(*  Title:      HOL/Library/RBT_Impl.thy

     Author:     Markus Reiter, TU Muenchen

     Author:     Alexander Krauss, TU Muenchen

 *)

 header {* Implementation of Red-Black Trees *}

 theory RBT_Impl

 imports Main

 begin

 text {*

   For applications, you should use theory @{text RBT} which defines

   an abstract type of red-black tree obeying the invariant.

 *}

 subsection {* Datatype of RB trees *}

 datatype color = R | B

 datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"

 lemma rbt_cases:

   obtains (Empty) "t = Empty" 

   | (Red) l k v r where "t = Branch R l k v r" 

   | (Black) l k v r where "t = Branch B l k v r"

 proof (cases t)

   case Empty with that show thesis by blast

 next

   case (Branch c) with that show thesis by (cases c) blast+

 qed

 subsection {* Tree properties *}

 subsubsection {* Content of a tree *}

 primrec entries :: "('a, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"

 where 

   "entries Empty = []"

 | "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"

 abbreviation (input) entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"

 where

   "entry_in_tree k v t \<equiv> (k, v) \<in> set (entries t)"

 definition keys :: "('a, 'b) rbt \<Rightarrow> 'a list" where

   "keys t = map fst (entries t)"

 lemma keys_simps [simp, code]:

   "keys Empty = []"

   "keys (Branch c l k v r) = keys l @ k # keys r"

   by (simp_all add: keys_def)

 lemma entry_in_tree_keys:

   assumes "(k, v) \<in> set (entries t)"

   shows "k \<in> set (keys t)"

 proof -

   from assms have "fst (k, v) \<in> fst ` set (entries t)" by (rule imageI)

   then show ?thesis by (simp add: keys_def)

 qed

 lemma keys_entries:

   "k \<in> set (keys t) \<longleftrightarrow> (\<exists>v. (k, v) \<in> set (entries t))"

   by (auto intro: entry_in_tree_keys) (auto simp add: keys_def)

 lemma non_empty_rbt_keys: 

   "t \<noteq> rbt.Empty \<Longrightarrow> keys t \<noteq> []"

   by (cases t) simp_all

 subsubsection {* Search tree properties *}

 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 sorted_Cons 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 sorted_Cons 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 {* Tree lookup *}

 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: split_if_asm)

   moreover from Branch have "finite (insert a (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2)))" by simp

   ultimately show ?case by (rule finite_subset)

 qed 

 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 = 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 {* Red-black properties *}

 primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"

 where

   "color_of Empty = B"

 | "color_of (Branch c _ _ _ _) = c"

 primrec bheight :: "('a,'b) rbt \<Rightarrow> nat"

 where

   "bheight Empty = 0"

 | "bheight (Branch c lt k v rt) = (if c = B then Suc (bheight lt) else bheight lt)"

 primrec inv1 :: "('a, 'b) rbt \<Rightarrow> bool"

 where

   "inv1 Empty = True"

 | "inv1 (Branch c lt k v rt) \<longleftrightarrow> inv1 lt \<and> inv1 rt \<and> (c = B \<or> color_of lt = B \<and> color_of rt = B)"

 primrec inv1l :: "('a, 'b) rbt \<Rightarrow> bool" -- {* Weaker version *}

 where

   "inv1l Empty = True"

 | "inv1l (Branch c l k v r) = (inv1 l \<and> inv1 r)"

 lemma [simp]: "inv1 t \<Longrightarrow> inv1l t" by (cases t) simp+

 primrec inv2 :: "('a, 'b) rbt \<Rightarrow> bool"

 where

   "inv2 Empty = True"

 | "inv2 (Branch c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bheight lt = bheight rt)"

 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 {* Insertion *}

 fun (* slow, due to massive case splitting *)

   balance :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"

 where

   "balance (Branch R a w x b) s t (Branch R c y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |

   "balance (Branch R (Branch R a w x b) s t c) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |

   "balance (Branch R a w x (Branch R b s t c)) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |

   "balance a w x (Branch R b s t (Branch R c y z d)) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |

   "balance a w x (Branch R (Branch R b s t c) y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |

