Theory HOL-Library.RBT_Impl

(*  Title:      HOL/Library/RBT_Impl.thy
    Author:     Markus Reiter, TU Muenchen
    Author:     Alexander Krauss, TU Muenchen
*)

section ‹Implementation of Red-Black Trees›

theory RBT_Impl
imports Main
begin

text ‹
  For applications, you should use theory ‹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 ⇒ ('a × 'b) list"
where 
  "entries Empty = []"
| "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"

abbreviation (input) entry_in_tree :: "'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ bool"
where
  "entry_in_tree k v t ≡ (k, v) ∈ set (entries t)"

definition keys :: "('a, 'b) rbt ⇒ '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) ∈ set (entries t)"
  shows "k ∈ set (keys t)"
proof -
  from assms have "fst (k, v) ∈ fst ` set (entries t)" by (rule imageI)
  then show ?thesis by (simp add: keys_def)
qed

lemma keys_entries:
  "k ∈ set (keys t) ⟷ (∃v. (k, v) ∈ set (entries t))"
  by (auto intro: entry_in_tree_keys) (auto simp add: keys_def)

lemma non_empty_rbt_keys: 
  "t ≠ rbt.Empty ⟹ keys t ≠ []"
  by (cases t) simp_all

subsubsection ‹Search tree properties›

context ord begin

definition rbt_less :: "'a ⇒ ('a, 'b) rbt ⇒ bool"
where
  rbt_less_prop: "rbt_less k t ⟷ (∀x∈set (keys t). x < k)"

abbreviation rbt_less_symbol (infix "|«" 50)
where "t |« x ≡ rbt_less x t"

definition rbt_greater :: "'a ⇒ ('a, 'b) rbt ⇒ bool" (infix "«|" 50) 
where
  rbt_greater_prop: "rbt_greater k t = (∀x∈set (keys t). k < x)"

lemma rbt_less_simps [simp]:
  "Empty |« k = True"
  "Branch c lt kt v rt |« k ⟷ kt < k ∧ lt |« k ∧ rt |« k"
  by (auto simp add: rbt_less_prop)

lemma rbt_greater_simps [simp]:
  "k «| Empty = True"
  "k «| (Branch c lt kt v rt) ⟷ k < kt ∧ k «| lt ∧ k «| 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 |« u ⟹ u ≤ v ⟹ l |« v"
  and rbt_less_trans: "t |« x ⟹ x < y ⟹ t |« y"
  and rbt_greater_eq_trans: "u ≤ v ⟹ v «| r ⟹ u «| r"
  and rbt_greater_trans: "x < y ⟹ y «| t ⟹ x «| t"
  by (auto simp: rbt_ord_props)

primrec rbt_sorted :: "('a, 'b) rbt ⇒ bool"
where
  "rbt_sorted Empty = True"
| "rbt_sorted (Branch c l k v r) = (l |« k ∧ k «| r ∧ rbt_sorted l ∧ rbt_sorted r)"

end

context linorder begin

lemma rbt_sorted_entries:
  "rbt_sorted t ⟹ List.sorted (map fst (entries t))"
by (induct t)  (force simp: sorted_append rbt_ord_props dest!: entry_in_tree_keys)+

lemma distinct_entries:
  "rbt_sorted t ⟹ distinct (map fst (entries t))"
by (induct t) (force simp: sorted_append rbt_ord_props dest!: entry_in_tree_keys)+

lemma distinct_keys:
  "rbt_sorted t ⟹ distinct (keys t)"
  by (simp add: distinct_entries keys_def)


subsubsection ‹Tree lookup›

primrec (in ord) rbt_lookup :: "('a, 'b) rbt ⇒ 'a ⇀ '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 ⟹ 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) ⟹ 
    dom (rbt_lookup (Branch c t1 k v t2)) 
    = Set.insert k (dom (rbt_lookup t1) ∪ 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) ∪ dom (rbt_lookup t2))"
  have "dom (rbt_lookup (Branch color t1 a b t2)) ⊆ ?A" by (auto split: if_split_asm)
  moreover from Branch have "finite (insert a (dom (rbt_lookup t1) ∪ 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 |« k ⟹ rbt_lookup t k = None" 
by (induct t) auto

lemma rbt_lookup_rbt_greater[simp]: "k «| t ⟹ rbt_lookup t k = None"
by (induct t) auto

