rbt_delete_def
   by (auto simp: rbt_del_in_tree)
 
 lemma rbt_lookup_rbt_delete:
   assumes is_rbt: "is_rbt t"
   shows "rbt_lookup (rbt_delete k t) = (rbt_lookup t)|`(-{k})"
 proof
   fix x
   show "rbt_lookup (rbt_delete k t) x = (rbt_lookup t |` (-{k})) x" 
   proof (cases "x = k")
     assume "x = k" 
     with is_rbt show ?thesis
       by (cases "rbt_lookup (rbt_delete k t) k") (auto simp: rbt_lookup_in_tree rbt_delete_in_tree)
   next
     assume "x \<noteq> k"
     thus ?thesis
       by auto (metis is_rbt rbt_delete_is_rbt rbt_delete_in_tree is_rbt_rbt_sorted rbt_lookup_from_in_tree)
   qed
 qed
 
 end
 
 subsection \<open>Modifying existing entries\<close>
 
 context ord begin
 
 primrec
   rbt_map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
 where
   "rbt_map_entry k f Empty = Empty"
 | "rbt_map_entry k f (Branch c lt x v rt) =
     (if k < x then Branch c (rbt_map_entry k f lt) x v rt
     else if k > x then (Branch c lt x v (rbt_map_entry k f rt))
     else Branch c lt x (f v) rt)"
 
 
 lemma rbt_map_entry_color_of: "color_of (rbt_map_entry k f t) = color_of t" by (induct t) simp+
 lemma rbt_map_entry_inv1: "inv1 (rbt_map_entry k f t) = inv1 t" by (induct t) (simp add: rbt_map_entry_color_of)+
 lemma rbt_map_entry_inv2: "inv2 (rbt_map_entry k f t) = inv2 t" "bheight (rbt_map_entry k f t) = bheight t" by (induct t) simp+
 lemma rbt_map_entry_rbt_greater: "rbt_greater a (rbt_map_entry k f t) = rbt_greater a t" by (induct t) simp+
 lemma rbt_map_entry_rbt_less: "rbt_less a (rbt_map_entry k f t) = rbt_less a t" by (induct t) simp+
 lemma rbt_map_entry_rbt_sorted: "rbt_sorted (rbt_map_entry k f t) = rbt_sorted t"
   by (induct t) (simp_all add: rbt_map_entry_rbt_less rbt_map_entry_rbt_greater)
 
 theorem rbt_map_entry_is_rbt [simp]: "is_rbt (rbt_map_entry k f t) = is_rbt t" 
 unfolding is_rbt_def by (simp add: rbt_map_entry_inv2 rbt_map_entry_color_of rbt_map_entry_rbt_sorted rbt_map_entry_inv1 )
 
 end
 
 theorem (in linorder) rbt_lookup_rbt_map_entry:
   "rbt_lookup (rbt_map_entry k f t) = (rbt_lookup t)(k := map_option f (rbt_lookup t k))"
   by (induct t) (auto split: option.splits simp add: fun_eq_iff)
 
 subsection \<open>Mapping all entries\<close>
 
 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_color_of: "color_of (map f t) = color_of t" by (induct t) simp+
 lemma map_inv1: "inv1 (map f t) = inv1 t" by (induct t) (simp add: map_color_of)+
 lemma map_inv2: "inv2 (map f t) = inv2 t" "bheight (map f t) = bheight t" by (induct t) simp+
 
 context ord begin
 
 lemma map_rbt_greater: "rbt_greater k (map f t) = rbt_greater k t" by (induct t) simp+
 lemma map_rbt_less: "rbt_less k (map f t) = rbt_less k t" by (induct t) simp+
 lemma map_rbt_sorted: "rbt_sorted (map f t) = rbt_sorted t"  by (induct t) (simp add: map_rbt_less map_rbt_greater)+
 theorem map_is_rbt [simp]: "is_rbt (map f t) = is_rbt t" 
 unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_rbt_sorted map_color_of)
 
 end
 
 theorem (in linorder) rbt_lookup_map: "rbt_lookup (map f t) x = map_option (f x) (rbt_lookup t x)"
   apply(induct t)
   apply auto
   apply(rename_tac a b c, subgoal_tac "x = a")
   apply auto
   done
  (* FIXME: simproc "antisym less" does not work for linorder context, only for linorder type class
     by (induct t) auto *)
 
 hide_const (open) map
 
 subsection \<open>Folding over entries\<close>
 
 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
   "fold f t = List.fold (case_prod f) (entries t)"
 
 lemma fold_simps [simp]:
   "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 fun_eq_iff)
 
 lemma fold_code [code]:
   "fold f Empty x = x"
   "fold f (Branch c lt k v rt) x = fold f rt (f k v (fold f lt x))"
 by(simp_all)
 
 \<comment> \<open>fold with continuation predicate\<close>
 fun foldi :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a :: linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" 
   where
   "foldi c f Empty s = s" |
   "foldi c f (Branch col l k v r) s = (
     if (c s) then
       let s' = foldi c f l s in
         if (c s') then
           foldi c f r (f k v s')
         else s'
     else 
       s
   )"
 
 subsection \<open>Bulkloading a tree\<close>
 
 definition (in ord) rbt_bulkload :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where
   "rbt_bulkload xs = foldr (\<lambda>(k, v). rbt_insert k v) xs Empty"
 
 context linorder begin
 
 lemma rbt_bulkload_is_rbt [simp, intro]:
   "is_rbt (rbt_bulkload xs)"
   unfolding rbt_bulkload_def by (induct xs) auto
 
 lemma rbt_lookup_rbt_bulkload:
   "rbt_lookup (rbt_bulkload xs) = map_of xs"
 proof -
   obtain ys where "ys = rev xs" by simp
   have "\<And>t. is_rbt t \<Longrightarrow>
     rbt_lookup (List.fold (case_prod rbt_insert) ys t) = rbt_lookup t ++ map_of (rev ys)"
       by (induct ys) (simp_all add: rbt_bulkload_def rbt_lookup_rbt_insert case_prod_beta)
   from this Empty_is_rbt have
     "rbt_lookup (List.fold (case_prod rbt_insert) (rev xs) Empty) = rbt_lookup Empty ++ map_of xs"
      by (simp add: \<open>ys = rev xs\<close>)
   then show ?thesis by (simp add: rbt_bulkload_def rbt_lookup_Empty foldr_conv_fold)
 qed
 
 end
 
 
 
 subsection \<open>Building a RBT from a sorted list\<close>
 
 text \<open>
   These functions have been adapted from 
   Andrew W. Appel, Efficient Verified Red-Black Trees (September 2011) 
 \<close>
 
 fun rbtreeify_f :: "nat \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt \<times> ('a \<times> 'b) list"
   and rbtreeify_g :: "nat \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt \<times> ('a \<times> 'b) list"
 where
   "rbtreeify_f n kvs =
    (if n = 0 then (Empty, kvs)
     else if n = 1 then
       case kvs of (k, v) # kvs' \<Rightarrow> (Branch R Empty k v Empty, kvs')
     else if (n mod 2 = 0) then
       case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
         apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs')
     else case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
         apfst (Branch B t1 k v) (rbtreeify_f (n div 2) kvs'))"
 
 | "rbtreeify_g n kvs =
    (if n = 0 \<or> n = 1 then (Empty, kvs)
     else if n mod 2 = 0 then
       case rbtreeify_g (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
         apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs')
     else case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
         apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs'))"
 
 definition rbtreeify :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt"
 where "rbtreeify kvs = fst (rbtreeify_g (Suc (length kvs)) kvs)"
 
 declare rbtreeify_f.simps [simp del] rbtreeify_g.simps [simp del]
 
 lemma rbtreeify_f_code [code]:
   "rbtreeify_f n kvs =
    (if n = 0 then (Empty, kvs)
     else if n = 1 then
       case kvs of (k, v) # kvs' \<Rightarrow> 
         (Branch R Empty k v Empty, kvs')
     else let (n', r) = Divides.divmod_nat n 2 in
       if r = 0 then
         case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
       else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_f n' kvs'))"
 by (subst rbtreeify_f.simps) (simp only: Let_def divmod_nat_def prod.case)
 
 lemma rbtreeify_g_code [code]:
   "rbtreeify_g n kvs =
    (if n = 0 \<or> n = 1 then (Empty, kvs)
     else let (n', r) = Divides.divmod_nat n 2 in
       if r = 0 then
         case rbtreeify_g n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
       else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_g n' kvs'))"
 by(subst rbtreeify_g.simps)(simp only: Let_def divmod_nat_def prod.case)
 
 lemma Suc_double_half: "Suc (2 * n) div 2 = n"
 by simp
 
 lemma div2_plus_div2: "n div 2 + n div 2 = (n :: nat) - n mod 2"
 by arith
 
 lemma rbtreeify_f_rec_aux_lemma:
   "\<lbrakk>k - n div 2 = Suc k'; n \<le> k; n mod 2 = Suc 0\<rbrakk>
   \<Longrightarrow> k' - n div 2 = k - n"
 apply(rule add_right_imp_eq[where a = "n - n div 2"])
 apply(subst add_diff_assoc2, arith)
 apply(simp add: div2_plus_div2)
 done
 
 lemma rbtreeify_f_simps:
   "rbtreeify_f 0 kvs = (Empty, kvs)"
   "rbtreeify_f (Suc 0) ((k, v) # kvs) = 
   (Branch R Empty k v Empty, kvs)"
   "0 < n \<Longrightarrow> rbtreeify_f (2 * n) kvs =
    (case rbtreeify_f n kvs of (t1, (k, v) # kvs') \<Rightarrow>
      apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
   "0 < n \<Longrightarrow> rbtreeify_f (Suc (2 * n)) kvs =
    (case rbtreeify_f n kvs of (t1, (k, v) # kvs') \<Rightarrow> 
      apfst (Branch B t1 k v) (rbtreeify_f n kvs'))"
 by(subst (1) rbtreeify_f.simps, simp add: Suc_double_half)+
 
 lemma rbtreeify_g_simps:
   "rbtreeify_g 0 kvs = (Empty, kvs)"
   "rbtreeify_g (Suc 0) kvs = (Empty, kvs)"
   "0 < n \<Longrightarrow> rbtreeify_g (2 * n) kvs =
    (case rbtreeify_g n kvs of (t1, (k, v) # kvs') \<Rightarrow> 
      apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
   "0 < n \<Longrightarrow> rbtreeify_g (Suc (2 * n)) kvs =
    (case rbtreeify_f n kvs of (t1, (k, v) # kvs') \<Rightarrow> 
      apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
 by(subst (1) rbtreeify_g.simps, simp add: Suc_double_half)+
 
 declare rbtreeify_f_simps[simp] rbtreeify_g_simps[simp]
 
 lemma length_rbtreeify_f: "n \<le> length kvs
   \<Longrightarrow> length (snd (rbtreeify_f n kvs)) = length kvs - n"
   and length_rbtreeify_g:"\<lbrakk> 0 < n; n \<le> Suc (length kvs) \<rbrakk>
   \<Longrightarrow> length (snd (rbtreeify_g n kvs)) = Suc (length kvs) - n"
 proof(induction n kvs and n kvs rule: rbtreeify_f_rbtreeify_g.induct)
   case (1 n kvs)
   show ?case
   proof(cases "n \<le> 1")
     case True thus ?thesis using "1.prems"
       by(cases n kvs rule: nat.exhaust[case_product list.exhaust]) auto
   next
     case False
     hence "n \<noteq> 0" "n \<noteq> 1" by simp_all
     note IH = "1.IH"[OF this]
     show ?thesis
     proof(cases "n mod 2 = 0")
       case True
       hence "length (snd (rbtreeify_f n kvs)) = 
         length (snd (rbtreeify_f (2 * (n div 2)) kvs))"
         by(metis minus_nat.diff_0 minus_mod_eq_mult_div [symmetric])
       also from "1.prems" False obtain k v kvs' 
         where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
       also have "0 < n div 2" using False by(simp) 
       note rbtreeify_f_simps(3)[OF this]
       also note kvs[symmetric] 
       also let ?rest1 = "snd (rbtreeify_f (n div 2) kvs)"
       from "1.prems" have "n div 2 \<le> length kvs" by simp
       with True have len: "length ?rest1 = length kvs - n div 2" by(rule IH)
       with "1.prems" False obtain t1 k' v' kvs''
         where kvs'': "rbtreeify_f (n div 2) kvs = (t1, (k', v') # kvs'')"
          by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm)
       note this also note prod.case also note list.simps(5) 
       also note prod.case also note snd_apfst
       also have "0 < n div 2" "n div 2 \<le> Suc (length kvs'')" 
         using len "1.prems" False unfolding kvs'' by simp_all
       with True kvs''[symmetric] refl refl
       have "length (snd (rbtreeify_g (n div 2) kvs'')) = 
         Suc (length kvs'') - n div 2" by(rule IH)
       finally show ?thesis using len[unfolded kvs''] "1.prems" True
         by(simp add: Suc_diff_le[symmetric] mult_2[symmetric] minus_mod_eq_mult_div [symmetric])
     next
       case False
       hence "length (snd (rbtreeify_f n kvs)) = 
         length (snd (rbtreeify_f (Suc (2 * (n div 2))) kvs))"
         by (simp add: mod_eq_0_iff_dvd)
       also from "1.prems" \<open>\<not> n \<le> 1\<close> obtain k v kvs' 
         where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
       also have "0 < n div 2" using \<open>\<not> n \<le> 1\<close> by(simp) 
       note rbtreeify_f_simps(4)[OF this]
       also note kvs[symmetric] 
       also let ?rest1 = "snd (rbtreeify_f (n div 2) kvs)"
       from "1.prems" have "n div 2 \<le> length kvs" by simp
       with False have len: "length ?rest1 = length kvs - n div 2" by(rule IH)
       with "1.prems" \<open>\<not> n \<le> 1\<close> obtain t1 k' v' kvs''
         where kvs'': "rbtreeify_f (n div 2) kvs = (t1, (k', v') # kvs'')"
         by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm)
       note this also note prod.case also note list.simps(5)
       also note prod.case also note snd_apfst
       also have "n div 2 \<le> length kvs''" 
         using len "1.prems" False unfolding kvs'' by simp arith
       with False kvs''[symmetric] refl refl
       have "length (snd (rbtreeify_f (n div 2) kvs'')) = length kvs'' - n div 2"
         by(rule IH)
       finally show ?thesis using len[unfolded kvs''] "1.prems" False
         by simp(rule rbtreeify_f_rec_aux_lemma[OF sym])
     qed
   qed
 next
   case (2 n kvs)
   show ?case
   proof(cases "n > 1")
     case False with \<open>0 < n\<close> show ?thesis
       by(cases n kvs rule: nat.exhaust[case_product list.exhaust]) simp_all
   next
     case True
     hence "\<not> (n = 0 \<or> n = 1)" by simp
     note IH = "2.IH"[OF this]
     show ?thesis
     proof(cases "n mod 2 = 0")
       case True
       hence "length (snd (rbtreeify_g n kvs)) =
         length (snd (rbtreeify_g (2 * (n div 2)) kvs))"
         by(metis minus_nat.diff_0 minus_mod_eq_mult_div [symmetric])
       also from "2.prems" True obtain k v kvs' 
         where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
       also have "0 < n div 2" using \<open>1 < n\<close> by(simp) 
       note rbtreeify_g_simps(3)[OF this]
       also note kvs[symmetric] 
       also let ?rest1 = "snd (rbtreeify_g (n div 2) kvs)"
       from "2.prems" \<open>1 < n\<close>
       have "0 < n div 2" "n div 2 \<le> Suc (length kvs)" by simp_all
       with True have len: "length ?rest1 = Suc (length kvs) - n div 2" by(rule IH)
       with "2.prems" obtain t1 k' v' kvs''
         where kvs'': "rbtreeify_g (n div 2) kvs = (t1, (k', v') # kvs'')"
         by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm)
       note this also note prod.case also note list.simps(5) 
       also note prod.case also note snd_apfst
       also have "n div 2 \<le> Suc (length kvs'')" 
         using len "2.prems" unfolding kvs'' by simp
       with True kvs''[symmetric] refl refl \<open>0 < n div 2\<close>
       have "length (snd (rbtreeify_g (n div 2) kvs'')) = Suc (length kvs'') - n div 2"
         by(rule IH)
       finally show ?thesis using len[unfolded kvs''] "2.prems" True
         by(simp add: Suc_diff_le[symmetric] mult_2[symmetric] minus_mod_eq_mult_div [symmetric])
     next
       case False
       hence "length (snd (rbtreeify_g n kvs)) = 
         length (snd (rbtreeify_g (Suc (2 * (n div 2))) kvs))"
         by (simp add: mod_eq_0_iff_dvd)
       also from "2.prems" \<open>1 < n\<close> obtain k v kvs'
         where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
       also have "0 < n div 2" using \<open>1 < n\<close> by(simp)
       note rbtreeify_g_simps(4)[OF this]
       also note kvs[symmetric] 
       also let ?rest1 = "snd (rbtreeify_f (n div 2) kvs)"
       from "2.prems" have "n div 2 \<le> length kvs" by simp
       with False have len: "length ?rest1 = length kvs - n div 2" by(rule IH)
       with "2.prems" \<open>1 < n\<close> False obtain t1 k' v' kvs'' 
         where kvs'': "rbtreeify_f (n div 2) kvs = (t1, (k', v') # kvs'')"
         by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm, arith)
       note this also note prod.case also note list.simps(5) 
       also note prod.case also note snd_apfst
       also have "n div 2 \<le> Suc (length kvs'')" 
         using len "2.prems" False unfolding kvs'' by simp arith
       with False kvs''[symmetric] refl refl \<open>0 < n div 2\<close>
       have "length (snd (rbtreeify_g (n div 2) kvs'')) = Suc (length kvs'') - n div 2"
         by(rule IH)
       finally show ?thesis using len[unfolded kvs''] "2.prems" False
         by(simp add: div2_plus_div2)
     qed
   qed
 qed
 