   "balance a s t b = Branch B a s t b"

 lemma balance_inv1: "\<lbrakk>inv1l l; inv1l r\<rbrakk> \<Longrightarrow> inv1 (balance l k v r)" 

   by (induct l k v r rule: balance.induct) auto

 lemma balance_bheight: "bheight l = bheight r \<Longrightarrow> bheight (balance l k v r) = Suc (bheight l)"

   by (induct l k v r rule: balance.induct) auto

 lemma balance_inv2: 

   assumes "inv2 l" "inv2 r" "bheight l = bheight r"

   shows "inv2 (balance l k v r)"

   using assms

   by (induct l k v r rule: balance.induct) auto

 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:

   assumes "is_rbt t"

   shows "rbt_lookup (rbt_insert_with f k v t) = (rbt_lookup t)(k \<mapsto> (if k:dom (rbt_lookup t) then f (the (rbt_lookup t k)) v else v))"

 using assms

 unfolding rbt_insertw_def

 by (rule_tac ext) (cases "rbt_lookup t k", auto simp:rbt_lookup_rbt_insertwk dom_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: 

   assumes "is_rbt t"

   shows "rbt_lookup (rbt_insert k v t) = (rbt_lookup t)(k\<mapsto>v)"

 unfolding rbt_insert_def

 using assms

 by (rule_tac ext) (simp add: rbt_lookup_rbt_insertwk split:option.split)

 end

 subsection {* Deletion *}

 lemma bheight_paintR'[simp]: "color_of t = B \<Longrightarrow> bheight (paint R t) = bheight t - 1"

 by (cases t rule: rbt_cases) auto

 fun

   balance_left :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"

 where

   "balance_left (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |

   "balance_left bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |

   "balance_left bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |

   "balance_left t k x s = Empty"

 lemma balance_left_inv2_with_inv1:

   assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"

   shows "bheight (balance_left lt k v rt) = bheight lt + 1"

   and   "inv2 (balance_left lt k v rt)"

 using assms 

 by (induct lt k v rt rule: balance_left.induct) (auto simp: balance_inv2 balance_bheight)

 lemma balance_left_inv2_app: 

   assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"

   shows "inv2 (balance_left lt k v rt)" 

         "bheight (balance_left lt k v rt) = bheight rt"

 using assms 

 by (induct lt k v rt rule: balance_left.induct) (auto simp add: balance_inv2 balance_bheight)+ 

 lemma balance_left_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balance_left a k x b)"

   by (induct a k x b rule: balance_left.induct) (simp add: balance_inv1)+

 lemma balance_left_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balance_left lt k x rt)"

 by (induct lt k x rt rule: balance_left.induct) (auto simp: balance_inv1)

 lemma (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 `xx = yy` `xx = k` have "rbt_sorted (Branch c aa yy ss bb) \<and> k = yy" by simp

       hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: rbt_less_nit rbt_greater_prop)

       with `xx = yy` 2 `xx = k` show ?thesis by (simp add: combine_in_tree)

     qed (simp add: combine_in_tree)

   qed simp+

 next    

   case (3 xx lta zz vv rta yy ss bb)

   def mt[simp]: mt == "Branch B lta zz vv rta"

   from 3 have "inv2 mt \<and> inv1 mt" by simp

   hence "inv2 (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" `xx = k` 

    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)

   def mt[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" `xx = k` show ?thesis by (auto simp: rbt_less_nit)

   qed simp

 next

   case ("6_2" xx aa yy ss vaa vbb vdd vc)

   thus ?case proof (cases "xx = k")

     case True

     with "6_2" have "k > yy \<and> 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 {* Modifying existing entries *}

 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 := Option.map f (rbt_lookup t k))"

   by (induct t) (auto split: option.splits simp add: fun_eq_iff)

 subsection {* Mapping all entries *}

 primrec

   map :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'c) rbt"

 where

   "map f Empty = Empty"

 | "map f (Branch c lt k v rt) = Branch c (map f lt) k (f k v) (map f rt)"

 lemma map_entries [simp]: "entries (map f t) = List.map (\<lambda>(k, v). (k, f k v)) (entries t)"

   by (induct t) auto

 lemma map_keys [simp]: "keys (map f t) = keys t" by (simp add: keys_def split_def)

 lemma map_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 = Option.map (f x) (rbt_lookup t x)"

   apply(induct t)

   apply auto

   apply(subgoal_tac "x = a")

   apply auto

   done

  (* FIXME: simproc "antisym less" does not work for linorder context, only for linorder type class

     by (induct t) auto *)

 hide_const (open) map

 subsection {* Folding over entries *}

 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where

   "fold f t = List.fold (prod_case 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)