lemma rbt_lookup_Empty: "rbt_lookup Empty = Map.empty"
by (rule ext) simp

end

context linorder begin

lemma map_of_entries:
  "rbt_sorted t ⟹ 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↦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 ↦ v] ++ rbt_lookup t1) x"

    have DOM_T1: "!!k'. k'∈dom (rbt_lookup t1) ⟹ k>k'"
    proof -
      fix k'
      from RBT_SORTED have "t1 |« k" by simp
      with rbt_less_prop have "∀k'∈set (keys t1). k>k'" by auto
      moreover assume "k'∈dom (rbt_lookup t1)"
      ultimately show "k>k'" using rbt_lookup_keys RBT_SORTED by auto
    qed
    
    have DOM_T2: "!!k'. k'∈dom (rbt_lookup t2) ⟹ k<k'"
    proof -
      fix k'
      from RBT_SORTED have "k «| t2" by simp
      with rbt_greater_prop have "∀k'∈set (keys t2). k<k'" by auto
      moreover assume "k'∈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∉dom [k↦v]" by simp
      moreover have "x ∉ dom (rbt_lookup t2)"
      proof
        assume "x ∈ 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 ↦ v] x" by simp
      moreover have "x ∉ dom (rbt_lookup t1)" 
      proof
        assume "x ∈ 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∉dom [k↦v]" by simp
      moreover have "x∉dom (rbt_lookup t1)" proof
        assume "x∈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 ↦ 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 ⟹ rbt_lookup t k = Some v ⟷ (k, v) ∈ 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) ⟷ 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 ⟷ 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 "⋀v. (k, v) ∈ set (entries t1) ⟷ (k, v) ∈ set (entries t2)" 
  shows "rbt_lookup t1 k = rbt_lookup t2 k"
proof -
  from assms have "k ∈ dom (rbt_lookup t1) ⟷ k ∈ 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 ⇒ color"
where
  "color_of Empty = B"
| "color_of (Branch c _ _ _ _) = c"

primrec bheight :: "('a,'b) rbt ⇒ 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 ⇒ bool"
where
  "inv1 Empty = True"
| "inv1 (Branch c lt k v rt) ⟷ inv1 lt ∧ inv1 rt ∧ (c = B ∨ color_of lt = B ∧ color_of rt = B)"

primrec inv1l :: "('a, 'b) rbt ⇒ bool" ― ‹Weaker version›
where
  "inv1l Empty = True"
| "inv1l (Branch c l k v r) = (inv1 l ∧ inv1 r)"
lemma [simp]: "inv1 t ⟹ inv1l t" by (cases t) simp+

primrec inv2 :: "('a, 'b) rbt ⇒ bool"
where
  "inv2 Empty = True"
| "inv2 (Branch c lt k v rt) = (inv2 lt ∧ inv2 rt ∧ bheight lt = bheight rt)"

context ord begin

definition is_rbt :: "('a, 'b) rbt ⇒ bool" where
  "is_rbt t ⟷ inv1 t ∧ inv2 t ∧ color_of t = B ∧ rbt_sorted t"

lemma is_rbt_rbt_sorted [simp]:
  "is_rbt t ⟹ 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›

text ‹The function definitions are based on the book by Okasaki.›

fun (* slow, due to massive case splitting *)
  balance :: "('a,'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('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: "⟦inv1l l; inv1l r⟧ ⟹ inv1 (balance l k v r)" 
  by (induct l k v r rule: balance.induct) auto

lemma balance_bheight: "bheight l = bheight r ⟹ 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 «| balance a k x b) = (v «| a ∧ v «| b ∧ v < k)" 
  by (induct a k x b rule: balance.induct) auto

