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'))"

| "rbtreeify_g n kvs =
   (if n = 0 ∨ n = 1 then (Empty, kvs)
    else if n mod 2 = 0 then
      case rbtreeify_g (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_g (n div 2) kvs'))"

definition rbtreeify :: "('a × 'b) list ⇒ ('a, 'b) rbt"
where "rbtreeify kvs = fst (rbtreeify_g (Suc (length kvs)) kvs)"

declare rbtreeify_f.simps [simp del] rbtreeify_g.simps [simp del]

lemma rbtreeify_f_code [code]:
  "rbtreeify_f n kvs =
   (if n = 0 then (Empty, kvs)
    else if n = 1 then
      case kvs of (k, v) # kvs' ⇒ 
        (Branch R Empty k v Empty, kvs')
    else let (n', r) = Euclidean_Rings.divmod_nat n 2 in
      if r = 0 then
        case rbtreeify_f n' kvs of (t1, (k, v) # kvs') ⇒
          apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
      else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') ⇒
          apfst (Branch B t1 k v) (rbtreeify_f n' kvs'))"
by (subst rbtreeify_f.simps) (simp only: Let_def Euclidean_Rings.divmod_nat_def prod.case)

lemma rbtreeify_g_code [code]:
  "rbtreeify_g n kvs =
   (if n = 0 ∨ n = 1 then (Empty, kvs)
    else let (n', r) = Euclidean_Rings.divmod_nat n 2 in
      if r = 0 then
        case rbtreeify_g n' kvs of (t1, (k, v) # kvs') ⇒
          apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
      else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') ⇒
          apfst (Branch B t1 k v) (rbtreeify_g n' kvs'))"
by(subst rbtreeify_g.simps)(simp only: Let_def Euclidean_Rings.divmod_nat_def prod.case)

lemma Suc_double_half: "Suc (2 * n) div 2 = n"
by simp

lemma div2_plus_div2: "n div 2 + n div 2 = (n :: nat) - n mod 2"
by arith

lemma rbtreeify_f_rec_aux_lemma:
  "⟦k - n div 2 = Suc k'; n ≤ k; n mod 2 = Suc 0⟧
  ⟹ k' - n div 2 = k - n"
apply(rule add_right_imp_eq[where a = "n - n div 2"])
apply(subst add_diff_assoc2, arith)
apply(simp add: div2_plus_div2)
done

lemma rbtreeify_f_simps:
  "rbtreeify_f 0 kvs = (Empty, kvs)"
  "rbtreeify_f (Suc 0) ((k, v) # kvs) = 
  (Branch R Empty k v Empty, kvs)"
  "0 < n ⟹ rbtreeify_f (2 * n) kvs =
   (case rbtreeify_f n kvs of (t1, (k, v) # kvs') ⇒
     apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
  "0 < n ⟹ rbtreeify_f (Suc (2 * n)) kvs =
   (case rbtreeify_f n kvs of (t1, (k, v) # kvs') ⇒ 
     apfst (Branch B t1 k v) (rbtreeify_f n kvs'))"
by(subst (1) rbtreeify_f.simps, simp add: Suc_double_half)+

lemma rbtreeify_g_simps:
  "rbtreeify_g 0 kvs = (Empty, kvs)"
  "rbtreeify_g (Suc 0) kvs = (Empty, kvs)"
  "0 < n ⟹ rbtreeify_g (2 * n) kvs =
   (case rbtreeify_g n kvs of (t1, (k, v) # kvs') ⇒ 
     apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
  "0 < n ⟹ rbtreeify_g (Suc (2 * n)) kvs =
   (case rbtreeify_f n kvs of (t1, (k, v) # kvs') ⇒ 
     apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
by(subst (1) rbtreeify_g.simps, simp add: Suc_double_half)+

declare rbtreeify_f_simps[simp] rbtreeify_g_simps[simp]