 (* fold with continuation predicate *)

 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 {* Bulkloading a tree *}

 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 (prod_case rbt_insert) ys t) = rbt_lookup t ++ map_of (rev ys)"

       by (induct ys) (simp_all add: rbt_bulkload_def rbt_lookup_rbt_insert prod_case_beta)

   from this Empty_is_rbt have

     "rbt_lookup (List.fold (prod_case rbt_insert) (rev xs) Empty) = rbt_lookup Empty ++ map_of xs"

      by (simp add: `ys = rev xs`)

   then show ?thesis by (simp add: rbt_bulkload_def rbt_lookup_Empty foldr_conv_fold)

 qed

 end

 subsection {* Building a RBT from a sorted list *}

 text {* 

   These functions have been adapted from 

   Andrew W. Appel, Efficient Verified Red-Black Trees (September 2011) 

 *}

 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) = 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_div_mod prod.simps)

 lemma rbtreeify_g_code [code]:

   "rbtreeify_g n kvs =

    (if n = 0 \<or> n = 1 then (Empty, kvs)

     else let (n', r) = 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_div_mod prod.simps)

 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 = (RBT_Impl.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 mult_div_cancel)

       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.simps(2) also note list.simps(5) 

       also note prod.simps(2) 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] mult_div_cancel)

     next

       case False

       hence "length (snd (rbtreeify_f n kvs)) = 

         length (snd (rbtreeify_f (Suc (2 * (n div 2))) kvs))"

         by(metis Suc_eq_plus1_left comm_semiring_1_class.normalizing_semiring_rules(7)

              mod_2_not_eq_zero_eq_one_nat semiring_div_class.mod_div_equality')

       also from "1.prems" `\<not> n \<le> 1` obtain k v kvs' 

         where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto

       also have "0 < n div 2" using `\<not> n \<le> 1` 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" `\<not> n \<le> 1` 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.simps(2) also note list.simps(5) 

       also note prod.simps(2) 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 `0 < n` 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 mult_div_cancel)

       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 `1 < n` 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" `1 < n`

       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.simps(2) also note list.simps(5) 

       also note prod.simps(2) 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 `0 < n div 2`

       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] mult_div_cancel)

     next

       case False

       hence "length (snd (rbtreeify_g n kvs)) = 

         length (snd (rbtreeify_g (Suc (2 * (n div 2))) kvs))"

         by(metis Suc_eq_plus1_left comm_semiring_1_class.normalizing_semiring_rules(7) 

             mod_2_not_eq_zero_eq_one_nat semiring_div_class.mod_div_equality')

       also from "2.prems" `1 < n` obtain k v kvs'

         where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto

       also have "0 < n div 2" using `1 < n` 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" `1 < n` 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.simps(2) also note list.simps(5) 

       also note prod.simps(2) 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 `0 < n div 2`

       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 div_mod_equality' minus_nat.diff_0 nat_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 mult_div_cancel)

       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 div_mod_equality' minus_nat.diff_0 nat_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 mult_div_cancel)

       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 `rbtreeify_f n kvs = (t, (k, v) # kvs')` `n \<le> length kvs`]

   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 `sorted (map fst kvs)` 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 sorted_Cons)

   moreover from `distinct (map fst kvs)` 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 `n \<le> Suc (length kvs')` `n \<le> length kvs` 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 `sorted (map fst kvs)` `distinct (map fst kvs)`

   have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule f_even.IH)