lemma balance_rbt_less[simp]: "(balance a k x b |« v) = (a |« v ∧ b |« v ∧ 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 |« k" "k «| 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 ∧ z «| Branch B va vb vd vc" 
    by (auto simp add: rbt_ord_props)
  hence "y «| (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 ∧ Branch B va vb vd vc |« x" 
    by simp
  hence "Branch B va vb vd vc |« 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 ∧ z «| Branch B va vb vd vc" by simp
  hence "y «| 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 ∧ Branch B vd ve vg vf |« x" by simp
  hence 1: "Branch B vd ve vg vf |« y" by (blast dest: rbt_less_trans)
  from "3_4" have "y < z ∧ z «| Branch B va vb vii vc" by simp
  hence "y «| 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 ∧ Branch B va vb vd vc |« x" by simp
  hence "Branch B va vb vd vc |« 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 ∧ z «| Branch B va vb vd vc" by simp
  hence "y «| 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 ∧ Branch B va vb vd vc |« x" by simp
  hence "Branch B va vb vd vc |« 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 ∧ Branch B va vb vg vc |« x" by simp
  hence 1: "Branch B va vb vg vc |« y" by (blast dest: rbt_less_trans)
  from "5_4" have "y < z ∧ z «| Branch B vd ve vii vf" by simp
  hence "y «| 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) ⟷ entry_in_tree k x l ∨ k = v ∧ x = y ∨ 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 |« k" "k «| 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 ⇒ ('a,'b) rbt ⇒ ('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 ⟹ inv1l (paint c t)" by (cases t) auto
lemma paint_inv1[simp]: "inv1l t ⟹ inv1 (paint B t)" by (cases t) auto
lemma paint_inv2[simp]: "inv2 t ⟹ 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 ⟹ 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 «| paint c t) = (v «| t)" by (cases t) auto
lemma paint_rbt_less[simp]: "(paint c t |« v) = (t |« v)" by (cases t) auto

fun
  rbt_ins :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('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 ⟹ 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 «| rbt_ins f (k :: 'a) x t) = (v «| t ∧ k > v)"
  by (induct f k x t rule: rbt_ins.induct) auto
lemma ins_rbt_less[simp]: "(rbt_ins f k x t |« v) = (t |« v ∧ k < v)"
  by (induct f k x t rule: rbt_ins.induct) auto
lemma ins_rbt_sorted[simp]: "rbt_sorted t ⟹ 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 } ∪ 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 ⇒ v 
                                                                | Some w ⇒ 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 ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('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 (λ_. f)"

definition rbt_insert :: "'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_insert = rbt_insert_with_key (λ_ _ nv. nv)"

end

context linorder begin

lemma rbt_insertwk_rbt_sorted: "rbt_sorted t ⟹ 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 ⇒ v 
                                                       | Some w ⇒ 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 ⟹ 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 ⟹ is_rbt (rbt_insert_with f k v t)"
  by (simp add: rbt_insertwk_is_rbt rbt_insertw_def)

lemma rbt_lookup_rbt_insertw:
  "is_rbt t ⟹
    rbt_lookup (rbt_insert_with f k v t) =
      (rbt_lookup t)(k ↦ (if k ∈ dom (rbt_lookup t) then f (the (rbt_lookup t k)) v else v))"
  by (rule ext, cases "rbt_lookup t k") (auto simp: rbt_lookup_rbt_insertwk dom_def rbt_insertw_def)

lemma rbt_insert_rbt_sorted: "rbt_sorted t ⟹ 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 ⟹ is_rbt (rbt_insert k v t)"
  by (simp add: rbt_insertwk_is_rbt rbt_insert_def)

lemma rbt_lookup_rbt_insert: "is_rbt t ⟹ rbt_lookup (rbt_insert k v t) = (rbt_lookup t)(k↦v)"
  by (rule ext) (simp add: rbt_insert_def rbt_lookup_rbt_insertwk split: option.split)

end

subsection ‹Deletion›

lemma bheight_paintR'[simp]: "color_of t = B ⟹ bheight (paint R t) = bheight t - 1"
by (cases t rule: rbt_cases) auto

text ‹
  The function definitions are based on the Haskell code by Stefan Kahrs
  at 🌐‹http://www.cs.ukc.ac.uk/people/staff/smk/redblack/rb.html›.
›