 lemma rbtreeify_induct [consumes 1, case_names f_0 f_1 f_even f_odd g_0 g_1 g_even g_odd]:
   fixes P Q
   defines "f0 == (\<And>kvs. P 0 kvs)"
   and "f1 == (\<And>k v kvs. P (Suc 0) ((k, v) # kvs))"
   and "feven ==
     (\<And>n kvs t k v kvs'. \<lbrakk> n > 0; n \<le> length kvs; P n kvs; 
        rbtreeify_f n kvs = (t, (k, v) # kvs'); n \<le> Suc (length kvs'); Q n kvs' \<rbrakk> 
      \<Longrightarrow> P (2 * n) kvs)"
   and "fodd == 
     (\<And>n kvs t k v kvs'. \<lbrakk> n > 0; n \<le> length kvs; P n kvs;
        rbtreeify_f n kvs = (t, (k, v) # kvs'); n \<le> length kvs'; P n kvs' \<rbrakk> 
     \<Longrightarrow> P (Suc (2 * n)) kvs)"
   and "g0 == (\<And>kvs. Q 0 kvs)"
   and "g1 == (\<And>kvs. Q (Suc 0) kvs)"
   and "geven == 
     (\<And>n kvs t k v kvs'. \<lbrakk> n > 0; n \<le> Suc (length kvs); Q n kvs; 
        rbtreeify_g n kvs = (t, (k, v) # kvs'); n \<le> Suc (length kvs'); Q n kvs' \<rbrakk>
     \<Longrightarrow> Q (2 * n) kvs)"
   and "godd == 
     (\<And>n kvs t k v kvs'. \<lbrakk> n > 0; n \<le> length kvs; P n kvs;
        rbtreeify_f n kvs = (t, (k, v) # kvs'); n \<le> Suc (length kvs'); Q n kvs' \<rbrakk>
     \<Longrightarrow> Q (Suc (2 * n)) kvs)"
   shows "\<lbrakk> n \<le> length kvs; 
            PROP f0; PROP f1; PROP feven; PROP fodd; 
            PROP g0; PROP g1; PROP geven; PROP godd \<rbrakk>
          \<Longrightarrow> P n kvs"
   and "\<lbrakk> n \<le> Suc (length kvs);
           PROP f0; PROP f1; PROP feven; PROP fodd; 
           PROP g0; PROP g1; PROP geven; PROP godd \<rbrakk>
        \<Longrightarrow> Q n kvs"
 proof -
   assume f0: "PROP f0" and f1: "PROP f1" and feven: "PROP feven" and fodd: "PROP fodd"
     and g0: "PROP g0" and g1: "PROP g1" and geven: "PROP geven" and godd: "PROP godd"
   show "n \<le> length kvs \<Longrightarrow> P n kvs" and "n \<le> Suc (length kvs) \<Longrightarrow> Q n kvs"
   proof(induction rule: rbtreeify_f_rbtreeify_g.induct)
     case (1 n kvs)
     show ?case
     proof(cases "n \<le> 1")
       case True thus ?thesis using "1.prems"
         by(cases n kvs rule: nat.exhaust[case_product list.exhaust])
           (auto simp add: f0[unfolded f0_def] f1[unfolded f1_def])
     next
       case False 
       hence ns: "n \<noteq> 0" "n \<noteq> 1" by simp_all
       hence ge0: "n div 2 > 0" by simp
       note IH = "1.IH"[OF ns]
       show ?thesis
       proof(cases "n mod 2 = 0")
         case True note ge0 
         moreover from "1.prems" have n2: "n div 2 \<le> length kvs" by simp
         moreover from True n2 have "P (n div 2) kvs" by(rule IH)
         moreover from length_rbtreeify_f[OF n2] ge0 "1.prems" obtain t k v kvs' 
           where kvs': "rbtreeify_f (n div 2) kvs = (t, (k, v) # kvs')"
           by(cases "snd (rbtreeify_f (n div 2) kvs)")
             (auto simp add: snd_def split: prod.split_asm)
         moreover from "1.prems" length_rbtreeify_f[OF n2] ge0
         have n2': "n div 2 \<le> Suc (length kvs')" by(simp add: kvs')
         moreover from True kvs'[symmetric] refl refl n2'
         have "Q (n div 2) kvs'" by(rule IH)
         moreover note feven[unfolded feven_def]
           (* FIXME: why does by(rule feven[unfolded feven_def]) not work? *)
         ultimately have "P (2 * (n div 2)) kvs" by -
         thus ?thesis using True by (metis minus_mod_eq_div_mult [symmetric] minus_nat.diff_0 mult.commute)
       next
         case False note ge0
         moreover from "1.prems" have n2: "n div 2 \<le> length kvs" by simp
         moreover from False n2 have "P (n div 2) kvs" by(rule IH)
         moreover from length_rbtreeify_f[OF n2] ge0 "1.prems" obtain t k v kvs' 
           where kvs': "rbtreeify_f (n div 2) kvs = (t, (k, v) # kvs')"
           by(cases "snd (rbtreeify_f (n div 2) kvs)")
             (auto simp add: snd_def split: prod.split_asm)
         moreover from "1.prems" length_rbtreeify_f[OF n2] ge0 False
         have n2': "n div 2 \<le> length kvs'" by(simp add: kvs') arith
         moreover from False kvs'[symmetric] refl refl n2' have "P (n div 2) kvs'" by(rule IH)
         moreover note fodd[unfolded fodd_def]
         ultimately have "P (Suc (2 * (n div 2))) kvs" by -
         thus ?thesis using False 
           by simp (metis One_nat_def Suc_eq_plus1_left le_add_diff_inverse mod_less_eq_dividend minus_mod_eq_mult_div [symmetric])
       qed
     qed
   next
     case (2 n kvs)
     show ?case
     proof(cases "n \<le> 1")
       case True thus ?thesis using "2.prems"
         by(cases n kvs rule: nat.exhaust[case_product list.exhaust])
           (auto simp add: g0[unfolded g0_def] g1[unfolded g1_def])
     next
       case False 
       hence ns: "\<not> (n = 0 \<or> n = 1)" by simp
       hence ge0: "n div 2 > 0" by simp
       note IH = "2.IH"[OF ns]
       show ?thesis
       proof(cases "n mod 2 = 0")
         case True note ge0
         moreover from "2.prems" have n2: "n div 2 \<le> Suc (length kvs)" by simp
         moreover from True n2 have "Q (n div 2) kvs" by(rule IH)
         moreover from length_rbtreeify_g[OF ge0 n2] ge0 "2.prems" obtain t k v kvs' 
           where kvs': "rbtreeify_g (n div 2) kvs = (t, (k, v) # kvs')"
           by(cases "snd (rbtreeify_g (n div 2) kvs)")
             (auto simp add: snd_def split: prod.split_asm)
         moreover from "2.prems" length_rbtreeify_g[OF ge0 n2] ge0
         have n2': "n div 2 \<le> Suc (length kvs')" by(simp add: kvs')
         moreover from True kvs'[symmetric] refl refl  n2'
         have "Q (n div 2) kvs'" by(rule IH)
         moreover note geven[unfolded geven_def]
         ultimately have "Q (2 * (n div 2)) kvs" by -
         thus ?thesis using True 
           by(metis minus_mod_eq_div_mult [symmetric] minus_nat.diff_0 mult.commute)
       next
         case False note ge0
         moreover from "2.prems" have n2: "n div 2 \<le> length kvs" by simp
         moreover from False n2 have "P (n div 2) kvs" by(rule IH)
         moreover from length_rbtreeify_f[OF n2] ge0 "2.prems" False obtain t k v kvs' 
           where kvs': "rbtreeify_f (n div 2) kvs = (t, (k, v) # kvs')"
           by(cases "snd (rbtreeify_f (n div 2) kvs)")
             (auto simp add: snd_def split: prod.split_asm, arith)
         moreover from "2.prems" length_rbtreeify_f[OF n2] ge0 False
         have n2': "n div 2 \<le> Suc (length kvs')" by(simp add: kvs') arith
         moreover from False kvs'[symmetric] refl refl n2'
         have "Q (n div 2) kvs'" by(rule IH)
         moreover note godd[unfolded godd_def]
         ultimately have "Q (Suc (2 * (n div 2))) kvs" by -
         thus ?thesis using False 
           by simp (metis One_nat_def Suc_eq_plus1_left le_add_diff_inverse mod_less_eq_dividend minus_mod_eq_mult_div [symmetric])
       qed
     qed
   qed
 qed
 
 lemma inv1_rbtreeify_f: "n \<le> length kvs 
   \<Longrightarrow> inv1 (fst (rbtreeify_f n kvs))"
   and inv1_rbtreeify_g: "n \<le> Suc (length kvs)
   \<Longrightarrow> inv1 (fst (rbtreeify_g n kvs))"
 by(induct n kvs and n kvs rule: rbtreeify_induct) simp_all
 
 fun plog2 :: "nat \<Rightarrow> nat" 
 where "plog2 n = (if n \<le> 1 then 0 else plog2 (n div 2) + 1)"
 
 declare plog2.simps [simp del]
 
 lemma plog2_simps [simp]:
   "plog2 0 = 0" "plog2 (Suc 0) = 0"
   "0 < n \<Longrightarrow> plog2 (2 * n) = 1 + plog2 n"
   "0 < n \<Longrightarrow> plog2 (Suc (2 * n)) = 1 + plog2 n"
 by(subst plog2.simps, simp add: Suc_double_half)+
 
 lemma bheight_rbtreeify_f: "n \<le> length kvs
   \<Longrightarrow> bheight (fst (rbtreeify_f n kvs)) = plog2 n"
   and bheight_rbtreeify_g: "n \<le> Suc (length kvs)
   \<Longrightarrow> bheight (fst (rbtreeify_g n kvs)) = plog2 n"
 by(induct n kvs and n kvs rule: rbtreeify_induct) simp_all
 
 lemma bheight_rbtreeify_f_eq_plog2I:
   "\<lbrakk> rbtreeify_f n kvs = (t, kvs'); n \<le> length kvs \<rbrakk> 
   \<Longrightarrow> bheight t = plog2 n"
 using bheight_rbtreeify_f[of n kvs] by simp
 
 lemma bheight_rbtreeify_g_eq_plog2I: 
   "\<lbrakk> rbtreeify_g n kvs = (t, kvs'); n \<le> Suc (length kvs) \<rbrakk>
   \<Longrightarrow> bheight t = plog2 n"
 using bheight_rbtreeify_g[of n kvs] by simp
 
 hide_const (open) plog2
 
 lemma inv2_rbtreeify_f: "n \<le> length kvs
   \<Longrightarrow> inv2 (fst (rbtreeify_f n kvs))"
   and inv2_rbtreeify_g: "n \<le> Suc (length kvs)
   \<Longrightarrow> inv2 (fst (rbtreeify_g n kvs))"
 by(induct n kvs and n kvs rule: rbtreeify_induct)
   (auto simp add: bheight_rbtreeify_f bheight_rbtreeify_g 
         intro: bheight_rbtreeify_f_eq_plog2I bheight_rbtreeify_g_eq_plog2I)
 
 lemma "n \<le> length kvs \<Longrightarrow> True"
   and color_of_rbtreeify_g:
   "\<lbrakk> n \<le> Suc (length kvs); 0 < n \<rbrakk> 
   \<Longrightarrow> color_of (fst (rbtreeify_g n kvs)) = B"
 by(induct n kvs and n kvs rule: rbtreeify_induct) simp_all
 
 lemma entries_rbtreeify_f_append:
   "n \<le> length kvs 
   \<Longrightarrow> entries (fst (rbtreeify_f n kvs)) @ snd (rbtreeify_f n kvs) = kvs"
   and entries_rbtreeify_g_append: 
   "n \<le> Suc (length kvs) 
   \<Longrightarrow> entries (fst (rbtreeify_g n kvs)) @ snd (rbtreeify_g n kvs) = kvs"
 by(induction rule: rbtreeify_induct) simp_all
 
 lemma length_entries_rbtreeify_f:
   "n \<le> length kvs \<Longrightarrow> length (entries (fst (rbtreeify_f n kvs))) = n"
   and length_entries_rbtreeify_g: 
   "n \<le> Suc (length kvs) \<Longrightarrow> length (entries (fst (rbtreeify_g n kvs))) = n - 1"
 by(induct rule: rbtreeify_induct) simp_all
 
 lemma rbtreeify_f_conv_drop: 
   "n \<le> length kvs \<Longrightarrow> snd (rbtreeify_f n kvs) = drop n kvs"
 using entries_rbtreeify_f_append[of n kvs]
 by(simp add: append_eq_conv_conj length_entries_rbtreeify_f)
 
 lemma rbtreeify_g_conv_drop: 
   "n \<le> Suc (length kvs) \<Longrightarrow> snd (rbtreeify_g n kvs) = drop (n - 1) kvs"
 using entries_rbtreeify_g_append[of n kvs]
 by(simp add: append_eq_conv_conj length_entries_rbtreeify_g)
 
 lemma entries_rbtreeify_f [simp]:
   "n \<le> length kvs \<Longrightarrow> entries (fst (rbtreeify_f n kvs)) = take n kvs"
 using entries_rbtreeify_f_append[of n kvs]
 by(simp add: append_eq_conv_conj length_entries_rbtreeify_f)
 
 lemma entries_rbtreeify_g [simp]:
   "n \<le> Suc (length kvs) \<Longrightarrow> 
   entries (fst (rbtreeify_g n kvs)) = take (n - 1) kvs"
 using entries_rbtreeify_g_append[of n kvs]
 by(simp add: append_eq_conv_conj length_entries_rbtreeify_g)
 
 lemma keys_rbtreeify_f [simp]: "n \<le> length kvs
   \<Longrightarrow> keys (fst (rbtreeify_f n kvs)) = take n (map fst kvs)"
 by(simp add: keys_def take_map)
 
 lemma keys_rbtreeify_g [simp]: "n \<le> Suc (length kvs)
   \<Longrightarrow> keys (fst (rbtreeify_g n kvs)) = take (n - 1) (map fst kvs)"
 by(simp add: keys_def take_map)
 
