src/HOL/Library/RBT_Impl.thy

remove duplicate lemma card_unit in favor of Finite_Set.card_UNIV_unit

(*  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)


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 (List.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 (List.map fst (entries t))"
by (induct t) 
  (force simp: sorted_append sorted_Cons rbt_ord_props 
      dest!: entry_in_tree_keys)+

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)"
  moreover from this have "rbt_sorted t1" "rbt_sorted t2" by simp_all
  ultimately show ?thesis by (simp add: rbt_lookup_keys)
qed

lemma finite_dom_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 {* Union *}

context ord begin

primrec rbt_union_with_key :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
  "rbt_union_with_key f t Empty = t"
| "rbt_union_with_key f t (Branch c lt k v rt) = rbt_union_with_key f (rbt_union_with_key f (rbt_insert_with_key f k v t) lt) rt"

definition rbt_union_with where
  "rbt_union_with f = rbt_union_with_key (\<lambda>_. f)"

definition rbt_union where
  "rbt_union = rbt_union_with_key (%_ _ rv. rv)"

end

context linorder begin

lemma rbt_unionwk_rbt_sorted: "rbt_sorted lt \<Longrightarrow> rbt_sorted (rbt_union_with_key f lt rt)" 
  by (induct rt arbitrary: lt) (auto simp: rbt_insertwk_rbt_sorted)
theorem rbt_unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (rbt_union_with_key f lt rt)" 
  by (induct rt arbitrary: lt) (simp add: rbt_insertwk_is_rbt)+

theorem rbt_unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (rbt_union_with f lt rt)" unfolding rbt_union_with_def by simp

theorem rbt_union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (rbt_union lt rt)" unfolding rbt_union_def by simp

lemma (in ord) rbt_union_Branch[simp]:
  "rbt_union t (Branch c lt k v rt) = rbt_union (rbt_union (rbt_insert k v t) lt) rt"
  unfolding rbt_union_def rbt_insert_def
  by simp

lemma rbt_lookup_rbt_union:
  assumes "is_rbt s" "rbt_sorted t"
  shows "rbt_lookup (rbt_union s t) = rbt_lookup s ++ rbt_lookup t"
using assms
proof (induct t arbitrary: s)
  case Empty thus ?case by (auto simp: rbt_union_def)
next
  case (Branch c l k v r s)
  then have "rbt_sorted r" "rbt_sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto

  have meq: "rbt_lookup s(k \<mapsto> v) ++ rbt_lookup l ++ rbt_lookup r =
    rbt_lookup s ++
    (\<lambda>a. if a < k then rbt_lookup l a
    else if k < a then rbt_lookup r a else Some v)" (is "?m1 = ?m2")
  proof (rule ext)
    fix a

   have "k < a \<or> k = a \<or> k > a" by auto
    thus "?m1 a = ?m2 a"
    proof (elim disjE)
      assume "k < a"
      with `l |\<guillemotleft> k` have "l |\<guillemotleft> a" by (rule rbt_less_trans)
      with `k < a` show ?thesis
        by (auto simp: map_add_def split: option.splits)
    next
      assume "k = a"
      with `l |\<guillemotleft> k` `k \<guillemotleft>| r` 
      show ?thesis by (auto simp: map_add_def)
    next
      assume "a < k"
      from this `k \<guillemotleft>| r` have "a \<guillemotleft>| r" by (rule rbt_greater_trans)
      with `a < k` show ?thesis
        by (auto simp: map_add_def split: option.splits)
    qed
  qed

  from Branch have is_rbt: "is_rbt (RBT_Impl.rbt_union (RBT_Impl.rbt_insert k v s) l)"
    by (auto intro: rbt_union_is_rbt rbt_insert_is_rbt)
  with Branch have IHs:
    "rbt_lookup (rbt_union (rbt_union (rbt_insert k v s) l) r) = rbt_lookup (rbt_union (rbt_insert k v s) l) ++ rbt_lookup r"
    "rbt_lookup (rbt_union (rbt_insert k v s) l) = rbt_lookup (rbt_insert k v s) ++ rbt_lookup l"
    by auto
  
  with meq show ?case
    by (auto simp: rbt_lookup_rbt_insert[OF Branch(3)])

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 *)

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, code]:
  "fold f Empty = id"
  "fold f (Branch c lt k v rt) = fold f rt \<circ> f k v \<circ> fold f lt"
  by (simp_all add: fold_def fun_eq_iff)


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

lemmas [code] =
  ord.rbt_less_prop
  ord.rbt_greater_prop
  ord.rbt_sorted.simps
  ord.rbt_lookup.simps
  ord.is_rbt_def
  ord.rbt_ins.simps
  ord.rbt_insert_with_key_def
  ord.rbt_insertw_def
  ord.rbt_insert_def
  ord.rbt_del_from_left.simps
  ord.rbt_del_from_right.simps
  ord.rbt_del.simps
  ord.rbt_delete_def
  ord.rbt_union_with_key.simps
  ord.rbt_union_with_def
  ord.rbt_union_def
  ord.rbt_map_entry.simps
  ord.rbt_bulkload_def

text {* Restore original type constraints for constants *}
setup {*
  fold Sign.add_const_constraint
    [(@{const_name rbt_less}, SOME @{typ "('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"}),
     (@{const_name rbt_greater}, SOME @{typ "('a :: order) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"}),
     (@{const_name rbt_sorted}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> bool"}),
     (@{const_name rbt_lookup}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"}),
     (@{const_name is_rbt}, SOME @{typ "('a :: linorder, 'b) rbt \<Rightarrow> bool"}),
     (@{const_name rbt_ins}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_insert_with_key}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_insert_with}, SOME @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_insert}, SOME @{typ "('a :: linorder) \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_del_from_left}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_del_from_right}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_del}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_delete}, SOME @{typ "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_union_with_key}, SOME @{typ "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_union_with}, SOME @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_union}, SOME @{typ "('a\<Colon>linorder,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_map_entry}, SOME @{typ "'a\<Colon>linorder \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"}),
     (@{const_name rbt_bulkload}, SOME @{typ "('a \<times> 'b) list \<Rightarrow> ('a\<Colon>linorder,'b) rbt"})]
*}

hide_const (open) R B Empty entries keys map fold

end