   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"

     using `sorted (map fst kvs)` `distinct (map fst kvs)`

     by(subst (asm) (1 2) unfold, simp add: sorted_append sorted_Cons)+

   hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule f_even.IH)

   ultimately show ?case

     using `0 < n` `rbtreeify_f n kvs = (t, (k, v) # kvs')` by simp

 next

   case (f_odd n kvs t k v kvs')

   from rbtreeify_fD[OF `rbtreeify_f n kvs = (t, (k, v) # kvs')` `n \<le> length kvs`]

   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 `sorted (map fst kvs)` 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 sorted_Cons)

   moreover from `distinct (map fst kvs)` 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 `n \<le> length kvs'` `n \<le> length kvs` set_take_subset[of n kvs']

     by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)

   moreover from `sorted (map fst kvs)` `distinct (map fst kvs)`

   have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule f_odd.IH)

   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"

     using `sorted (map fst kvs)` `distinct (map fst kvs)`

     by(subst (asm) (1 2) unfold, simp add: sorted_append sorted_Cons)+

   hence "rbt_sorted (fst (rbtreeify_f n kvs'))" by(rule f_odd.IH)

   ultimately show ?case 

     using `0 < n` `rbtreeify_f n kvs = (t, (k, v) # kvs')` by simp

 next

   case (g_even n kvs t k v kvs')

   from rbtreeify_gD[OF `rbtreeify_g n kvs = (t, (k, v) # kvs')` `n \<le> Suc (length kvs)`]

   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 `sorted (map fst kvs)` 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 sorted_Cons)

   moreover from `distinct (map fst kvs)` 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 `n \<le> Suc (length kvs')` `n \<le> Suc (length kvs)` set_take_subset[of "n - 1" kvs']

     by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)

   moreover from `sorted (map fst kvs)` `distinct (map fst kvs)`

   have "rbt_sorted (fst (rbtreeify_g n kvs))" by(rule g_even.IH)

   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"

     using `sorted (map fst kvs)` `distinct (map fst kvs)`

     by(subst (asm) (1 2) unfold, simp add: sorted_append sorted_Cons)+

   hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule g_even.IH)

   ultimately show ?case using `0 < n` `rbtreeify_g n kvs = (t, (k, v) # kvs')` by simp

 next

   case (g_odd n kvs t k v kvs')

   from rbtreeify_fD[OF `rbtreeify_f n kvs = (t, (k, v) # kvs')` `n \<le> length kvs`]

   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 `sorted (map fst kvs)` 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 sorted_Cons)

   moreover from `distinct (map fst kvs)` 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 `n \<le> Suc (length kvs')` `n \<le> length kvs` set_take_subset[of "n - 1" kvs']

     by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)

   moreover from `sorted (map fst kvs)` `distinct (map fst kvs)`

   have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule g_odd.IH)

   moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"

     using `sorted (map fst kvs)` `distinct (map fst kvs)`

     by(subst (asm) (1 2) unfold, simp add: sorted_append sorted_Cons)+

   hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule g_odd.IH)

   ultimately show ?case

     using `0 < n` `rbtreeify_f n kvs = (t, (k, v) # kvs')` 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 {* 

   Functions to compare the height of two rbt trees, taken from 

   Andrew W. Appel, Efficient Verified Red-Black Trees (September 2011)

 *}

 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 compare_case compare_rec 

   Abs_compare Rep_compare compare_rep_set

 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.recs compare.simps

   compare.size compare.case_cong compare.weak_case_cong compare.cases 

   compare.nchotomy compare.split compare.split_asm compare_rec_def

   compare.eq.refl compare.eq.simps

   compare.EQ_def compare.GT_def compare.LT_def

   equal_compare_def

   skip_red_def skip_red.simps skip_red.cases skip_red.induct 

   skip_black_def

   compare_height_def compare_height.simps

 subsection {* union and intersection of sorted associative lists *}

 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: sorted_Cons 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 sorted_Cons 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 simp add: sorted_Cons 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 add: sorted_Cons 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 add: sorted_Cons 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> Option.map (f k v) (map_of ys k))"

 apply(induct f xs ys rule: sinter_with.induct)

 apply(auto simp add: sorted_Cons Option.map_def 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 (Option.map f \<circ> g) xs"

 by(auto simp add: List.map_filter_def)

 lemma map_filter_option_map_const: 