fun
  balance_left :: "('a,'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('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: "⟦inv1l a; inv1 b; color_of b = B⟧ ⟹ 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: "⟦ inv1l lt; inv1 rt ⟧ ⟹ 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: 
  "⟦ rbt_sorted l; rbt_sorted r; rbt_less k l; k «| r ⟧ ⟹ 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 «| a" "k «| b" "k < x" 
  shows "k «| 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 |« k" "b |« k" "x < k" 
  shows "balance_left a x t b |« 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 ∨ k = a ∧ v = b ∨ 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 ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('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) ∧ 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: "⟦inv1 a; inv1l b; color_of a = B⟧ ⟹ 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: "⟦ inv1 lt; inv1l rt ⟧ ⟹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:
  "⟦ rbt_sorted l; rbt_sorted r; rbt_less k l; k «| r ⟧ ⟹ 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 «| a" "k «| b" "k < x" 
  shows "k «| 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 |« k" "b |« k" "x < k" 
  shows "balance_right a x t b |« 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 ∨ x = k ∧ y = v ∨ 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 ⇒ ('a,'b) rbt ⇒ ('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 ⇒ (Branch R (Branch R a k x b2) t z (Branch R c2 s y d)) |
                                    bc ⇒ 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 ⇒ Branch R (Branch B a k x b2) t z (Branch B c2 s y d) |
                                    bc ⇒ 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 ⟹ color_of rt = B ⟹ 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 «| l" "k «| r" 
  shows "k «| 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 |« k" "r |« k" 
  shows "combine l r |« 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 |« k" "k «| 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 |« x" "x «| b" "b |« k" "k «| c" "c |« y" "y «| 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 ∧ 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 ∧ 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 ∧ 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 ∧ 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 ∧ 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 ∧ 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 ∧ 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 ∧ 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 ∨ 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 ⇒ ('a,'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt" and
  rbt_del_from_right :: "'a ⇒ ('a,'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt" and
  rbt_del :: "'a⇒ ('a,'b) rbt ⇒ ('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
  "⟦inv2 rt; bheight lt = bheight rt; inv1 rt⟧ ⟹
   inv2 (rbt_del_from_left x lt k v rt) ∧ 
   bheight (rbt_del_from_left x lt k v rt) = bheight lt ∧ 
   (color_of lt = B ∧ color_of rt = B ∧ inv1 (rbt_del_from_left x lt k v rt) ∨ 
    (color_of lt ≠ B ∨ color_of rt ≠ B) ∧ inv1l (rbt_del_from_left x lt k v rt))"
  and "⟦inv2 rt; bheight lt = bheight rt; inv1 rt⟧ ⟹
  inv2 (rbt_del_from_right x lt k v rt) ∧ 
  bheight (rbt_del_from_right x lt k v rt) = bheight lt ∧ 
  (color_of lt = B ∧ color_of rt = B ∧ inv1 (rbt_del_from_right x lt k v rt) ∨ 
   (color_of lt ≠ B ∨ color_of rt ≠ B) ∧ inv1l (rbt_del_from_right x lt k v rt))"
  and rbt_del_inv1_inv2: "inv2 (rbt_del x lt) ∧ (color_of lt = R ∧ bheight (rbt_del x lt) = bheight lt ∧ inv1 (rbt_del x lt) 
  ∨ color_of lt = B ∧ bheight (rbt_del x lt) = bheight lt - 1 ∧ 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' ∨ y < y' ∨ 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 ∧ 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 ∧ 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 ∧ color_of Empty = B") simp+
qed auto