lemma length_rbtreeify_f: "n ≤ length kvs
  ⟹ length (snd (rbtreeify_f n kvs)) = length kvs - n"
  and length_rbtreeify_g:"⟦ 0 < n; n ≤ Suc (length kvs) ⟧
  ⟹ length (snd (rbtreeify_g n kvs)) = Suc (length kvs) - n"
proof(induction n kvs and n kvs rule: rbtreeify_f_rbtreeify_g.induct)
  case (1 n kvs)
  show ?case
  proof(cases "n ≤ 1")
    case True thus ?thesis using "1.prems"
      by(cases n kvs rule: nat.exhaust[case_product list.exhaust]) auto
  next
    case False
    hence "n ≠ 0" "n ≠ 1" by simp_all
    note IH = "1.IH"[OF this]
    show ?thesis
    proof(cases "n mod 2 = 0")
      case True
      hence "length (snd (rbtreeify_f n kvs)) = 
        length (snd (rbtreeify_f (2 * (n div 2)) kvs))"
        by(metis minus_nat.diff_0 minus_mod_eq_mult_div [symmetric])
      also from "1.prems" False obtain k v kvs' 
        where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
      also have "0 < n div 2" using False by(simp) 
      note rbtreeify_f_simps(3)[OF this]
      also note kvs[symmetric] 
      also let ?rest1 = "snd (rbtreeify_f (n div 2) kvs)"
      from "1.prems" have "n div 2 ≤ length kvs" by simp
      with True have len: "length ?rest1 = length kvs - n div 2" by(rule IH)
      with "1.prems" False obtain t1 k' v' kvs''
        where kvs'': "rbtreeify_f (n div 2) kvs = (t1, (k', v') # kvs'')"
         by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm)
      note this also note prod.case also note list.simps(5) 
      also note prod.case also note snd_apfst
      also have "0 < n div 2" "n div 2 ≤ Suc (length kvs'')" 
        using len "1.prems" False unfolding kvs'' by simp_all
      with True kvs''[symmetric] refl refl
      have "length (snd (rbtreeify_g (n div 2) kvs'')) = 
        Suc (length kvs'') - n div 2" by(rule IH)
      finally show ?thesis using len[unfolded kvs''] "1.prems" True
        by(simp add: Suc_diff_le[symmetric] mult_2[symmetric] minus_mod_eq_mult_div [symmetric])
    next
      case False
      hence "length (snd (rbtreeify_f n kvs)) = 
        length (snd (rbtreeify_f (Suc (2 * (n div 2))) kvs))"
        by (simp add: mod_eq_0_iff_dvd)
      also from "1.prems" ‹¬ n ≤ 1› obtain k v kvs' 
        where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
      also have "0 < n div 2" using ‹¬ n ≤ 1› by(simp) 
      note rbtreeify_f_simps(4)[OF this]
      also note kvs[symmetric] 
      also let ?rest1 = "snd (rbtreeify_f (n div 2) kvs)"
      from "1.prems" have "n div 2 ≤ length kvs" by simp
      with False have len: "length ?rest1 = length kvs - n div 2" by(rule IH)
      with "1.prems" ‹¬ n ≤ 1› obtain t1 k' v' kvs''
        where kvs'': "rbtreeify_f (n div 2) kvs = (t1, (k', v') # kvs'')"
        by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm)
      note this also note prod.case also note list.simps(5)
      also note prod.case also note snd_apfst
      also have "n div 2 ≤ length kvs''" 
        using len "1.prems" False unfolding kvs'' by simp arith
      with False kvs''[symmetric] refl refl
      have "length (snd (rbtreeify_f (n div 2) kvs'')) = length kvs'' - n div 2"
        by(rule IH)
      finally show ?thesis using len[unfolded kvs''] "1.prems" False
        by simp(rule rbtreeify_f_rec_aux_lemma[OF sym])
    qed
  qed
next
  case (2 n kvs)
  show ?case
  proof(cases "n > 1")
    case False with ‹0 < n› show ?thesis
      by(cases n kvs rule: nat.exhaust[case_product list.exhaust]) simp_all
  next
    case True
    hence "¬ (n = 0 ∨ n = 1)" by simp
    note IH = "2.IH"[OF this]
    show ?thesis
    proof(cases "n mod 2 = 0")
      case True
      hence "length (snd (rbtreeify_g n kvs)) =
        length (snd (rbtreeify_g (2 * (n div 2)) kvs))"
        by(metis minus_nat.diff_0 minus_mod_eq_mult_div [symmetric])
      also from "2.prems" True obtain k v kvs' 
        where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
      also have "0 < n div 2" using ‹1 < n› by(simp) 
      note rbtreeify_g_simps(3)[OF this]
      also note kvs[symmetric] 
      also let ?rest1 = "snd (rbtreeify_g (n div 2) kvs)"
      from "2.prems" ‹1 < n›
      have "0 < n div 2" "n div 2 ≤ Suc (length kvs)" by simp_all
      with True have len: "length ?rest1 = Suc (length kvs) - n div 2" by(rule IH)
      with "2.prems" obtain t1 k' v' kvs''
        where kvs'': "rbtreeify_g (n div 2) kvs = (t1, (k', v') # kvs'')"
        by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm)
      note this also note prod.case also note list.simps(5) 
      also note prod.case also note snd_apfst
      also have "n div 2 ≤ Suc (length kvs'')" 
        using len "2.prems" unfolding kvs'' by simp
      with True kvs''[symmetric] refl refl ‹0 < n div 2›
      have "length (snd (rbtreeify_g (n div 2) kvs'')) = Suc (length kvs'') - n div 2"
        by(rule IH)
      finally show ?thesis using len[unfolded kvs''] "2.prems" True
        by(simp add: Suc_diff_le[symmetric] mult_2[symmetric] minus_mod_eq_mult_div [symmetric])
    next
      case False
      hence "length (snd (rbtreeify_g n kvs)) = 
        length (snd (rbtreeify_g (Suc (2 * (n div 2))) kvs))"
        by (simp add: mod_eq_0_iff_dvd)
      also from "2.prems" ‹1 < n› obtain k v kvs'
        where kvs: "kvs = (k, v) # kvs'" by(cases kvs) auto
      also have "0 < n div 2" using ‹1 < n› by(simp)
      note rbtreeify_g_simps(4)[OF this]
      also note kvs[symmetric] 
      also let ?rest1 = "snd (rbtreeify_f (n div 2) kvs)"
      from "2.prems" have "n div 2 ≤ length kvs" by simp
      with False have len: "length ?rest1 = length kvs - n div 2" by(rule IH)
      with "2.prems" ‹1 < n› False obtain t1 k' v' kvs'' 
        where kvs'': "rbtreeify_f (n div 2) kvs = (t1, (k', v') # kvs'')"
        by(cases ?rest1)(auto simp add: snd_def split: prod.split_asm, arith)
      note this also note prod.case also note list.simps(5) 
      also note prod.case also note snd_apfst
      also have "n div 2 ≤ Suc (length kvs'')" 
        using len "2.prems" False unfolding kvs'' by simp arith
      with False kvs''[symmetric] refl refl ‹0 < n div 2›
      have "length (snd (rbtreeify_g (n div 2) kvs'')) = Suc (length kvs'') - n div 2"
        by(rule IH)
      finally show ?thesis using len[unfolded kvs''] "2.prems" False
        by(simp add: div2_plus_div2)
    qed
  qed
qed