   "List.map_filter (\<lambda>x. Option.map (\<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)

 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 =

   (case RBT_Impl.compare_height t1 t1 t2 t2

    of compare.EQ \<Rightarrow> rbtreeify (sunion_with f (entries t1) (entries t2))

     | compare.LT \<Rightarrow> fold (rbt_insert_with_key (\<lambda>k v w. f k w v)) t1 t2

     | compare.GT \<Rightarrow> fold (rbt_insert_with_key f) t2 t1)"

 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 =

   (case RBT_Impl.compare_height t1 t1 t2 t2 

    of compare.EQ \<Rightarrow> rbtreeify (sinter_with f (entries t1) (entries t2))

     | compare.LT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). Option.map (\<lambda>w. (k, f k v w)) (rbt_lookup t2 k)) (entries t1))

     | compare.GT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). Option.map (\<lambda>w. (k, f k w v)) (rbt_lookup t1 k)) (entries 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)"

 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 -

   def xs \<equiv> "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_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 -

   def xs \<equiv> "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 is_rbt_rbt_unionwk [simp]:

   "\<lbrakk> is_rbt t1; is_rbt t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_union_with_key f t1 t2)"

 by(simp add: rbt_union_with_key_def Let_def is_rbt_fold_rbt_insertwk is_rbt_rbtreeify rbt_sorted_entries distinct_entries split: compare.split)

 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))"

 by(auto simp add: rbt_union_with_key_def Let_def rbt_lookup_fold_rbt_insertwk rbt_sorted_entries distinct_entries map_of_sunion_with map_of_entries rbt_lookup_rbtreeify split: option.split compare.split)

 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> rbt_sorted t1; rbt_sorted t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_inter_with_key f t1 t2)"

 by(auto simp add: rbt_inter_with_key_def Let_def map_map_filter split_def o_def option_map_comp map_filter_option_map_const sorted_filter[where f=id, simplified] rbt_sorted_entries distinct_entries intro: is_rbt_rbtreeify split: compare.split)

 lemma rbt_interw_is_rbt:

   "\<lbrakk> rbt_sorted t1; rbt_sorted t2 \<rbrakk> \<Longrightarrow> is_rbt (rbt_inter_with f t1 t2)"

 by(simp add: rbt_inter_with_def)

 lemma rbt_inter_is_rbt:

   "\<lbrakk> rbt_sorted t1; rbt_sorted 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))"

 by(auto 4 3 simp add: rbt_inter_with_key_def Let_def map_of_entries[symmetric] rbt_lookup_rbtreeify map_map_filter split_def o_def option_map_comp map_filter_option_map_const sorted_filter[where f=id, simplified] rbt_sorted_entries distinct_entries map_of_sinter_with map_of_eq_None_iff set_map_filter split: option.split compare.split intro: rev_image_eqI dest: rbt_sorted_entries_right_unique)

 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)

 end

 subsection {* Code generator setup *}

 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.sunion_with.simps

   ord.sinter_with.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_map_entry.simps

   ord.rbt_bulkload_def

 text {* More efficient implementations for @{term entries} and @{term keys} *}

 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 {* Restore original type constraints for constants *}

 setup {*

   fold Sign.add_const_constraint

     [(@{const_name rbt_less}, SOME @{typ "('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"}),

      (@{const_name rbt_greater}, SOME @{typ "('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"}),

      (@{const_name rbt_sorted}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> bool"}),

      (@{const_name rbt_lookup}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"}),

      (@{const_name is_rbt}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> bool"}),

      (@{const_name rbt_ins}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_insert_with_key}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_insert_with}, SOME @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_insert}, SOME @{typ "('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_del_from_left}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_del_from_right}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_del}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_delete}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_union_with_key}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_union_with}, SOME @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_union}, SOME @{typ "('a\<Colon>linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_map_entry}, SOME @{typ "'a\<Colon>linorder \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),

      (@{const_name rbt_bulkload}, SOME @{typ "('a \<times> 'b) list \<Rightarrow> ('a\<Colon>linorder,'b) rbt"})]

 *}

 hide_const (open) R B Empty entries keys fold gen_keys gen_entries

 end