 lemma rbtreeify_fD: 
   "\<lbrakk> rbtreeify_f n kvs = (t, kvs'); n \<le> length kvs \<rbrakk> 
   \<Longrightarrow> entries t = take n kvs \<and> kvs' = drop n kvs"
 using rbtreeify_f_conv_drop[of n kvs] entries_rbtreeify_f[of n kvs] by simp
 
 lemma rbtreeify_gD: 
   "\<lbrakk> rbtreeify_g n kvs = (t, kvs'); n \<le> Suc (length kvs) \<rbrakk>
   \<Longrightarrow> entries t = take (n - 1) kvs \<and> kvs' = drop (n - 1) kvs"
 using rbtreeify_g_conv_drop[of n kvs] entries_rbtreeify_g[of n kvs] by simp
 
 lemma entries_rbtreeify [simp]: "entries (rbtreeify kvs) = kvs"
 by(simp add: rbtreeify_def entries_rbtreeify_g)
 
 context linorder begin
 
 lemma rbt_sorted_rbtreeify_f: 
   "\<lbrakk> n \<le> length kvs; sorted (map fst kvs); distinct (map fst kvs) \<rbrakk> 
   \<Longrightarrow> rbt_sorted (fst (rbtreeify_f n kvs))"
   and rbt_sorted_rbtreeify_g: 
   "\<lbrakk> n \<le> Suc (length kvs); sorted (map fst kvs); distinct (map fst kvs) \<rbrakk>
   \<Longrightarrow> rbt_sorted (fst (rbtreeify_g n kvs))"
 proof(induction n kvs and n kvs rule: rbtreeify_induct)
   case (f_even n kvs t k v kvs')
   from rbtreeify_fD[OF \<open>rbtreeify_f n kvs = (t, (k, v) # kvs')\<close> \<open>n \<le> length kvs\<close>]
   have "entries t = take n kvs"
     and kvs': "drop n kvs = (k, v) # kvs'" by simp_all
   hence unfold: "kvs = take n kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
   from \<open>sorted (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take n kvs). x \<le> k) \<and> (\<forall>(x, y) \<in> set kvs'. k \<le> x)"
     by(subst (asm) unfold)(auto simp add: sorted_append)
   moreover from \<open>distinct (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take n kvs). x \<noteq> k) \<and> (\<forall>(x, y) \<in> set kvs'. x \<noteq> k)"
     by(subst (asm) unfold)(auto intro: rev_image_eqI)
   ultimately have "(\<forall>(x, y) \<in> set (take n kvs). x < k) \<and> (\<forall>(x, y) \<in> set kvs'. k < x)"
     by fastforce
   hence "fst (rbtreeify_f n kvs) |\<guillemotleft> k" "k \<guillemotleft>| fst (rbtreeify_g n kvs')"
     using \<open>n \<le> Suc (length kvs')\<close> \<open>n \<le> length kvs\<close> set_take_subset[of "n - 1" kvs']
     by(auto simp add: ord.rbt_greater_prop ord.rbt_less_prop take_map split_def)
   moreover from \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
   have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule f_even.IH)
   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
     using \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
     by(subst (asm) (1 2) unfold, simp add: sorted_append)+
   hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule f_even.IH)
   ultimately show ?case
     using \<open>0 < n\<close> \<open>rbtreeify_f n kvs = (t, (k, v) # kvs')\<close> by simp
 next
   case (f_odd n kvs t k v kvs')
   from rbtreeify_fD[OF \<open>rbtreeify_f n kvs = (t, (k, v) # kvs')\<close> \<open>n \<le> length kvs\<close>]
   have "entries t = take n kvs" 
     and kvs': "drop n kvs = (k, v) # kvs'" by simp_all
   hence unfold: "kvs = take n kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
   from \<open>sorted (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take n kvs). x \<le> k) \<and> (\<forall>(x, y) \<in> set kvs'. k \<le> x)"
     by(subst (asm) unfold)(auto simp add: sorted_append)
   moreover from \<open>distinct (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take n kvs). x \<noteq> k) \<and> (\<forall>(x, y) \<in> set kvs'. x \<noteq> k)"
     by(subst (asm) unfold)(auto intro: rev_image_eqI)
   ultimately have "(\<forall>(x, y) \<in> set (take n kvs). x < k) \<and> (\<forall>(x, y) \<in> set kvs'. k < x)"
     by fastforce
   hence "fst (rbtreeify_f n kvs) |\<guillemotleft> k" "k \<guillemotleft>| fst (rbtreeify_f n kvs')"
     using \<open>n \<le> length kvs'\<close> \<open>n \<le> length kvs\<close> set_take_subset[of n kvs']
     by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)
   moreover from \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
   have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule f_odd.IH)
   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
     using \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
     by(subst (asm) (1 2) unfold, simp add: sorted_append)+
   hence "rbt_sorted (fst (rbtreeify_f n kvs'))" by(rule f_odd.IH)
   ultimately show ?case 
     using \<open>0 < n\<close> \<open>rbtreeify_f n kvs = (t, (k, v) # kvs')\<close> by simp
 next
   case (g_even n kvs t k v kvs')
   from rbtreeify_gD[OF \<open>rbtreeify_g n kvs = (t, (k, v) # kvs')\<close> \<open>n \<le> Suc (length kvs)\<close>]
   have t: "entries t = take (n - 1) kvs" 
     and kvs': "drop (n - 1) kvs = (k, v) # kvs'" by simp_all
   hence unfold: "kvs = take (n - 1) kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
   from \<open>sorted (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take (n - 1) kvs). x \<le> k) \<and> (\<forall>(x, y) \<in> set kvs'. k \<le> x)"
     by(subst (asm) unfold)(auto simp add: sorted_append)
   moreover from \<open>distinct (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take (n - 1) kvs). x \<noteq> k) \<and> (\<forall>(x, y) \<in> set kvs'. x \<noteq> k)"
     by(subst (asm) unfold)(auto intro: rev_image_eqI)
   ultimately have "(\<forall>(x, y) \<in> set (take (n - 1) kvs). x < k) \<and> (\<forall>(x, y) \<in> set kvs'. k < x)"
     by fastforce
   hence "fst (rbtreeify_g n kvs) |\<guillemotleft> k" "k \<guillemotleft>| fst (rbtreeify_g n kvs')"
     using \<open>n \<le> Suc (length kvs')\<close> \<open>n \<le> Suc (length kvs)\<close> set_take_subset[of "n - 1" kvs']
     by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)
   moreover from \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
   have "rbt_sorted (fst (rbtreeify_g n kvs))" by(rule g_even.IH)
   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
     using \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
     by(subst (asm) (1 2) unfold, simp add: sorted_append)+
   hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule g_even.IH)
   ultimately show ?case using \<open>0 < n\<close> \<open>rbtreeify_g n kvs = (t, (k, v) # kvs')\<close> by simp
 next
   case (g_odd n kvs t k v kvs')
   from rbtreeify_fD[OF \<open>rbtreeify_f n kvs = (t, (k, v) # kvs')\<close> \<open>n \<le> length kvs\<close>]
   have "entries t = take n kvs"
     and kvs': "drop n kvs = (k, v) # kvs'" by simp_all
   hence unfold: "kvs = take n kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
   from \<open>sorted (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take n kvs). x \<le> k) \<and> (\<forall>(x, y) \<in> set kvs'. k \<le> x)"
     by(subst (asm) unfold)(auto simp add: sorted_append)
   moreover from \<open>distinct (map fst kvs)\<close> kvs'
   have "(\<forall>(x, y) \<in> set (take n kvs). x \<noteq> k) \<and> (\<forall>(x, y) \<in> set kvs'. x \<noteq> k)"
     by(subst (asm) unfold)(auto intro: rev_image_eqI)
   ultimately have "(\<forall>(x, y) \<in> set (take n kvs). x < k) \<and> (\<forall>(x, y) \<in> set kvs'. k < x)"
     by fastforce
   hence "fst (rbtreeify_f n kvs) |\<guillemotleft> k" "k \<guillemotleft>| fst (rbtreeify_g n kvs')"
     using \<open>n \<le> Suc (length kvs')\<close> \<open>n \<le> length kvs\<close> set_take_subset[of "n - 1" kvs']
     by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)
   moreover from \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
   have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule g_odd.IH)
   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
     using \<open>sorted (map fst kvs)\<close> \<open>distinct (map fst kvs)\<close>
     by(subst (asm) (1 2) unfold, simp add: sorted_append)+
   hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule g_odd.IH)
   ultimately show ?case
     using \<open>0 < n\<close> \<open>rbtreeify_f n kvs = (t, (k, v) # kvs')\<close> by simp
 qed simp_all
 
 lemma rbt_sorted_rbtreeify: 
   "\<lbrakk> sorted (map fst kvs); distinct (map fst kvs) \<rbrakk> \<Longrightarrow> rbt_sorted (rbtreeify kvs)"
 by(simp add: rbtreeify_def rbt_sorted_rbtreeify_g)
 
 lemma is_rbt_rbtreeify: 
   "\<lbrakk> sorted (map fst kvs); distinct (map fst kvs) \<rbrakk>
   \<Longrightarrow> is_rbt (rbtreeify kvs)"
 by(simp add: is_rbt_def rbtreeify_def inv1_rbtreeify_g inv2_rbtreeify_g rbt_sorted_rbtreeify_g color_of_rbtreeify_g)
 
 lemma rbt_lookup_rbtreeify:
   "\<lbrakk> sorted (map fst kvs); distinct (map fst kvs) \<rbrakk> \<Longrightarrow> 
   rbt_lookup (rbtreeify kvs) = map_of kvs"
 by(simp add: map_of_entries[symmetric] rbt_sorted_rbtreeify)
 
 end
 
 text \<open>
   Functions to compare the height of two rbt trees, taken from 
   Andrew W. Appel, Efficient Verified Red-Black Trees (September 2011)
 \<close>
 
 fun skip_red :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
 where
   "skip_red (Branch color.R l k v r) = l"
 | "skip_red t = t"
 
 definition skip_black :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
 where
   "skip_black t = (let t' = skip_red t in case t' of Branch color.B l k v r \<Rightarrow> l | _ \<Rightarrow> t')"
 
 datatype compare = LT | GT | EQ
 
 partial_function (tailrec) compare_height :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> compare"
 where
   "compare_height sx s t tx =
   (case (skip_red sx, skip_red s, skip_red t, skip_red tx) of
      (Branch _ sx' _ _ _, Branch _ s' _ _ _, Branch _ t' _ _ _, Branch _ tx' _ _ _) \<Rightarrow> 
        compare_height (skip_black sx') s' t' (skip_black tx')
    | (_, rbt.Empty, _, Branch _ _ _ _ _) \<Rightarrow> LT
    | (Branch _ _ _ _ _, _, rbt.Empty, _) \<Rightarrow> GT
    | (Branch _ sx' _ _ _, Branch _ s' _ _ _, Branch _ t' _ _ _, rbt.Empty) \<Rightarrow>
        compare_height (skip_black sx') s' t' rbt.Empty
    | (rbt.Empty, Branch _ s' _ _ _, Branch _ t' _ _ _, Branch _ tx' _ _ _) \<Rightarrow>
        compare_height rbt.Empty s' t' (skip_black tx')
    | _ \<Rightarrow> EQ)"
 
 declare compare_height.simps [code]
 
 hide_type (open) compare
 hide_const (open)
   compare_height skip_black skip_red LT GT EQ case_compare rec_compare
   Abs_compare Rep_compare
 hide_fact (open)
   Abs_compare_cases Abs_compare_induct Abs_compare_inject Abs_compare_inverse
   Rep_compare Rep_compare_cases Rep_compare_induct Rep_compare_inject Rep_compare_inverse
   compare.simps compare.exhaust compare.induct compare.rec compare.simps
   compare.size compare.case_cong compare.case_cong_weak compare.case
   compare.nchotomy compare.split compare.split_asm compare.eq.refl compare.eq.simps
   equal_compare_def
   skip_red.simps skip_red.cases skip_red.induct 
   skip_black_def
   compare_height.simps
 
 subsection \<open>union and intersection of sorted associative lists\<close>
 
 context ord begin
 
 function sunion_with :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list" 
 where
   "sunion_with f ((k, v) # as) ((k', v') # bs) =
    (if k > k' then (k', v') # sunion_with f ((k, v) # as) bs
     else if k < k' then (k, v) # sunion_with f as ((k', v') # bs)
     else (k, f k v v') # sunion_with f as bs)"
 | "sunion_with f [] bs = bs"
 | "sunion_with f as [] = as"
 by pat_completeness auto
 termination by lexicographic_order
 
 function sinter_with :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"
 where
   "sinter_with f ((k, v) # as) ((k', v') # bs) =
   (if k > k' then sinter_with f ((k, v) # as) bs
    else if k < k' then sinter_with f as ((k', v') # bs)
    else (k, f k v v') # sinter_with f as bs)"
 | "sinter_with f [] _ = []"
 | "sinter_with f _ [] = []"
 by pat_completeness auto
 termination by lexicographic_order
 
 end
 
 declare ord.sunion_with.simps [code] ord.sinter_with.simps[code]
 
 context linorder begin
 
 lemma set_fst_sunion_with: 
   "set (map fst (sunion_with f xs ys)) = set (map fst xs) \<union> set (map fst ys)"
 by(induct f xs ys rule: sunion_with.induct) auto
 
 lemma sorted_sunion_with [simp]:
   "\<lbrakk> sorted (map fst xs); sorted (map fst ys) \<rbrakk> 
   \<Longrightarrow> sorted (map fst (sunion_with f xs ys))"
 by(induct f xs ys rule: sunion_with.induct)
   (auto simp add: set_fst_sunion_with simp del: set_map)
 
 lemma distinct_sunion_with [simp]:
   "\<lbrakk> distinct (map fst xs); distinct (map fst ys); sorted (map fst xs); sorted (map fst ys) \<rbrakk>
   \<Longrightarrow> distinct (map fst (sunion_with f xs ys))"
 proof(induct f xs ys rule: sunion_with.induct)
   case (1 f k v xs k' v' ys)
   have "\<lbrakk> \<not> k < k'; \<not> k' < k \<rbrakk> \<Longrightarrow> k = k'" by simp
   thus ?case using "1"
     by(auto simp add: set_fst_sunion_with simp del: set_map)
 qed simp_all
 
 lemma map_of_sunion_with: 
   "\<lbrakk> sorted (map fst xs); sorted (map fst ys) \<rbrakk>
   \<Longrightarrow> map_of (sunion_with f xs ys) k = 
   (case map_of xs k of None \<Rightarrow> map_of ys k 
   | Some v \<Rightarrow> case map_of ys k of None \<Rightarrow> Some v 
               | Some w \<Rightarrow> Some (f k v w))"
 by(induct f xs ys rule: sunion_with.induct)(auto split: option.split dest: map_of_SomeD bspec)
 
 lemma set_fst_sinter_with [simp]:
   "\<lbrakk> sorted (map fst xs); sorted (map fst ys) \<rbrakk>
   \<Longrightarrow> set (map fst (sinter_with f xs ys)) = set (map fst xs) \<inter> set (map fst ys)"
 by(induct f xs ys rule: sinter_with.induct)(auto simp del: set_map)
 
 lemma set_fst_sinter_with_subset1:
   "set (map fst (sinter_with f xs ys)) \<subseteq> set (map fst xs)"
 by(induct f xs ys rule: sinter_with.induct) auto
 
 lemma set_fst_sinter_with_subset2:
   "set (map fst (sinter_with f xs ys)) \<subseteq> set (map fst ys)"
 by(induct f xs ys rule: sinter_with.induct)(auto simp del: set_map)
 
 lemma sorted_sinter_with [simp]:
   "\<lbrakk> sorted (map fst xs); sorted (map fst ys) \<rbrakk>
   \<Longrightarrow> sorted (map fst (sinter_with f xs ys))"
 by(induct f xs ys rule: sinter_with.induct)(auto simp del: set_map)
 
 lemma distinct_sinter_with [simp]:
   "\<lbrakk> distinct (map fst xs); distinct (map fst ys) \<rbrakk>
   \<Longrightarrow> distinct (map fst (sinter_with f xs ys))"
 proof(induct f xs ys rule: sinter_with.induct)
   case (1 f k v as k' v' bs)
   have "\<lbrakk> \<not> k < k'; \<not> k' < k \<rbrakk> \<Longrightarrow> k = k'" by simp
   thus ?case using "1" set_fst_sinter_with_subset1[of f as bs]
     set_fst_sinter_with_subset2[of f as bs]
     by(auto simp del: set_map)
 qed simp_all
 
 lemma map_of_sinter_with:
   "\<lbrakk> sorted (map fst xs); sorted (map fst ys) \<rbrakk>
   \<Longrightarrow> map_of (sinter_with f xs ys) k = 
   (case map_of xs k of None \<Rightarrow> None | Some v \<Rightarrow> map_option (f k v) (map_of ys k))"
 apply(induct f xs ys rule: sinter_with.induct)
 apply(auto simp add: map_option_case split: option.splits dest: map_of_SomeD bspec)
 done
 
 end
 
 lemma distinct_map_of_rev: "distinct (map fst xs) \<Longrightarrow> map_of (rev xs) = map_of xs"
 by(induct xs)(auto 4 3 simp add: map_add_def intro!: ext split: option.split intro: rev_image_eqI)
 
 lemma map_map_filter: 
   "map f (List.map_filter g xs) = List.map_filter (map_option f \<circ> g) xs"
 by(auto simp add: List.map_filter_def)
 
 lemma map_filter_map_option_const: 
   "List.map_filter (\<lambda>x. map_option (\<lambda>y. f x) (g (f x))) xs = filter (\<lambda>x. g x \<noteq> None) (map f xs)"
 by(auto simp add: map_filter_def filter_map o_def)
 
 lemma set_map_filter: "set (List.map_filter P xs) = the ` (P ` set xs - {None})"
 by(auto simp add: List.map_filter_def intro: rev_image_eqI)
 