lemma 
  rbt_del_from_left_rbt_less: "⟦ lt |« v; rt |« v; k < v⟧ ⟹ rbt_del_from_left x lt k y rt |« v"
  and rbt_del_from_right_rbt_less: "⟦lt |« v; rt |« v; k < v⟧ ⟹ rbt_del_from_right x lt k y rt |« v"
  and rbt_del_rbt_less: "lt |« v ⟹ rbt_del x lt |« 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: "⟦v «| lt; v «| rt; k > v⟧ ⟹ v «| rbt_del_from_left x lt k y rt"
  and rbt_del_from_right_rbt_greater: "⟦v «| lt; v «| rt; k > v⟧ ⟹ v «| rbt_del_from_right x lt k y rt"
  and rbt_del_rbt_greater: "v «| lt ⟹ v «| 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 "⟦rbt_sorted lt; rbt_sorted rt; lt |« k; k «| rt⟧ ⟹ rbt_sorted (rbt_del_from_left x lt k y rt)"
  and "⟦rbt_sorted lt; rbt_sorted rt; lt |« k; k «| rt⟧ ⟹ rbt_sorted (rbt_del_from_right x lt k y rt)"
  and rbt_del_rbt_sorted: "rbt_sorted lt ⟹ 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 |« yy" by simp
  hence "rbt_del x (Branch B lta zz v rta) |« 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 |« yy" by simp
  hence "rbt_del x (Branch R vaa vbb vdd vc) |« 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 «| Branch B lta zz v rta" by simp
  hence "yy «| 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 «| Branch R vaa vbb vdd vc" by simp
  hence "yy «| 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 "⟦rbt_sorted lt; rbt_sorted rt; lt |« kt; kt «| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt⟧ ⟹ entry_in_tree k v (rbt_del_from_left x lt kt y rt) = (False ∨ (x ≠ k ∧ entry_in_tree k v (Branch c lt kt y rt)))"
  and "⟦rbt_sorted lt; rbt_sorted rt; lt |« kt; kt «| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt⟧ ⟹ entry_in_tree k v (rbt_del_from_right x lt kt y rt) = (False ∨ (x ≠ k ∧ entry_in_tree k v (Branch c lt kt y rt)))"
  and rbt_del_in_tree: "⟦rbt_sorted t; inv1 t; inv2 t⟧ ⟹ entry_in_tree k v (rbt_del x t) = (False ∨ (x ≠ k ∧ 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 ∨ xx < yy ∨ 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) ∧ k = yy" by simp
      hence "¬ entry_in_tree k v aa" "¬ 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)
  define mt where [simp]: "mt = Branch B lta zz vv rta"
  from 3 have "inv2 mt ∧ inv1 mt" by simp
  hence "inv2 (rbt_del xx mt) ∧ (color_of mt = R ∧ bheight (rbt_del xx mt) = bheight mt ∧ inv1 (rbt_del xx mt) ∨ color_of mt = B ∧ bheight (rbt_del xx mt) = bheight mt - 1 ∧ 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 ∨ xx ≠ k ∧ entry_in_tree k v mt ∨ (k = yy ∧ v = ss) ∨ 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 «| bb ∧ yy > k" by simp
    hence "k «| 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 «| bb ∧ k < yy" by simp
    hence "k «| 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 ∧ yy «| bb" by simp
    hence "k «| bb" by (blast dest: rbt_greater_trans)
    with True "4_2" show ?thesis by (auto simp: rbt_greater_nit)
  qed auto
next
  case (5 xx aa yy ss lta zz vv rta)
  define mt where [simp]: "mt = Branch B lta zz vv rta"
  from 5 have "inv2 mt ∧ inv1 mt" by simp
  hence "inv2 (rbt_del xx mt) ∧ (color_of mt = R ∧ bheight (rbt_del xx mt) = bheight mt ∧ inv1 (rbt_del xx mt) ∨ color_of mt = B ∧ bheight (rbt_del xx mt) = bheight mt - 1 ∧ 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 ∨ (k = yy ∧ v = ss) ∨ False ∨ xx ≠ k ∧ 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 |« yy ∧ yy < k" by simp
    hence "aa |« 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 |« yy ∧ k > yy" by simp
    hence "aa |« 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 ∧ aa |« yy" by simp
    hence "aa |« 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) ∧ (color_of t = R ∧ bheight (rbt_del k t) = bheight t ∧ inv1 (rbt_del k t) ∨ color_of t = B ∧ bheight (rbt_del k t) = bheight t - 1 ∧ inv1l (rbt_del k t))" by (rule rbt_del_inv1_inv2)
  hence "inv2 (rbt_del k t) ∧ 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 ≠ k ∧ 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 ≠ 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 ⇒ ('b ⇒ 'b) ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt"
where
  "rbt_map_entry k f Empty = Empty"
| "rbt_map_entry k f (Branch c lt x v rt) =
    (if k < x then Branch c (rbt_map_entry k f lt) x v rt
    else if k > x then (Branch c lt x v (rbt_map_entry k f rt))
    else Branch c lt x (f v) rt)"