 (* Split and Join *)
 
 definition is_rbt_empty :: "('a, 'b) rbt \<Rightarrow> bool" where
   "is_rbt_empty t \<longleftrightarrow> (case t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
 
 lemma is_rbt_empty_prop[simp]: "is_rbt_empty t \<longleftrightarrow> t = RBT_Impl.Empty"
   by (auto simp: is_rbt_empty_def split: RBT_Impl.rbt.splits)
 
 definition small_rbt :: "('a, 'b) rbt \<Rightarrow> bool" where
   "small_rbt t \<longleftrightarrow> bheight t < 4"
 
 definition flip_rbt :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" where
   "flip_rbt t1 t2 \<longleftrightarrow> bheight t2 < bheight t1"
 
-abbreviation MR where "MR l a b r \<equiv> Branch RBT_Impl.R l a b r"
-abbreviation MB where "MB l a b r \<equiv> Branch RBT_Impl.B l a b r"
+abbreviation (input) MR where "MR l a b r \<equiv> Branch RBT_Impl.R l a b r"
+abbreviation (input) MB where "MB l a b r \<equiv> Branch RBT_Impl.B l a b r"
 
 fun rbt_baliL :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_baliL (MR (MR t1 a b t2) a' b' t3) a'' b'' t4 = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
 | "rbt_baliL (MR t1 a b (MR t2 a' b' t3)) a'' b'' t4 = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
 | "rbt_baliL t1 a b t2 = MB t1 a b t2"
 
 fun rbt_baliR :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_baliR t1 a b (MR t2 a' b' (MR t3 a'' b'' t4)) = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
 | "rbt_baliR t1 a b (MR (MR t2 a' b' t3) a'' b'' t4) = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
 | "rbt_baliR t1 a b t2 = MB t1 a b t2"
 
 fun rbt_baldL :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_baldL (MR t1 a b t2) a' b' t3 = MR (MB t1 a b t2) a' b' t3"
 | "rbt_baldL t1 a b (MB t2 a' b' t3) = rbt_baliR t1 a b (MR t2 a' b' t3)"
 | "rbt_baldL t1 a b (MR (MB t2 a' b' t3) a'' b'' t4) =
   MR (MB t1 a b t2) a' b' (rbt_baliR t3 a'' b'' (paint RBT_Impl.R t4))"
 | "rbt_baldL t1 a b t2 = MR t1 a b t2"
 
 fun rbt_baldR :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_baldR t1 a b (MR t2 a' b' t3) = MR t1 a b (MB t2 a' b' t3)"
 | "rbt_baldR (MB t1 a b t2) a' b' t3 = rbt_baliL (MR t1 a b t2) a' b' t3"
 | "rbt_baldR (MR t1 a b (MB t2 a' b' t3)) a'' b'' t4 =
   MR (rbt_baliL (paint RBT_Impl.R t1) a b t2) a' b' (MB t3 a'' b'' t4)"
 | "rbt_baldR t1 a b t2 = MR t1 a b t2"
 
 fun rbt_app :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_app RBT_Impl.Empty t = t"
 | "rbt_app t RBT_Impl.Empty = t"
 | "rbt_app (MR t1 a b t2) (MR t3 a'' b'' t4) = (case rbt_app t2 t3 of
     MR u2 a' b' u3 \<Rightarrow> (MR (MR t1 a b u2) a' b' (MR u3 a'' b'' t4))
   | t23 \<Rightarrow> MR t1 a b (MR t23 a'' b'' t4))"
 | "rbt_app (MB t1 a b t2) (MB t3 a'' b'' t4) = (case rbt_app t2 t3 of
     MR u2 a' b' u3 \<Rightarrow> MR (MB t1 a b u2) a' b' (MB u3 a'' b'' t4)
   | t23 \<Rightarrow> rbt_baldL t1 a b (MB t23 a'' b'' t4))"
 | "rbt_app t1 (MR t2 a b t3) = MR (rbt_app t1 t2) a b t3"
 | "rbt_app (MR t1 a b t2) t3 = MR t1 a b (rbt_app t2 t3)"
 
 fun rbt_joinL :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_joinL l a b r = (if bheight l \<ge> bheight r then MR l a b r
     else case r of MB l' a' b' r' \<Rightarrow> rbt_baliL (rbt_joinL l a b l') a' b' r'
     | MR l' a' b' r' \<Rightarrow> MR (rbt_joinL l a b l') a' b' r')"
 
 declare rbt_joinL.simps[simp del]
 
 fun rbt_joinR :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_joinR l a b r = (if bheight l \<le> bheight r then MR l a b r
     else case l of MB l' a' b' r' \<Rightarrow> rbt_baliR l' a' b' (rbt_joinR r' a b r)
     | MR l' a' b' r' \<Rightarrow> MR l' a' b' (rbt_joinR r' a b r))"
 
 declare rbt_joinR.simps[simp del]
 
 definition rbt_join :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_join l a b r =
     (let bhl = bheight l; bhr = bheight r
     in if bhl > bhr
     then paint RBT_Impl.B (rbt_joinR l a b r)
     else if bhl < bhr
     then paint RBT_Impl.B (rbt_joinL l a b r)
     else MB l a b r)"
 
 lemma size_paint[simp]: "size (paint c t) = size t"
   by (cases t) auto
 
 lemma size_baliL[simp]: "size (rbt_baliL t1 a b t2) = Suc (size t1 + size t2)"
   by (induction t1 a b t2 rule: rbt_baliL.induct) auto
 
 lemma size_baliR[simp]: "size (rbt_baliR t1 a b t2) = Suc (size t1 + size t2)"
   by (induction t1 a b t2 rule: rbt_baliR.induct) auto
 
 lemma size_baldL[simp]: "size (rbt_baldL t1 a b t2) = Suc (size t1 + size t2)"
   by (induction t1 a b t2 rule: rbt_baldL.induct) auto
 
 lemma size_baldR[simp]: "size (rbt_baldR t1 a b t2) = Suc (size t1 + size t2)"
   by (induction t1 a b t2 rule: rbt_baldR.induct) auto
 
 lemma size_rbt_app[simp]: "size (rbt_app t1 t2) = size t1 + size t2"
   by (induction t1 t2 rule: rbt_app.induct)
      (auto split: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 
 lemma size_rbt_joinL[simp]: "size (rbt_joinL t1 a b t2) = Suc (size t1 + size t2)"
   by (induction t1 a b t2 rule: rbt_joinL.induct)
      (auto simp: rbt_joinL.simps split: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 
 lemma size_rbt_joinR[simp]: "size (rbt_joinR t1 a b t2) = Suc (size t1 + size t2)"
   by (induction t1 a b t2 rule: rbt_joinR.induct)
      (auto simp: rbt_joinR.simps split: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 
 lemma size_rbt_join[simp]: "size (rbt_join t1 a b t2) = Suc (size t1 + size t2)"
   by (auto simp: rbt_join_def Let_def)
 
 definition "inv_12 t \<longleftrightarrow> inv1 t \<and> inv2 t"
 
 lemma rbt_Node: "inv_12 (RBT_Impl.Branch c l a b r) \<Longrightarrow> inv_12 l \<and> inv_12 r"
   by (auto simp: inv_12_def)
 
 lemma paint2: "paint c2 (paint c1 t) = paint c2 t"
   by (cases t) auto
 
 lemma inv1_rbt_baliL: "inv1l l \<Longrightarrow> inv1 r \<Longrightarrow> inv1 (rbt_baliL l a b r)"
   by (induct l a b r rule: rbt_baliL.induct) auto
 
 lemma inv1_rbt_baliR: "inv1 l \<Longrightarrow> inv1l r \<Longrightarrow> inv1 (rbt_baliR l a b r)"
   by (induct l a b r rule: rbt_baliR.induct) auto
 
 lemma rbt_bheight_rbt_baliL: "bheight l = bheight r \<Longrightarrow> bheight (rbt_baliL l a b r) = Suc (bheight l)"
   by (induct l a b r rule: rbt_baliL.induct) auto
 
 lemma rbt_bheight_rbt_baliR: "bheight l = bheight r \<Longrightarrow> bheight (rbt_baliR l a b r) = Suc (bheight l)"
   by (induct l a b r rule: rbt_baliR.induct) auto
 
 lemma inv2_rbt_baliL: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l = bheight r \<Longrightarrow> inv2 (rbt_baliL l a b r)"
   by (induct l a b r rule: rbt_baliL.induct) auto
 
 lemma inv2_rbt_baliR: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l = bheight r \<Longrightarrow> inv2 (rbt_baliR l a b r)"
   by (induct l a b r rule: rbt_baliR.induct) auto
 
 lemma inv_rbt_baliR: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> inv1 l \<Longrightarrow> inv1l r \<Longrightarrow> bheight l = bheight r \<Longrightarrow>
   inv1 (rbt_baliR l a b r) \<and> inv2 (rbt_baliR l a b r) \<and> bheight (rbt_baliR l a b r) = Suc (bheight l)"
   by (induct l a b r rule: rbt_baliR.induct) auto
 
 lemma inv_rbt_baliL: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> inv1l l \<Longrightarrow> inv1 r \<Longrightarrow> bheight l = bheight r \<Longrightarrow>
   inv1 (rbt_baliL l a b r) \<and> inv2 (rbt_baliL l a b r) \<and> bheight (rbt_baliL l a b r) = Suc (bheight l)"
   by (induct l a b r rule: rbt_baliL.induct) auto
 
 lemma inv2_rbt_baldL_inv1: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l + 1 = bheight r \<Longrightarrow> inv1 r \<Longrightarrow>
   inv2 (rbt_baldL l a b r) \<and> bheight (rbt_baldL l a b r) = bheight r"
   by (induct l a b r rule: rbt_baldL.induct) (auto simp: inv2_rbt_baliR rbt_bheight_rbt_baliR)
 
 lemma inv2_rbt_baldL_B: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l + 1 = bheight r \<Longrightarrow> color_of r = RBT_Impl.B \<Longrightarrow>
   inv2 (rbt_baldL l a b r) \<and> bheight (rbt_baldL l a b r) = bheight r"
   by (induct l a b r rule: rbt_baldL.induct) (auto simp add: inv2_rbt_baliR rbt_bheight_rbt_baliR)
 
 lemma inv1_rbt_baldL: "inv1l l \<Longrightarrow> inv1 r \<Longrightarrow> color_of r = RBT_Impl.B \<Longrightarrow> inv1 (rbt_baldL l a b r)"
   by (induct l a b r rule: rbt_baldL.induct) (simp_all add: inv1_rbt_baliR)
 
 lemma inv1lI: "inv1 t \<Longrightarrow> inv1l t"
   by (cases t) auto
 
 lemma neq_Black[simp]: "(c \<noteq> RBT_Impl.B) = (c = RBT_Impl.R)"
   by (cases c) auto
 
 lemma inv1l_rbt_baldL: "inv1l l \<Longrightarrow> inv1 r \<Longrightarrow> inv1l (rbt_baldL l a b r)"
   by (induct l a b r rule: rbt_baldL.induct) (auto simp: inv1_rbt_baliR paint2)
 
 lemma inv2_rbt_baldR_inv1: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l = bheight r + 1 \<Longrightarrow> inv1 l \<Longrightarrow>
   inv2 (rbt_baldR l a b r) \<and> bheight (rbt_baldR l a b r) = bheight l"
   by (induct l a b r rule: rbt_baldR.induct) (auto simp: inv2_rbt_baliL rbt_bheight_rbt_baliL)
 
 lemma inv1_rbt_baldR: "inv1 l \<Longrightarrow> inv1l r \<Longrightarrow> color_of l = RBT_Impl.B \<Longrightarrow> inv1 (rbt_baldR l a b r)"
   by (induct l a b r rule: rbt_baldR.induct) (simp_all add: inv1_rbt_baliL)
 
 lemma inv1l_rbt_baldR: "inv1 l \<Longrightarrow> inv1l r \<Longrightarrow>inv1l (rbt_baldR l a b r)"
   by (induct l a b r rule: rbt_baldR.induct) (auto simp: inv1_rbt_baliL paint2)
 
 lemma inv2_rbt_app: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l = bheight r \<Longrightarrow>
   inv2 (rbt_app l r) \<and> bheight (rbt_app l r) = bheight l"
   by (induct l r rule: rbt_app.induct)
      (auto simp: inv2_rbt_baldL_B split: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 
 lemma inv1_rbt_app: "inv1 l \<Longrightarrow> inv1 r \<Longrightarrow> (color_of l = RBT_Impl.B \<and>
   color_of r = RBT_Impl.B \<longrightarrow> inv1 (rbt_app l r)) \<and> inv1l (rbt_app l r)"
   by (induct l r rule: rbt_app.induct)
      (auto simp: inv1_rbt_baldL split: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 
 lemma inv_rbt_baldL: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l + 1 = bheight r \<Longrightarrow> inv1l l \<Longrightarrow> inv1 r \<Longrightarrow>
   inv2 (rbt_baldL l a b r) \<and> bheight (rbt_baldL l a b r) = bheight r \<and>
   inv1l (rbt_baldL l a b r) \<and> (color_of r = RBT_Impl.B \<longrightarrow> inv1 (rbt_baldL l a b r))"
   by (induct l a b r rule: rbt_baldL.induct) (auto simp: inv_rbt_baliR rbt_bheight_rbt_baliR paint2)
 
 lemma inv_rbt_baldR: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l = bheight r + 1 \<Longrightarrow> inv1 l \<Longrightarrow> inv1l r \<Longrightarrow>
   inv2 (rbt_baldR l a b r) \<and> bheight (rbt_baldR l a b r) = bheight l \<and>
   inv1l (rbt_baldR l a b r) \<and> (color_of l = RBT_Impl.B \<longrightarrow> inv1 (rbt_baldR l a b r))"
   by (induct l a b r rule: rbt_baldR.induct) (auto simp: inv_rbt_baliL rbt_bheight_rbt_baliL paint2)
 
 lemma inv_rbt_app: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l = bheight r \<Longrightarrow> inv1 l \<Longrightarrow> inv1 r \<Longrightarrow>
   inv2 (rbt_app l r) \<and> bheight (rbt_app l r) = bheight l \<and>
   inv1l (rbt_app l r) \<and> (color_of l = RBT_Impl.B \<and> color_of r = RBT_Impl.B \<longrightarrow> inv1 (rbt_app l r))"
   by (induct l r rule: rbt_app.induct)
      (auto simp: inv2_rbt_baldL_B inv_rbt_baldL split: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 
 lemma inv1l_rbt_joinL: "inv1 l \<Longrightarrow> inv1 r \<Longrightarrow> bheight l \<le> bheight r \<Longrightarrow>
   inv1l (rbt_joinL l a b r) \<and>
   (bheight l \<noteq> bheight r \<and> color_of r = RBT_Impl.B \<longrightarrow> inv1 (rbt_joinL l a b r))"
 proof (induct l a b r rule: rbt_joinL.induct)
   case (1 l a b r)
   then show ?case
     by (auto simp: inv1_rbt_baliL rbt_joinL.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma inv1l_rbt_joinR: "inv1 l \<Longrightarrow> inv2 l \<Longrightarrow> inv1 r \<Longrightarrow> inv2 r \<Longrightarrow> bheight l \<ge> bheight r \<Longrightarrow>
   inv1l (rbt_joinR l a b r) \<and>
   (bheight l \<noteq> bheight r \<and> color_of l = RBT_Impl.B \<longrightarrow> inv1 (rbt_joinR l a b r))"
 proof (induct l a b r rule: rbt_joinR.induct)
   case (1 l a b r)
   then show ?case
     by (fastforce simp: inv1_rbt_baliR rbt_joinR.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma bheight_rbt_joinL: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l \<le> bheight r \<Longrightarrow>
   bheight (rbt_joinL l a b r) = bheight r"
 proof (induct l a b r rule: rbt_joinL.induct)
   case (1 l a b r)
   then show ?case
     by (auto simp: rbt_bheight_rbt_baliL rbt_joinL.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma inv2_rbt_joinL: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l \<le> bheight r \<Longrightarrow> inv2 (rbt_joinL l a b r)"
 proof (induct l a b r rule: rbt_joinL.induct)
   case (1 l a b r)
   then show ?case
     by (auto simp: inv2_rbt_baliL bheight_rbt_joinL rbt_joinL.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma bheight_rbt_joinR: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l \<ge> bheight r \<Longrightarrow>
   bheight (rbt_joinR l a b r) = bheight l"
 proof (induct l a b r rule: rbt_joinR.induct)
   case (1 l a b r)
   then show ?case
     by (fastforce simp: rbt_bheight_rbt_baliR rbt_joinR.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma inv2_rbt_joinR: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l \<ge> bheight r \<Longrightarrow> inv2 (rbt_joinR l a b r)"
 proof (induct l a b r rule: rbt_joinR.induct)
   case (1 l a b r)
   then show ?case
     by (fastforce simp: inv2_rbt_baliR bheight_rbt_joinR rbt_joinR.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma keys_paint[simp]: "RBT_Impl.keys (paint c t) = RBT_Impl.keys t"
   by (cases t) auto
 
 lemma keys_rbt_baliL: "RBT_Impl.keys (rbt_baliL l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
   by (cases "(l,a,b,r)" rule: rbt_baliL.cases) auto
 
 lemma keys_rbt_baliR: "RBT_Impl.keys (rbt_baliR l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
   by (cases "(l,a,b,r)" rule: rbt_baliR.cases) auto
 
 lemma keys_rbt_baldL: "RBT_Impl.keys (rbt_baldL l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
   by (cases "(l,a,b,r)" rule: rbt_baldL.cases) (auto simp: keys_rbt_baliL keys_rbt_baliR)
 
 lemma keys_rbt_baldR: "RBT_Impl.keys (rbt_baldR l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
   by (cases "(l,a,b,r)" rule: rbt_baldR.cases) (auto simp: keys_rbt_baliL keys_rbt_baliR)
 
 lemma keys_rbt_app: "RBT_Impl.keys (rbt_app l r) = RBT_Impl.keys l @ RBT_Impl.keys r"
   by (induction l r rule: rbt_app.induct)
      (auto simp: keys_rbt_baldL keys_rbt_baldR split: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 
 lemma keys_rbt_joinL: "bheight l \<le> bheight r \<Longrightarrow>
   RBT_Impl.keys (rbt_joinL l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
 proof (induction l a b r rule: rbt_joinL.induct)
   case (1 l a b r)
   thus ?case
     by (auto simp: keys_rbt_baliL rbt_joinL.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma keys_rbt_joinR: "RBT_Impl.keys (rbt_joinR l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
 proof (induction l a b r rule: rbt_joinR.induct)
   case (1 l a b r)
   thus ?case
     by (force simp: keys_rbt_baliR rbt_joinR.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma rbt_set_rbt_baliL: "set (RBT_Impl.keys (rbt_baliL l a b r)) =
   set (RBT_Impl.keys l) \<union> {a} \<union> set (RBT_Impl.keys r)"
   by (cases "(l,a,b,r)" rule: rbt_baliL.cases) auto
 
 lemma set_rbt_joinL: "set (RBT_Impl.keys (rbt_joinL l a b r)) =
   set (RBT_Impl.keys l) \<union> {a} \<union> set (RBT_Impl.keys r)"
 proof (induction l a b r rule: rbt_joinL.induct)
   case (1 l a b r)
   thus ?case
     by (auto simp: rbt_set_rbt_baliL rbt_joinL.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma rbt_set_rbt_baliR: "set (RBT_Impl.keys (rbt_baliR l a b r)) =
   set (RBT_Impl.keys l) \<union> {a} \<union> set (RBT_Impl.keys r)"
   by (cases "(l,a,b,r)" rule: rbt_baliR.cases) auto
 
 lemma set_rbt_joinR: "set (RBT_Impl.keys (rbt_joinR l a b r)) =
   set (RBT_Impl.keys l) \<union> {a} \<union> set (RBT_Impl.keys r)"
 proof (induction l a b r rule: rbt_joinR.induct)
   case (1 l a b r)
   thus ?case
     by (force simp: rbt_set_rbt_baliR rbt_joinR.simps[of l a b r]
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma set_keys_paint: "set (RBT_Impl.keys (paint c t)) = set (RBT_Impl.keys t)"
   by (cases t) auto
 
 lemma set_rbt_join: "set (RBT_Impl.keys (rbt_join l a b r)) =
   set (RBT_Impl.keys l) \<union> {a} \<union> set (RBT_Impl.keys r)"
   by (simp add: set_rbt_joinL set_rbt_joinR set_keys_paint rbt_join_def Let_def)
 
 lemma inv_rbt_join: "inv_12 l \<Longrightarrow> inv_12 r \<Longrightarrow> inv_12 (rbt_join l a b r)"
   by (auto simp: rbt_join_def Let_def inv1l_rbt_joinL inv1l_rbt_joinR
       inv2_rbt_joinL inv2_rbt_joinR inv_12_def)
 
 fun rbt_recolor :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_recolor (Branch RBT_Impl.R t1 k v t2) =
     (if color_of t1 = RBT_Impl.B \<and> color_of t2 = RBT_Impl.B then Branch RBT_Impl.B t1 k v t2
     else Branch RBT_Impl.R t1 k v t2)"
 | "rbt_recolor t = t"
 
 lemma rbt_recolor: "inv_12 t \<Longrightarrow> inv_12 (rbt_recolor t)"
   by (induction t rule: rbt_recolor.induct) (auto simp: inv_12_def)
 
 fun rbt_split_min :: "('a, 'b) rbt \<Rightarrow> 'a \<times> 'b \<times> ('a, 'b) rbt" where
   "rbt_split_min RBT_Impl.Empty = undefined"
 | "rbt_split_min (RBT_Impl.Branch _ l a b r) =
     (if is_rbt_empty l then (a,b,r) else let (a',b',l') = rbt_split_min l in (a',b',rbt_join l' a b r))"
 
 lemma rbt_split_min_set:
   "rbt_split_min t = (a,b,t') \<Longrightarrow> t \<noteq> RBT_Impl.Empty \<Longrightarrow>
   a \<in> set (RBT_Impl.keys t) \<and> set (RBT_Impl.keys t) = {a} \<union> set (RBT_Impl.keys t')"
   by (induction t arbitrary: t') (auto simp: set_rbt_join split: prod.splits if_splits)
 
 lemma rbt_split_min_inv: "rbt_split_min t = (a,b,t') \<Longrightarrow> inv_12 t \<Longrightarrow> t \<noteq> RBT_Impl.Empty \<Longrightarrow> inv_12 t'"
   by (induction t arbitrary: t')
      (auto simp: inv_rbt_join split: if_splits prod.splits dest: rbt_Node)
 
 definition rbt_join2 :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_join2 l r = (if is_rbt_empty r then l else let (a,b,r') = rbt_split_min r in rbt_join l a b r')"
 
 lemma set_rbt_join2[simp]: "set (RBT_Impl.keys (rbt_join2 l r)) =
   set (RBT_Impl.keys l) \<union> set (RBT_Impl.keys r)"
   by (simp add: rbt_join2_def rbt_split_min_set set_rbt_join split: prod.split)
 
 lemma inv_rbt_join2: "inv_12 l \<Longrightarrow> inv_12 r \<Longrightarrow> inv_12 (rbt_join2 l r)"
   by (simp add: rbt_join2_def inv_rbt_join rbt_split_min_set rbt_split_min_inv split: prod.split)
 
 context ord begin
 
 fun rbt_split :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> ('a, 'b) rbt \<times> 'b option \<times> ('a, 'b) rbt" where
   "rbt_split RBT_Impl.Empty k = (RBT_Impl.Empty, None, RBT_Impl.Empty)"
 | "rbt_split (RBT_Impl.Branch _ l a b r) x =
   (if x < a then (case rbt_split l x of (l1, \<beta>, l2) \<Rightarrow> (l1, \<beta>, rbt_join l2 a b r))
   else if a < x then (case rbt_split r x of (r1, \<beta>, r2) \<Rightarrow> (rbt_join l a b r1, \<beta>, r2))
   else (l, Some b, r))"
 
 lemma rbt_split: "rbt_split t x = (l,\<beta>,r) \<Longrightarrow> inv_12 t \<Longrightarrow> inv_12 l \<and> inv_12 r"
   by (induction t arbitrary: l r)
      (auto simp: set_rbt_join inv_rbt_join rbt_greater_prop rbt_less_prop
       split: if_splits prod.splits dest!: rbt_Node)
 
 lemma rbt_split_size: "(l2,\<beta>,r2) = rbt_split t2 a \<Longrightarrow> size l2 + size r2 \<le> size t2"
   by (induction t2 a arbitrary: l2 r2 rule: rbt_split.induct) (auto split: if_splits prod.splits)
 
 function rbt_union_rec :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_union_rec f t1 t2 = (let (f, t2, t1) =
     if flip_rbt t2 t1 then (\<lambda>k v v'. f k v' v, t1, t2) else (f, t2, t1) in
     if small_rbt t2 then RBT_Impl.fold (rbt_insert_with_key f) t2 t1
     else (case t1 of RBT_Impl.Empty \<Rightarrow> t2
       | RBT_Impl.Branch _ l1 a b r1 \<Rightarrow>
         case rbt_split t2 a of (l2, \<beta>, r2) \<Rightarrow>
           rbt_join (rbt_union_rec f l1 l2) a (case \<beta> of None \<Rightarrow> b | Some b' \<Rightarrow> f a b b') (rbt_union_rec f r1 r2)))"
   by pat_completeness auto
 termination
   using rbt_split_size
   by (relation "measure (\<lambda>(f,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+
 
 declare rbt_union_rec.simps[simp del]
 
 function rbt_union_swap_rec :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_union_swap_rec f \<gamma> t1 t2 = (let (\<gamma>, t2, t1) =
     if flip_rbt t2 t1 then (\<not>\<gamma>, t1, t2) else (\<gamma>, t2, t1);
     f' = (if \<gamma> then (\<lambda>k v v'. f k v' v) else f) in
     if small_rbt t2 then RBT_Impl.fold (rbt_insert_with_key f') t2 t1
     else (case t1 of RBT_Impl.Empty \<Rightarrow> t2
       | RBT_Impl.Branch _ l1 a b r1 \<Rightarrow>
         case rbt_split t2 a of (l2, \<beta>, r2) \<Rightarrow>
           rbt_join (rbt_union_swap_rec f \<gamma> l1 l2) a (case \<beta> of None \<Rightarrow> b | Some b' \<Rightarrow> f' a b b') (rbt_union_swap_rec f \<gamma> r1 r2)))"
   by pat_completeness auto
 termination
   using rbt_split_size
   by (relation "measure (\<lambda>(f,\<gamma>,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+
 
 declare rbt_union_swap_rec.simps[simp del]
 
 lemma rbt_union_swap_rec: "rbt_union_swap_rec f \<gamma> t1 t2 =
   rbt_union_rec (if \<gamma> then (\<lambda>k v v'. f k v' v) else f) t1 t2"
 proof (induction f \<gamma> t1 t2 rule: rbt_union_swap_rec.induct)
   case (1 f \<gamma> t1 t2)
   show ?case
     using 1[OF refl _ refl refl _ refl _ refl]
     unfolding rbt_union_swap_rec.simps[of _ _ t1] rbt_union_rec.simps[of _ t1]
     by (auto simp: Let_def split: rbt.splits prod.splits option.splits) (* slow *)
 qed
 
 lemma rbt_fold_rbt_insert:
   assumes "inv_12 t2"
   shows "inv_12 (RBT_Impl.fold (rbt_insert_with_key f) t1 t2)"
 proof -
   define xs where "xs = RBT_Impl.entries t1"
   from assms show ?thesis
     unfolding RBT_Impl.fold_def xs_def[symmetric]
     by (induct xs rule: rev_induct)
        (auto simp: inv_12_def rbt_insert_with_key_def ins_inv1_inv2)
 qed
 
 lemma rbt_union_rec: "inv_12 t1 \<Longrightarrow> inv_12 t2 \<Longrightarrow> inv_12 (rbt_union_rec f t1 t2)"
 proof (induction f t1 t2 rule: rbt_union_rec.induct)
   case (1 t1 t2)
   thus ?case
     by (auto simp: rbt_union_rec.simps[of t1 t2] inv_rbt_join rbt_split rbt_fold_rbt_insert
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits prod.split if_splits dest: rbt_Node)
 qed
 
 definition "map_filter_inter f t1 t2 = List.map_filter (\<lambda>(k, v).
   case rbt_lookup t1 k of None \<Rightarrow> None
   | Some v' \<Rightarrow> Some (k, f k v' v)) (RBT_Impl.entries t2)"
 
 function rbt_inter_rec :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_inter_rec f t1 t2 = (let (f, t2, t1) =
     if flip_rbt t2 t1 then (\<lambda>k v v'. f k v' v, t1, t2) else (f, t2, t1) in
     if small_rbt t2 then rbtreeify (map_filter_inter f t1 t2)
     else case t1 of RBT_Impl.Empty \<Rightarrow> RBT_Impl.Empty
     | RBT_Impl.Branch _ l1 a b r1 \<Rightarrow>
       case rbt_split t2 a of (l2, \<beta>, r2) \<Rightarrow> let l' = rbt_inter_rec f l1 l2; r' = rbt_inter_rec f r1 r2 in
       (case \<beta> of None \<Rightarrow> rbt_join2 l' r' | Some b' \<Rightarrow> rbt_join l' a (f a b b') r'))"
   by pat_completeness auto
 termination
   using rbt_split_size
   by (relation "measure (\<lambda>(f,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+
 
 declare rbt_inter_rec.simps[simp del]
 
 function rbt_inter_swap_rec :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_inter_swap_rec f \<gamma> t1 t2 = (let (\<gamma>, t2, t1) =
     if flip_rbt t2 t1 then (\<not>\<gamma>, t1, t2) else (\<gamma>, t2, t1);
     f' = (if \<gamma> then (\<lambda>k v v'. f k v' v) else f) in
     if small_rbt t2 then rbtreeify (map_filter_inter f' t1 t2)
     else case t1 of RBT_Impl.Empty \<Rightarrow> RBT_Impl.Empty
     | RBT_Impl.Branch _ l1 a b r1 \<Rightarrow>
       case rbt_split t2 a of (l2, \<beta>, r2) \<Rightarrow> let l' = rbt_inter_swap_rec f \<gamma> l1 l2; r' = rbt_inter_swap_rec f \<gamma> r1 r2 in
       (case \<beta> of None \<Rightarrow> rbt_join2 l' r' | Some b' \<Rightarrow> rbt_join l' a (f' a b b') r'))"
   by pat_completeness auto
 termination
   using rbt_split_size
   by (relation "measure (\<lambda>(f,\<gamma>,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+
 