lemma rbt_map_entry_color_of: "color_of (rbt_map_entry k f t) = color_of t" by (induct t) simp+
lemma rbt_map_entry_inv1: "inv1 (rbt_map_entry k f t) = inv1 t" by (induct t) (simp add: rbt_map_entry_color_of)+
lemma rbt_map_entry_inv2: "inv2 (rbt_map_entry k f t) = inv2 t" "bheight (rbt_map_entry k f t) = bheight t" by (induct t) simp+
lemma rbt_map_entry_rbt_greater: "rbt_greater a (rbt_map_entry k f t) = rbt_greater a t" by (induct t) simp+
lemma rbt_map_entry_rbt_less: "rbt_less a (rbt_map_entry k f t) = rbt_less a t" by (induct t) simp+
lemma rbt_map_entry_rbt_sorted: "rbt_sorted (rbt_map_entry k f t) = rbt_sorted t"
  by (induct t) (simp_all add: rbt_map_entry_rbt_less rbt_map_entry_rbt_greater)

theorem rbt_map_entry_is_rbt [simp]: "is_rbt (rbt_map_entry k f t) = is_rbt t" 
unfolding is_rbt_def by (simp add: rbt_map_entry_inv2 rbt_map_entry_color_of rbt_map_entry_rbt_sorted rbt_map_entry_inv1 )

end

theorem (in linorder) rbt_lookup_rbt_map_entry:
  "rbt_lookup (rbt_map_entry k f t) = (rbt_lookup t)(k := map_option f (rbt_lookup t k))"
  by (induct t) (auto split: option.splits simp add: fun_eq_iff)

subsection ‹Mapping all entries›

primrec
  map :: "('a ⇒ 'b ⇒ 'c) ⇒ ('a, 'b) rbt ⇒ ('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 (λ(k, v). (k, f k v)) (entries t)"
  by (induct t) auto
lemma map_keys [simp]: "keys (map f t) = keys t" by (simp add: keys_def split_def)
lemma map_color_of: "color_of (map f t) = color_of t" by (induct t) simp+
lemma map_inv1: "inv1 (map f t) = inv1 t" by (induct t) (simp add: map_color_of)+
lemma map_inv2: "inv2 (map f t) = inv2 t" "bheight (map f t) = bheight t" by (induct t) simp+

context ord begin

lemma map_rbt_greater: "rbt_greater k (map f t) = rbt_greater k t" by (induct t) simp+
lemma map_rbt_less: "rbt_less k (map f t) = rbt_less k t" by (induct t) simp+
lemma map_rbt_sorted: "rbt_sorted (map f t) = rbt_sorted t"  by (induct t) (simp add: map_rbt_less map_rbt_greater)+
theorem map_is_rbt [simp]: "is_rbt (map f t) = is_rbt t" 
unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_rbt_sorted map_color_of)

end

theorem (in linorder) rbt_lookup_map: "rbt_lookup (map f t) x = map_option (f x) (rbt_lookup t x)"
  by (induct t) (auto simp: antisym_conv3)
 (* 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 ⇒ 'b ⇒ 'c ⇒ 'c) ⇒ ('a, 'b) rbt ⇒ 'c ⇒ 'c" where
  "fold f t = List.fold (case_prod f) (entries t)"

lemma fold_simps [simp]:
  "fold f Empty = id"
  "fold f (Branch c lt k v rt) = fold f rt ∘ f k v ∘ 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 ⇒ bool) ⇒ ('a ⇒ 'b ⇒ 'c ⇒ 'c) ⇒ ('a :: linorder, 'b) rbt ⇒ 'c ⇒ '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 × 'b) list ⇒ ('a, 'b) rbt" where
  "rbt_bulkload xs = foldr (λ(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 "⋀t. is_rbt t ⟹
    rbt_lookup (List.fold (case_prod rbt_insert) ys t) = rbt_lookup t ++ map_of (rev ys)"
      by (induct ys) (simp_all add: rbt_bulkload_def rbt_lookup_rbt_insert case_prod_beta)
  from this Empty_is_rbt have
    "rbt_lookup (List.fold (case_prod rbt_insert) (rev xs) Empty) = rbt_lookup Empty ++ map_of xs"
     by (simp add: ‹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 ⇒ ('a × 'b) list ⇒ ('a, 'b) rbt × ('a × 'b) list"
  and rbtreeify_g :: "nat ⇒ ('a × 'b) list ⇒ ('a, 'b) rbt × ('a × 'b) list"
where
  "rbtreeify_f n kvs =
   (if n = 0 then (Empty, kvs)
    else if n = 1 then
      case kvs of (k, v) # kvs' ⇒ (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') ⇒
        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') ⇒
        apfst (Branch B t1 k v) (rbtreeify_f (n div 2) kvs'))"