 declare rbt_inter_swap_rec.simps[simp del]
 
 lemma rbt_inter_swap_rec: "rbt_inter_swap_rec f \<gamma> t1 t2 =
   rbt_inter_rec (if \<gamma> then (\<lambda>k v v'. f k v' v) else f) t1 t2"
 proof (induction f \<gamma> t1 t2 rule: rbt_inter_swap_rec.induct)
   case (1 f \<gamma> t1 t2)
   show ?case
     using 1[OF refl _ refl refl _ refl _ refl]
     unfolding rbt_inter_swap_rec.simps[of _ _ t1] rbt_inter_rec.simps[of _ t1]
     by (auto simp add: Let_def split: rbt.splits prod.splits option.splits)
 qed
 
 lemma rbt_rbtreeify[simp]: "inv_12 (rbtreeify kvs)"
   by (simp add: inv_12_def rbtreeify_def inv1_rbtreeify_g inv2_rbtreeify_g)
 
 lemma rbt_inter_rec: "inv_12 t1 \<Longrightarrow> inv_12 t2 \<Longrightarrow> inv_12 (rbt_inter_rec f t1 t2)"
 proof(induction f t1 t2 rule: rbt_inter_rec.induct)
   case (1 t1 t2)
   thus ?case
     by (auto simp: rbt_inter_rec.simps[of t1 t2] inv_rbt_join inv_rbt_join2 rbt_split Let_def
         split!: RBT_Impl.rbt.splits RBT_Impl.color.splits prod.split if_splits
         option.splits dest!: rbt_Node)
 qed
 
 definition "filter_minus t1 t2 = filter (\<lambda>(k, _). rbt_lookup t2 k = None) (RBT_Impl.entries t1)"
 
 fun rbt_minus_rec :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
   "rbt_minus_rec t1 t2 = (if small_rbt t2 then RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t2 t1
     else if small_rbt t1 then rbtreeify (filter_minus t1 t2)
     else case t2 of RBT_Impl.Empty \<Rightarrow> t1
       | RBT_Impl.Branch _ l2 a b r2 \<Rightarrow>
         case rbt_split t1 a of (l1, _, r1) \<Rightarrow> rbt_join2 (rbt_minus_rec l1 l2) (rbt_minus_rec r1 r2))"
 
 declare rbt_minus_rec.simps[simp del]
 
 end
 
 context linorder begin
 
 lemma rbt_sorted_entries_right_unique:
   "\<lbrakk> (k, v) \<in> set (entries t); (k, v') \<in> set (entries t); 
      rbt_sorted t \<rbrakk> \<Longrightarrow> v = v'"
 by(auto dest!: distinct_entries inj_onD[where x="(k, v)" and y="(k, v')"] simp add: distinct_map)
 
 lemma rbt_sorted_fold_rbt_insertwk:
   "rbt_sorted t \<Longrightarrow> rbt_sorted (List.fold (\<lambda>(k, v). rbt_insert_with_key f k v) xs t)"
 by(induct xs rule: rev_induct)(auto simp add: rbt_insertwk_rbt_sorted)
 
 lemma is_rbt_fold_rbt_insertwk:
   assumes "is_rbt t1"
   shows "is_rbt (fold (rbt_insert_with_key f) t2 t1)"
 proof -
   define xs where "xs = entries t2"
   from assms show ?thesis unfolding fold_def xs_def[symmetric]
     by(induct xs rule: rev_induct)(auto simp add: rbt_insertwk_is_rbt)
 qed
 
 lemma rbt_delete: "inv_12 t \<Longrightarrow> inv_12 (rbt_delete x t)"
   using rbt_del_inv1_inv2[of t x]
   by (auto simp: inv_12_def rbt_delete_def rbt_del_inv1_inv2)
 
 lemma rbt_sorted_delete: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_delete x t)"
   by (auto simp: rbt_delete_def rbt_del_rbt_sorted)
 
 lemma rbt_fold_rbt_delete:
   assumes "inv_12 t2"
   shows "inv_12 (RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t1 t2)"
 proof -
   define xs where "xs = RBT_Impl.entries t1"
   from assms show ?thesis
     unfolding RBT_Impl.fold_def xs_def[symmetric]
     by (induct xs rule: rev_induct) (auto simp: rbt_delete)
 qed
 
 lemma rbt_minus_rec: "inv_12 t1 \<Longrightarrow> inv_12 t2 \<Longrightarrow> inv_12 (rbt_minus_rec t1 t2)"
 proof(induction t1 t2 rule: rbt_minus_rec.induct)
   case (1 t1 t2)
   thus ?case
     by (auto simp: rbt_minus_rec.simps[of t1 t2] inv_rbt_join inv_rbt_join2 rbt_split
         rbt_fold_rbt_delete split!: RBT_Impl.rbt.splits RBT_Impl.color.splits prod.split if_splits
         dest: rbt_Node)
 qed
 
 end
 
 context linorder begin
 
 lemma rbt_sorted_rbt_baliL: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_sorted (rbt_baliL l a b r)"
   using rbt_greater_trans rbt_less_trans
   by (cases "(l,a,b,r)" rule: rbt_baliL.cases) fastforce+
 
 lemma rbt_lookup_rbt_baliL: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_lookup (rbt_baliL l a b r) k =
   (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
   by (cases "(l,a,b,r)" rule: rbt_baliL.cases) (auto split!: if_splits)
 
 lemma rbt_sorted_rbt_baliR: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_sorted (rbt_baliR l a b r)"
   using rbt_greater_trans rbt_less_trans
   by (cases "(l,a,b,r)" rule: rbt_baliR.cases) fastforce+
 
 lemma rbt_lookup_rbt_baliR: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_lookup (rbt_baliR l a b r) k =
   (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
   by (cases "(l,a,b,r)" rule: rbt_baliR.cases) (auto split!: if_splits)
 
 lemma rbt_sorted_rbt_joinL: "rbt_sorted (RBT_Impl.Branch c l a b r) \<Longrightarrow> bheight l \<le> bheight r \<Longrightarrow>
   rbt_sorted (rbt_joinL l a b r)"
 proof (induction l a b r arbitrary: c rule: rbt_joinL.induct)
   case (1 l a b r)
   thus ?case
     by (auto simp: rbt_set_rbt_baliL rbt_joinL.simps[of l a b r] set_rbt_joinL rbt_less_prop
         intro!: rbt_sorted_rbt_baliL split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma rbt_lookup_rbt_joinL: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_lookup (rbt_joinL l a b r) k =
   (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
 proof (induction l a b r rule: rbt_joinL.induct)
   case (1 l a b r)
   have less_rbt_joinL:
     "rbt_sorted r1 \<Longrightarrow> r1 |\<guillemotleft> x \<Longrightarrow> a \<guillemotleft>| r1 \<Longrightarrow> a < x \<Longrightarrow> rbt_joinL l a b r1 |\<guillemotleft> x" for x r1
     using 1(5)
     by (auto simp: rbt_less_prop rbt_greater_prop set_rbt_joinL)
   show ?case
     using 1 less_rbt_joinL rbt_lookup_rbt_baliL[OF rbt_sorted_rbt_joinL[of _ l a b], where ?k=k]
     by (auto simp: rbt_joinL.simps[of l a b r] split!: if_splits rbt.splits color.splits)
 qed
 
 lemma rbt_sorted_rbt_joinR: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_sorted (rbt_joinR l a b r)"
 proof (induction l a b r rule: rbt_joinR.induct)
   case (1 l a b r)
   thus ?case
     by (auto simp: rbt_set_rbt_baliR rbt_joinR.simps[of l a b r] set_rbt_joinR rbt_greater_prop
         intro!: rbt_sorted_rbt_baliR split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
 qed
 
 lemma rbt_lookup_rbt_joinR: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_lookup (rbt_joinR l a b r) k =
   (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
 proof (induction l a b r rule: rbt_joinR.induct)
   case (1 l a b r)
   have less_rbt_joinR:
     "rbt_sorted l1 \<Longrightarrow> x \<guillemotleft>| l1 \<Longrightarrow> l1 |\<guillemotleft> a \<Longrightarrow> x < a \<Longrightarrow> x \<guillemotleft>| rbt_joinR l1 a b r" for x l1
     using 1(6)
     by (auto simp: rbt_less_prop rbt_greater_prop set_rbt_joinR)
   show ?case
     using 1 less_rbt_joinR rbt_lookup_rbt_baliR[OF _ rbt_sorted_rbt_joinR[of _ r a b], where ?k=k]
     by (auto simp: rbt_joinR.simps[of l a b r] split!: if_splits rbt.splits color.splits)
 qed
 
 lemma rbt_sorted_paint: "rbt_sorted (paint c t) = rbt_sorted t"
   by (cases t) auto
 
 lemma rbt_sorted_rbt_join: "rbt_sorted (RBT_Impl.Branch c l a b r) \<Longrightarrow>
   rbt_sorted (rbt_join l a b r)"
   by (auto simp: rbt_sorted_paint rbt_sorted_rbt_joinL rbt_sorted_rbt_joinR rbt_join_def Let_def)
 
 lemma rbt_lookup_rbt_join: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> l |\<guillemotleft> a \<Longrightarrow> a \<guillemotleft>| r \<Longrightarrow>
   rbt_lookup (rbt_join l a b r) k =
   (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
   by (auto simp: rbt_join_def Let_def rbt_lookup_rbt_joinL rbt_lookup_rbt_joinR)
 
 lemma rbt_split_min_rbt_sorted: "rbt_split_min t = (a,b,t') \<Longrightarrow> rbt_sorted t \<Longrightarrow> t \<noteq> RBT_Impl.Empty \<Longrightarrow>
   rbt_sorted t' \<and> (\<forall>x \<in> set (RBT_Impl.keys t'). a < x)"
   by (induction t arbitrary: t')
      (fastforce simp: rbt_split_min_set rbt_sorted_rbt_join set_rbt_join rbt_less_prop rbt_greater_prop
       split: if_splits prod.splits)+
 
 lemma rbt_split_min_rbt_lookup: "rbt_split_min t = (a,b,t') \<Longrightarrow> rbt_sorted t \<Longrightarrow> t \<noteq> RBT_Impl.Empty \<Longrightarrow>
   rbt_lookup t k = (if k < a then None else if k = a then Some b else rbt_lookup t' k)"
   by (induction t arbitrary: a b t')
      (auto simp: rbt_less_trans antisym_conv3 rbt_less_prop rbt_split_min_set rbt_lookup_rbt_join
       rbt_split_min_rbt_sorted split!: if_splits prod.splits)
 
 lemma rbt_sorted_rbt_join2: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow>
   \<forall>x \<in> set (RBT_Impl.keys l). \<forall>y \<in> set (RBT_Impl.keys r). x < y \<Longrightarrow> rbt_sorted (rbt_join2 l r)"
   by (simp add: rbt_join2_def rbt_sorted_rbt_join rbt_split_min_set rbt_split_min_rbt_sorted set_rbt_join
       rbt_greater_prop rbt_less_prop split: prod.split)
 
 lemma rbt_lookup_rbt_join2: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow>
   \<forall>x \<in> set (RBT_Impl.keys l). \<forall>y \<in> set (RBT_Impl.keys r). x < y \<Longrightarrow>
   rbt_lookup (rbt_join2 l r) k = (case rbt_lookup l k of None \<Rightarrow> rbt_lookup r k | Some v \<Rightarrow> Some v)"
   using rbt_lookup_keys
   by (fastforce simp: rbt_join2_def rbt_greater_prop rbt_less_prop rbt_lookup_rbt_join
       rbt_split_min_rbt_lookup rbt_split_min_rbt_sorted rbt_split_min_set split: option.splits prod.splits)
 
 lemma rbt_split_props: "rbt_split t x = (l,\<beta>,r) \<Longrightarrow> rbt_sorted t \<Longrightarrow>
   set (RBT_Impl.keys l) = {a \<in> set (RBT_Impl.keys t). a < x} \<and>
   set (RBT_Impl.keys r) = {a \<in> set (RBT_Impl.keys t). x < a} \<and>
   rbt_sorted l \<and> rbt_sorted r"
   by (induction t arbitrary: l r)
      (force simp: Let_def set_rbt_join rbt_greater_prop rbt_less_prop
       split: if_splits prod.splits intro!: rbt_sorted_rbt_join)+
 
 lemma rbt_split_lookup: "rbt_split t x = (l,\<beta>,r) \<Longrightarrow> rbt_sorted t \<Longrightarrow>
   rbt_lookup t k = (if k < x then rbt_lookup l k else if k = x then \<beta> else rbt_lookup r k)"
 proof (induction t arbitrary: x l \<beta> r)
   case (Branch c t1 a b t2)
   have "rbt_sorted r1" "r1 |\<guillemotleft> a" if "rbt_split t1 x = (l, \<beta>, r1)" for r1
     using rbt_split_props Branch(4) that
     by (fastforce simp: rbt_less_prop)+
   moreover have "rbt_sorted l1" "a \<guillemotleft>| l1" if "rbt_split t2 x = (l1, \<beta>, r)" for l1
     using rbt_split_props Branch(4) that
     by (fastforce simp: rbt_greater_prop)+
   ultimately show ?case
     using Branch rbt_lookup_rbt_join[of t1 _ a b k] rbt_lookup_rbt_join[of _ t2 a b k]
     by (auto split!: if_splits prod.splits)
 qed simp
 
 lemma rbt_sorted_fold_insertwk: "rbt_sorted t \<Longrightarrow>
   rbt_sorted (RBT_Impl.fold (rbt_insert_with_key f) t' t)"
   by (induct t' arbitrary: t)
      (simp_all add: rbt_insertwk_rbt_sorted)
 
 lemma rbt_lookup_iff_keys:
   "rbt_sorted t \<Longrightarrow> set (RBT_Impl.keys t) = {k. \<exists>v. rbt_lookup t k = Some v}"
   "rbt_sorted t \<Longrightarrow> rbt_lookup t k = None \<longleftrightarrow> k \<notin> set (RBT_Impl.keys t)"
   "rbt_sorted t \<Longrightarrow> (\<exists>v. rbt_lookup t k = Some v) \<longleftrightarrow> k \<in> set (RBT_Impl.keys t)"
   using entry_in_tree_keys rbt_lookup_keys[of t]
   by force+
 
 lemma rbt_lookup_fold_rbt_insertwk:
   assumes t1: "rbt_sorted t1" and t2: "rbt_sorted t2"
   shows "rbt_lookup (fold (rbt_insert_with_key f) t1 t2) k =
   (case rbt_lookup t1 k of None \<Rightarrow> rbt_lookup t2 k
    | Some v \<Rightarrow> case rbt_lookup t2 k of None \<Rightarrow> Some v
                | Some w \<Rightarrow> Some (f k w v))"
 proof -
   define xs where "xs = entries t1"
   hence dt1: "distinct (map fst xs)" using t1 by(simp add: distinct_entries)
   with t2 show ?thesis
     unfolding fold_def map_of_entries[OF t1, symmetric]
       xs_def[symmetric] distinct_map_of_rev[OF dt1, symmetric]
     apply(induct xs rule: rev_induct)
     apply(auto simp add: rbt_lookup_rbt_insertwk rbt_sorted_fold_rbt_insertwk split: option.splits)
     apply(auto simp add: distinct_map_of_rev intro: rev_image_eqI)
     done
 qed
 
 lemma rbt_lookup_union_rec: "rbt_sorted t1 \<Longrightarrow> rbt_sorted t2 \<Longrightarrow>
   rbt_sorted (rbt_union_rec f t1 t2) \<and> rbt_lookup (rbt_union_rec f t1 t2) k =
   (case rbt_lookup t1 k of None \<Rightarrow> rbt_lookup t2 k
   | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> Some v
               | Some w \<Rightarrow> Some (f k v w)))"
 proof(induction f t1 t2 arbitrary: k rule: rbt_union_rec.induct)
   case (1 f t1 t2)
   obtain f' t1' t2' where flip: "(f', t2', t1') =
     (if flip_rbt t2 t1 then (\<lambda>k v v'. f k v' v, t1, t2) else (f, t2, t1))"
     by fastforce
   have rbt_sorted': "rbt_sorted t1'" "rbt_sorted t2'"
     using 1(3,4) flip
     by (auto split: if_splits)
   show ?case
   proof (cases t1')
     case Empty
     show ?thesis
       unfolding rbt_union_rec.simps[of _ t1] flip[symmetric]
       using flip rbt_sorted' rbt_split_props[of t2]
       by (auto simp: Empty rbt_lookup_fold_rbt_insertwk
           intro!: rbt_sorted_fold_insertwk split: if_splits option.splits)
   next
     case (Branch c l1 a b r1)
     {
       assume not_small: "\<not>small_rbt t2'"
       obtain l2 \<beta> r2 where rbt_split_t2': "rbt_split t2' a = (l2, \<beta>, r2)"
         by (cases "rbt_split t2' a") auto
       have rbt_sort: "rbt_sorted l1" "rbt_sorted r1"
         using 1(3,4) flip
         by (auto simp: Branch split: if_splits)
       note rbt_split_t2'_props = rbt_split_props[OF rbt_split_t2' rbt_sorted'(2)]
       have union_l1_l2: "rbt_sorted (rbt_union_rec f' l1 l2)" "rbt_lookup (rbt_union_rec f' l1 l2) k =
         (case rbt_lookup l1 k of None \<Rightarrow> rbt_lookup l2 k
         | Some v \<Rightarrow> (case rbt_lookup l2 k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f' k v w)))" for k
         using 1(1)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
         by (auto simp: not_small)
       have union_r1_r2: "rbt_sorted (rbt_union_rec f' r1 r2)" "rbt_lookup (rbt_union_rec f' r1 r2) k =
         (case rbt_lookup r1 k of None \<Rightarrow> rbt_lookup r2 k
         | Some v \<Rightarrow> (case rbt_lookup r2 k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f' k v w)))" for k
         using 1(2)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
         by (auto simp: not_small)
       have union_l1_l2_keys: "set (RBT_Impl.keys (rbt_union_rec f' l1 l2)) =
        set (RBT_Impl.keys l1) \<union> set (RBT_Impl.keys l2)"
         using rbt_sorted'(1) rbt_split_t2'_props
         by (auto simp: Branch rbt_lookup_iff_keys(1) union_l1_l2 split: option.splits)
       have union_r1_r2_keys: "set (RBT_Impl.keys (rbt_union_rec f' r1 r2)) =
         set (RBT_Impl.keys r1) \<union> set (RBT_Impl.keys r2)"
         using rbt_sorted'(1) rbt_split_t2'_props
         by (auto simp: Branch rbt_lookup_iff_keys(1) union_r1_r2 split: option.splits)
       have union_l1_l2_less: "rbt_union_rec f' l1 l2 |\<guillemotleft> a"
         using rbt_sorted'(1) rbt_split_t2'_props
         by (auto simp: Branch rbt_less_prop union_l1_l2_keys)
       have union_r1_r2_greater: "a \<guillemotleft>| rbt_union_rec f' r1 r2"
         using rbt_sorted'(1) rbt_split_t2'_props
         by (auto simp: Branch rbt_greater_prop union_r1_r2_keys)
       have "rbt_lookup (rbt_union_rec f t1 t2) k =
        (case rbt_lookup t1' k of None \<Rightarrow> rbt_lookup t2' k
        | Some v \<Rightarrow> (case rbt_lookup t2' k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f' k v w)))"
         using rbt_sorted' union_l1_l2 union_r1_r2 rbt_split_t2'_props
           union_l1_l2_less union_r1_r2_greater not_small
         by (auto simp: rbt_union_rec.simps[of _ t1] flip[symmetric] Branch
             rbt_split_t2' rbt_lookup_rbt_join rbt_split_lookup[OF rbt_split_t2'] split: option.splits)
       moreover have "rbt_sorted (rbt_union_rec f t1 t2)"
         using rbt_sorted' rbt_split_t2'_props not_small
         by (auto simp: rbt_union_rec.simps[of _ t1] flip[symmetric] Branch rbt_split_t2'
             union_l1_l2 union_r1_r2 union_l1_l2_keys union_r1_r2_keys rbt_less_prop
             rbt_greater_prop intro!: rbt_sorted_rbt_join)
       ultimately have ?thesis
         using flip
         by (auto split: if_splits option.splits)
     }
     then show ?thesis
       unfolding rbt_union_rec.simps[of _ t1] flip[symmetric]
       using rbt_sorted' flip
       by (auto simp: rbt_sorted_fold_insertwk rbt_lookup_fold_rbt_insertwk split: option.splits)
   qed
 qed
 
 lemma rbtreeify_map_filter_inter:
   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"
   assumes "rbt_sorted t2"
   shows "rbt_sorted (rbtreeify (map_filter_inter f t1 t2))"
     "rbt_lookup (rbtreeify (map_filter_inter f t1 t2)) k =
     (case rbt_lookup t1 k of None \<Rightarrow> None
     | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f k v w)))"
 proof -
   have map_of_map_filter: "map_of (List.map_filter (\<lambda>(k, v).
     case rbt_lookup t1 k of None \<Rightarrow> None | Some v' \<Rightarrow> Some (k, f k v' v)) xs) k =
     (case rbt_lookup t1 k of None \<Rightarrow> None
     | Some v \<Rightarrow> (case map_of xs k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f k v w)))" for xs k
     by (induction xs) (auto simp: List.map_filter_def split: option.splits) (* slow *)
   have map_fst_map_filter: "map fst (List.map_filter (\<lambda>(k, v).
     case rbt_lookup t1 k of None \<Rightarrow> None | Some v' \<Rightarrow> Some (k, f k v' v)) xs) =
     filter (\<lambda>k. rbt_lookup t1 k \<noteq> None) (map fst xs)" for xs
     by (induction xs) (auto simp: List.map_filter_def split: option.splits)
   have "sorted (map fst (map_filter_inter f t1 t2))"
     using sorted_filter[of id] rbt_sorted_entries[OF assms]
     by (auto simp: map_filter_inter_def map_fst_map_filter)
   moreover have "distinct (map fst (map_filter_inter f t1 t2))"
     using distinct_filter distinct_entries[OF assms]
     by (auto simp: map_filter_inter_def map_fst_map_filter)
   ultimately show
     "rbt_sorted (rbtreeify (map_filter_inter f t1 t2))"
     "rbt_lookup (rbtreeify (map_filter_inter f t1 t2)) k =
       (case rbt_lookup t1 k of None \<Rightarrow> None
       | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f k v w)))"
     using rbt_sorted_rbtreeify
     by (auto simp: rbt_lookup_rbtreeify map_filter_inter_def map_of_map_filter
         map_of_entries[OF assms] split: option.splits)
 qed
 
 lemma rbt_lookup_inter_rec: "rbt_sorted t1 \<Longrightarrow> rbt_sorted t2 \<Longrightarrow>
   rbt_sorted (rbt_inter_rec f t1 t2) \<and> rbt_lookup (rbt_inter_rec f t1 t2) k =
   (case rbt_lookup t1 k of None \<Rightarrow> None
   | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f k v w)))"
 proof(induction f t1 t2 arbitrary: k rule: rbt_inter_rec.induct)
   case (1 f t1 t2)
   obtain f' t1' t2' where flip: "(f', t2', t1') =
     (if flip_rbt t2 t1 then (\<lambda>k v v'. f k v' v, t1, t2) else (f, t2, t1))"
     by fastforce
   have rbt_sorted': "rbt_sorted t1'" "rbt_sorted t2'"
     using 1(3,4) flip
     by (auto split: if_splits)
   show ?case
   proof (cases t1')
     case Empty
     show ?thesis
       unfolding rbt_inter_rec.simps[of _ t1] flip[symmetric]
       using flip rbt_sorted' rbt_split_props[of t2] rbtreeify_map_filter_inter[OF rbt_sorted'(2)]
       by (auto simp: Empty split: option.splits)
   next
     case (Branch c l1 a b r1)
     {
       assume not_small: "\<not>small_rbt t2'"
       obtain l2 \<beta> r2 where rbt_split_t2': "rbt_split t2' a = (l2, \<beta>, r2)"
         by (cases "rbt_split t2' a") auto
       note rbt_split_t2'_props = rbt_split_props[OF rbt_split_t2' rbt_sorted'(2)]
       have rbt_sort: "rbt_sorted l1" "rbt_sorted r1" "rbt_sorted l2" "rbt_sorted r2"
         using 1(3,4) flip
         by (auto simp: Branch rbt_split_t2'_props split: if_splits)
       have inter_l1_l2: "rbt_sorted (rbt_inter_rec f' l1 l2)" "rbt_lookup (rbt_inter_rec f' l1 l2) k =
         (case rbt_lookup l1 k of None \<Rightarrow> None
         | Some v \<Rightarrow> (case rbt_lookup l2 k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f' k v w)))" for k
         using 1(1)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
         by (auto simp: not_small)
       have inter_r1_r2: "rbt_sorted (rbt_inter_rec f' r1 r2)" "rbt_lookup (rbt_inter_rec f' r1 r2) k =
         (case rbt_lookup r1 k of None \<Rightarrow> None
         | Some v \<Rightarrow> (case rbt_lookup r2 k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f' k v w)))" for k
         using 1(2)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
         by (auto simp: not_small)
       have inter_l1_l2_keys: "set (RBT_Impl.keys (rbt_inter_rec f' l1 l2)) =
         set (RBT_Impl.keys l1) \<inter> set (RBT_Impl.keys l2)"
         using inter_l1_l2(1)
         by (auto simp: rbt_lookup_iff_keys(1) inter_l1_l2(2) rbt_sort split: option.splits)
       have inter_r1_r2_keys: "set (RBT_Impl.keys (rbt_inter_rec f' r1 r2)) =
         set (RBT_Impl.keys r1) \<inter> set (RBT_Impl.keys r2)"
         using inter_r1_r2(1)
         by (auto simp: rbt_lookup_iff_keys(1) inter_r1_r2(2) rbt_sort split: option.splits)
       have inter_l1_l2_less: "rbt_inter_rec f' l1 l2 |\<guillemotleft> a"
         using rbt_sorted'(1) rbt_split_t2'_props
         by (auto simp: Branch rbt_less_prop inter_l1_l2_keys)
       have inter_r1_r2_greater: "a \<guillemotleft>| rbt_inter_rec f' r1 r2"
         using rbt_sorted'(1) rbt_split_t2'_props
         by (auto simp: Branch rbt_greater_prop inter_r1_r2_keys)
       have rbt_lookup_join2: "rbt_lookup (rbt_join2 (rbt_inter_rec f' l1 l2) (rbt_inter_rec f' r1 r2)) k =
         (case rbt_lookup (rbt_inter_rec f' l1 l2) k of None \<Rightarrow> rbt_lookup (rbt_inter_rec f' r1 r2) k
         | Some v \<Rightarrow> Some v)" for k
         using rbt_lookup_rbt_join2[OF inter_l1_l2(1) inter_r1_r2(1)] rbt_sorted'(1)
         by (fastforce simp: Branch inter_l1_l2_keys inter_r1_r2_keys rbt_less_prop rbt_greater_prop)
       have rbt_lookup_l1_k: "rbt_lookup l1 k = Some v \<Longrightarrow> k < a" for k v
         using rbt_sorted'(1) rbt_lookup_iff_keys(3)
         by (auto simp: Branch rbt_less_prop)
       have rbt_lookup_r1_k: "rbt_lookup r1 k = Some v \<Longrightarrow> a < k" for k v
         using rbt_sorted'(1) rbt_lookup_iff_keys(3)
         by (auto simp: Branch rbt_greater_prop)
       have "rbt_lookup (rbt_inter_rec f t1 t2) k =
         (case rbt_lookup t1' k of None \<Rightarrow> None
         | Some v \<Rightarrow> (case rbt_lookup t2' k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f' k v w)))"
         by (auto simp: Let_def rbt_inter_rec.simps[of _ t1] flip[symmetric] not_small Branch
             rbt_split_t2' rbt_lookup_join2 rbt_lookup_rbt_join inter_l1_l2_less inter_r1_r2_greater
             rbt_split_lookup[OF rbt_split_t2' rbt_sorted'(2)] inter_l1_l2 inter_r1_r2
             split!: if_splits option.splits dest: rbt_lookup_l1_k rbt_lookup_r1_k)
       moreover have "rbt_sorted (rbt_inter_rec f t1 t2)"
         using rbt_sorted' inter_l1_l2 inter_r1_r2 rbt_split_t2'_props not_small
         by (auto simp: Let_def rbt_inter_rec.simps[of _ t1] flip[symmetric] Branch rbt_split_t2'
             rbt_less_prop rbt_greater_prop inter_l1_l2_less inter_r1_r2_greater
             inter_l1_l2_keys inter_r1_r2_keys intro!: rbt_sorted_rbt_join rbt_sorted_rbt_join2
             split: if_splits option.splits dest!: bspec)
       ultimately have ?thesis
         using flip
         by (auto split: if_splits split: option.splits)
     }
     then show ?thesis
       unfolding rbt_inter_rec.simps[of _ t1] flip[symmetric]
       using rbt_sorted' flip rbtreeify_map_filter_inter[OF rbt_sorted'(2)]
       by (auto split: option.splits)
   qed
 qed
 
 lemma rbt_lookup_delete:
   assumes "inv_12 t" "rbt_sorted t"
   shows "rbt_lookup (rbt_delete x t) k = (if x = k then None else rbt_lookup t k)"
 proof -
   note rbt_sorted_del = rbt_del_rbt_sorted[OF assms(2), of x]
   show ?thesis
     using assms rbt_sorted_del rbt_del_in_tree rbt_lookup_from_in_tree[OF assms(2) rbt_sorted_del]
     by (fastforce simp: inv_12_def rbt_delete_def rbt_lookup_iff_keys(2) keys_entries)
 qed
 
 lemma fold_rbt_delete:
   assumes "inv_12 t1" "rbt_sorted t1" "rbt_sorted t2"
   shows "inv_12 (RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t2 t1) \<and>
     rbt_sorted (RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t2 t1) \<and>
     rbt_lookup (RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t2 t1) k =
     (case rbt_lookup t1 k of None \<Rightarrow> None
     | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> Some v | _ \<Rightarrow> None))"
 proof -
   define xs where "xs = RBT_Impl.entries t2"
   show "inv_12 (RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t2 t1) \<and>
     rbt_sorted (RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t2 t1) \<and>
     rbt_lookup (RBT_Impl.fold (\<lambda>k _ t. rbt_delete k t) t2 t1) k =
     (case rbt_lookup t1 k of None \<Rightarrow> None
     | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> Some v | _ \<Rightarrow> None))"
     using assms(1,2)
     unfolding map_of_entries[OF assms(3), symmetric] RBT_Impl.fold_def xs_def[symmetric]
     by (induction xs arbitrary: t1 rule: rev_induct)
        (auto simp: rbt_delete rbt_sorted_delete rbt_lookup_delete split!: option.splits)
 qed
 
 lemma rbtreeify_filter_minus:
   assumes "rbt_sorted t1"
   shows "rbt_sorted (rbtreeify (filter_minus t1 t2)) \<and>
     rbt_lookup (rbtreeify (filter_minus t1 t2)) k =
     (case rbt_lookup t1 k of None \<Rightarrow> None
     | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> Some v | _ \<Rightarrow> None))"
 proof -
   have map_of_filter: "map_of (filter (\<lambda>(k, _). rbt_lookup t2 k = None) xs) k =
     (case map_of xs k of None \<Rightarrow> None
     | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> Some v | Some x \<Rightarrow> Map.empty x))"
       for xs :: "('a \<times> 'b) list"
     by (induction xs) (auto split: option.splits)
   have map_fst_filter_minus: "map fst (filter_minus t1 t2) =
     filter (\<lambda>k. rbt_lookup t2 k = None) (map fst (RBT_Impl.entries t1))"
     by (auto simp: filter_minus_def filter_map comp_def case_prod_unfold)
   have "sorted (map fst (filter_minus t1 t2))" "distinct (map fst (filter_minus t1 t2))"
     using distinct_filter distinct_entries[OF assms]
       sorted_filter[of id] rbt_sorted_entries[OF assms]
     by (auto simp: map_fst_filter_minus intro!: rbt_sorted_rbtreeify)
   then show ?thesis
     by (auto simp: rbt_lookup_rbtreeify filter_minus_def map_of_filter map_of_entries[OF assms]
         intro!: rbt_sorted_rbtreeify)
 qed
 
 lemma rbt_lookup_minus_rec: "inv_12 t1 \<Longrightarrow> rbt_sorted t1 \<Longrightarrow> rbt_sorted t2 \<Longrightarrow>
   rbt_sorted (rbt_minus_rec t1 t2) \<and> rbt_lookup (rbt_minus_rec t1 t2) k =
   (case rbt_lookup t1 k of None \<Rightarrow> None
   | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> Some v | _ \<Rightarrow> None))"
 proof(induction t1 t2 arbitrary: k rule: rbt_minus_rec.induct)
   case (1 t1 t2)
   show ?case
   proof (cases t2)
     case Empty
     show ?thesis
       using rbtreeify_filter_minus[OF 1(4)] 1(4)
       by (auto simp: rbt_minus_rec.simps[of t1] Empty split: option.splits)
   next
     case (Branch c l2 a b r2)
     {
       assume not_small: "\<not>small_rbt t2" "\<not>small_rbt t1"
       obtain l1 \<beta> r1 where rbt_split_t1: "rbt_split t1 a = (l1, \<beta>, r1)"
         by (cases "rbt_split t1 a") auto
       note rbt_split_t1_props = rbt_split_props[OF rbt_split_t1 1(4)]
       have minus_l1_l2: "rbt_sorted (rbt_minus_rec l1 l2)"
         "rbt_lookup (rbt_minus_rec l1 l2) k =
         (case rbt_lookup l1 k of None \<Rightarrow> None
         | Some v \<Rightarrow> (case rbt_lookup l2 k of None \<Rightarrow> Some v | Some x \<Rightarrow> None))" for k
         using 1(1)[OF not_small Branch rbt_split_t1[symmetric] refl] 1(5) rbt_split_t1_props
           rbt_split[OF rbt_split_t1 1(3)]
         by (auto simp: Branch)
       have minus_r1_r2: "rbt_sorted (rbt_minus_rec r1 r2)"
         "rbt_lookup (rbt_minus_rec r1 r2) k =
         (case rbt_lookup r1 k of None \<Rightarrow> None
         | Some v \<Rightarrow> (case rbt_lookup r2 k of None \<Rightarrow> Some v | Some x \<Rightarrow> None))" for k
         using 1(2)[OF not_small Branch rbt_split_t1[symmetric] refl] 1(5) rbt_split_t1_props
           rbt_split[OF rbt_split_t1 1(3)]
         by (auto simp: Branch)
       have minus_l1_l2_keys: "set (RBT_Impl.keys (rbt_minus_rec l1 l2)) =
         set (RBT_Impl.keys l1) - set (RBT_Impl.keys l2)"
         using minus_l1_l2(1) 1(5) rbt_lookup_iff_keys(3) rbt_split_t1_props
         by (auto simp: Branch rbt_lookup_iff_keys(1) minus_l1_l2(2) split: option.splits)
       have minus_r1_r2_keys: "set (RBT_Impl.keys (rbt_minus_rec r1 r2)) =
         set (RBT_Impl.keys r1) - set (RBT_Impl.keys r2)"
         using minus_r1_r2(1) 1(5) rbt_lookup_iff_keys(3) rbt_split_t1_props
         by (auto simp: Branch rbt_lookup_iff_keys(1) minus_r1_r2(2) split: option.splits)
       have rbt_lookup_join2: "rbt_lookup (rbt_join2 (rbt_minus_rec l1 l2) (rbt_minus_rec r1 r2)) k =
         (case rbt_lookup (rbt_minus_rec l1 l2) k of None \<Rightarrow> rbt_lookup (rbt_minus_rec r1 r2) k
         | Some v \<Rightarrow> Some v)" for k
         using rbt_lookup_rbt_join2[OF minus_l1_l2(1) minus_r1_r2(1)] rbt_split_t1_props
         by (fastforce simp: minus_l1_l2_keys minus_r1_r2_keys)
       have lookup_l1_r1_a: "rbt_lookup l1 a = None" "rbt_lookup r1 a = None"
         using rbt_split_t1_props
         by (auto simp: rbt_lookup_iff_keys(2))
       have "rbt_lookup (rbt_minus_rec t1 t2) k =
         (case rbt_lookup t1 k of None \<Rightarrow> None
         | Some v \<Rightarrow> (case rbt_lookup t2 k of None \<Rightarrow> Some v | _ \<Rightarrow> None))"
         using not_small rbt_lookup_iff_keys(2)[of l1] rbt_lookup_iff_keys(3)[of l1]
           rbt_lookup_iff_keys(3)[of r1] rbt_split_t1_props
         by (auto simp: rbt_minus_rec.simps[of t1] Branch rbt_split_t1 rbt_lookup_join2
             minus_l1_l2(2) minus_r1_r2(2) rbt_split_lookup[OF rbt_split_t1 1(4)] lookup_l1_r1_a
             split: option.splits)
       moreover have "rbt_sorted (rbt_minus_rec t1 t2)"
         using not_small minus_l1_l2(1) minus_r1_r2(1) rbt_split_t1_props rbt_sorted_rbt_join2
         by (fastforce simp: rbt_minus_rec.simps[of t1] Branch rbt_split_t1 minus_l1_l2_keys minus_r1_r2_keys)
       ultimately have ?thesis
         by (auto split: if_splits split: option.splits)
     }
     then show ?thesis
       using fold_rbt_delete[OF 1(3,4,5)] rbtreeify_filter_minus[OF 1(4)]
       by (auto simp: rbt_minus_rec.simps[of t1])
   qed
 qed
 
 end
 
 context ord begin
 
 definition rbt_union_with_key :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
 where
   "rbt_union_with_key f t1 t2 = paint B (rbt_union_swap_rec f False t1 t2)"
 
 definition rbt_union_with where
   "rbt_union_with f = rbt_union_with_key (\<lambda>_. f)"
 
 definition rbt_union where
   "rbt_union = rbt_union_with_key (%_ _ rv. rv)"
 
 definition rbt_inter_with_key :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
 where
   "rbt_inter_with_key f t1 t2 = paint B (rbt_inter_swap_rec f False t1 t2)"
 
 definition rbt_inter_with where
   "rbt_inter_with f = rbt_inter_with_key (\<lambda>_. f)"
 
 definition rbt_inter where
   "rbt_inter = rbt_inter_with_key (\<lambda>_ _ rv. rv)"
 
 definition rbt_minus where
   "rbt_minus t1 t2 = paint B (rbt_minus_rec t1 t2)"
 
 end
 
 context linorder begin
 
 lemma is_rbt_rbt_unionwk [simp]:
   "\<lbrakk> is_rbt t1; is_rbt t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_union_with_key f t1 t2)"
   using rbt_union_rec rbt_lookup_union_rec
   by (fastforce simp: rbt_union_with_key_def rbt_union_swap_rec is_rbt_def inv_12_def)
 
 lemma rbt_lookup_rbt_unionwk:
   "\<lbrakk> rbt_sorted t1; rbt_sorted t2 \<rbrakk> 
   \<Longrightarrow> rbt_lookup (rbt_union_with_key f t1 t2) k = 
   (case rbt_lookup t1 k of None \<Rightarrow> rbt_lookup t2 k 
    | Some v \<Rightarrow> case rbt_lookup t2 k of None \<Rightarrow> Some v 
               | Some w \<Rightarrow> Some (f k v w))"
   using rbt_lookup_union_rec
   by (auto simp: rbt_union_with_key_def rbt_union_swap_rec)
 
 lemma rbt_unionw_is_rbt: "\<lbrakk> is_rbt lt; is_rbt rt \<rbrakk> \<Longrightarrow> is_rbt (rbt_union_with f lt rt)"
 by(simp add: rbt_union_with_def)
 
 lemma rbt_union_is_rbt: "\<lbrakk> is_rbt lt; is_rbt rt \<rbrakk> \<Longrightarrow> is_rbt (rbt_union lt rt)"
 by(simp add: rbt_union_def)
 
 lemma rbt_lookup_rbt_union:
   "\<lbrakk> rbt_sorted s; rbt_sorted t \<rbrakk> \<Longrightarrow>
   rbt_lookup (rbt_union s t) = rbt_lookup s ++ rbt_lookup t"
 by(rule ext)(simp add: rbt_lookup_rbt_unionwk rbt_union_def map_add_def split: option.split)
 
 lemma rbt_interwk_is_rbt [simp]:
   "\<lbrakk> is_rbt t1; is_rbt t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_inter_with_key f t1 t2)"
   using rbt_inter_rec rbt_lookup_inter_rec
   by (fastforce simp: rbt_inter_with_key_def rbt_inter_swap_rec is_rbt_def inv_12_def rbt_sorted_paint)
 
 lemma rbt_interw_is_rbt:
   "\<lbrakk> is_rbt t1; is_rbt t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_inter_with f t1 t2)"
 by(simp add: rbt_inter_with_def)
 
 lemma rbt_inter_is_rbt:
   "\<lbrakk> is_rbt t1; is_rbt t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_inter t1 t2)"
 by(simp add: rbt_inter_def)
 
 lemma rbt_lookup_rbt_interwk:
   "\<lbrakk> rbt_sorted t1; rbt_sorted t2 \<rbrakk>
   \<Longrightarrow> rbt_lookup (rbt_inter_with_key f t1 t2) k =
   (case rbt_lookup t1 k of None \<Rightarrow> None 
    | Some v \<Rightarrow> case rbt_lookup t2 k of None \<Rightarrow> None
                | Some w \<Rightarrow> Some (f k v w))"
   using rbt_lookup_inter_rec
   by (auto simp: rbt_inter_with_key_def rbt_inter_swap_rec)
 
 lemma rbt_lookup_rbt_inter:
   "\<lbrakk> rbt_sorted t1; rbt_sorted t2 \<rbrakk>
   \<Longrightarrow> rbt_lookup (rbt_inter t1 t2) = rbt_lookup t2 |` dom (rbt_lookup t1)"
 by(auto simp add: rbt_inter_def rbt_lookup_rbt_interwk restrict_map_def split: option.split)
 
 lemma rbt_minus_is_rbt:
   "\<lbrakk> is_rbt t1; is_rbt t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_minus t1 t2)"
   using rbt_minus_rec[of t1 t2] rbt_lookup_minus_rec[of t1 t2]
   by (auto simp: rbt_minus_def is_rbt_def inv_12_def)
 
 lemma rbt_lookup_rbt_minus:
   "\<lbrakk> is_rbt t1; is_rbt t2 \<rbrakk>
   \<Longrightarrow> rbt_lookup (rbt_minus t1 t2) = rbt_lookup t1 |` (- dom (rbt_lookup t2))"
   by (rule ext)
      (auto simp: rbt_minus_def is_rbt_def inv_12_def restrict_map_def rbt_lookup_minus_rec
       split: option.splits)
 
 end
 
 
 subsection \<open>Code generator setup\<close>
 
 lemmas [code] =
   ord.rbt_less_prop
   ord.rbt_greater_prop
   ord.rbt_sorted.simps
   ord.rbt_lookup.simps
   ord.is_rbt_def
   ord.rbt_ins.simps
   ord.rbt_insert_with_key_def
   ord.rbt_insertw_def
   ord.rbt_insert_def
   ord.rbt_del_from_left.simps
   ord.rbt_del_from_right.simps
   ord.rbt_del.simps
   ord.rbt_delete_def
   ord.rbt_split.simps
   ord.rbt_union_swap_rec.simps
   ord.map_filter_inter_def
   ord.rbt_inter_swap_rec.simps
   ord.filter_minus_def
   ord.rbt_minus_rec.simps
   ord.rbt_union_with_key_def
   ord.rbt_union_with_def
   ord.rbt_union_def
   ord.rbt_inter_with_key_def
   ord.rbt_inter_with_def
   ord.rbt_inter_def
   ord.rbt_minus_def
   ord.rbt_map_entry.simps
   ord.rbt_bulkload_def
 
 text \<open>More efficient implementations for \<^term>\<open>entries\<close> and \<^term>\<open>keys\<close>\<close>
 
 definition gen_entries :: 
   "(('a \<times> 'b) \<times> ('a, 'b) rbt) list \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
 where
   "gen_entries kvts t = entries t @ concat (map (\<lambda>(kv, t). kv # entries t) kvts)"
 
 lemma gen_entries_simps [simp, code]:
   "gen_entries [] Empty = []"
   "gen_entries ((kv, t) # kvts) Empty = kv # gen_entries kvts t"
   "gen_entries kvts (Branch c l k v r) = gen_entries (((k, v), r) # kvts) l"
 by(simp_all add: gen_entries_def)
 
 lemma entries_code [code]:
   "entries = gen_entries []"
 by(simp add: gen_entries_def fun_eq_iff)
 
 definition gen_keys :: "('a \<times> ('a, 'b) rbt) list \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'a list"
 where "gen_keys kts t = RBT_Impl.keys t @ concat (List.map (\<lambda>(k, t). k # keys t) kts)"
 
 lemma gen_keys_simps [simp, code]:
   "gen_keys [] Empty = []"
   "gen_keys ((k, t) # kts) Empty = k # gen_keys kts t"
   "gen_keys kts (Branch c l k v r) = gen_keys ((k, r) # kts) l"
 by(simp_all add: gen_keys_def)
 
 lemma keys_code [code]:
   "keys = gen_keys []"
 by(simp add: gen_keys_def fun_eq_iff)
 
 text \<open>Restore original type constraints for constants\<close>
 setup \<open>
   fold Sign.add_const_constraint
     [(\<^const_name>\<open>rbt_less\<close>, SOME \<^typ>\<open>('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool\<close>),
      (\<^const_name>\<open>rbt_greater\<close>, SOME \<^typ>\<open>('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool\<close>),
      (\<^const_name>\<open>rbt_sorted\<close>, SOME \<^typ>\<open>('a :: linorder, 'b) rbt \<Rightarrow> bool\<close>),
      (\<^const_name>\<open>rbt_lookup\<close>, SOME \<^typ>\<open>('a :: linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b\<close>),
      (\<^const_name>\<open>is_rbt\<close>, SOME \<^typ>\<open>('a :: linorder, 'b) rbt \<Rightarrow> bool\<close>),
      (\<^const_name>\<open>rbt_ins\<close>, SOME \<^typ>\<open>('a::linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_insert_with_key\<close>, SOME \<^typ>\<open>('a::linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_insert_with\<close>, SOME \<^typ>\<open>('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_insert\<close>, SOME \<^typ>\<open>('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_del_from_left\<close>, SOME \<^typ>\<open>('a::linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_del_from_right\<close>, SOME \<^typ>\<open>('a::linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_del\<close>, SOME \<^typ>\<open>('a::linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_delete\<close>, SOME \<^typ>\<open>('a::linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_union_with_key\<close>, SOME \<^typ>\<open>('a::linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_union_with\<close>, SOME \<^typ>\<open>('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a::linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_union\<close>, SOME \<^typ>\<open>('a::linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_map_entry\<close>, SOME \<^typ>\<open>'a::linorder \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt\<close>),
      (\<^const_name>\<open>rbt_bulkload\<close>, SOME \<^typ>\<open>('a \<times> 'b) list \<Rightarrow> ('a::linorder,'b) rbt\<close>)]
 \<close>
 
-hide_const (open) R B Empty entries keys fold gen_keys gen_entries
+hide_const (open) MR MB R B Empty entries keys fold gen_keys gen_entries
 
 end