lemma rbtreeify_induct [consumes 1, case_names f_0 f_1 f_even f_odd g_0 g_1 g_even g_odd]:
  fixes P Q
  defines "f0 == (⋀kvs. P 0 kvs)"
  and "f1 == (⋀k v kvs. P (Suc 0) ((k, v) # kvs))"
  and "feven ==
    (⋀n kvs t k v kvs'. ⟦ n > 0; n ≤ length kvs; P n kvs; 
       rbtreeify_f n kvs = (t, (k, v) # kvs'); n ≤ Suc (length kvs'); Q n kvs' ⟧ 
     ⟹ P (2 * n) kvs)"
  and "fodd == 
    (⋀n kvs t k v kvs'. ⟦ n > 0; n ≤ length kvs; P n kvs;
       rbtreeify_f n kvs = (t, (k, v) # kvs'); n ≤ length kvs'; P n kvs' ⟧ 
    ⟹ P (Suc (2 * n)) kvs)"
  and "g0 == (⋀kvs. Q 0 kvs)"
  and "g1 == (⋀kvs. Q (Suc 0) kvs)"
  and "geven == 
    (⋀n kvs t k v kvs'. ⟦ n > 0; n ≤ Suc (length kvs); Q n kvs; 
       rbtreeify_g n kvs = (t, (k, v) # kvs'); n ≤ Suc (length kvs'); Q n kvs' ⟧
    ⟹ Q (2 * n) kvs)"
  and "godd == 
    (⋀n kvs t k v kvs'. ⟦ n > 0; n ≤ length kvs; P n kvs;
       rbtreeify_f n kvs = (t, (k, v) # kvs'); n ≤ Suc (length kvs'); Q n kvs' ⟧
    ⟹ Q (Suc (2 * n)) kvs)"
  shows "⟦ n ≤ length kvs; 
           PROP f0; PROP f1; PROP feven; PROP fodd; 
           PROP g0; PROP g1; PROP geven; PROP godd ⟧
         ⟹ P n kvs"
  and "⟦ n ≤ Suc (length kvs);
          PROP f0; PROP f1; PROP feven; PROP fodd; 
          PROP g0; PROP g1; PROP geven; PROP godd ⟧
       ⟹ Q n kvs"
proof -
  assume f0: "PROP f0" and f1: "PROP f1" and feven: "PROP feven" and fodd: "PROP fodd"
    and g0: "PROP g0" and g1: "PROP g1" and geven: "PROP geven" and godd: "PROP godd"
  show "n ≤ length kvs ⟹ P n kvs" and "n ≤ Suc (length kvs) ⟹ Q n kvs"
  proof(induction rule: rbtreeify_f_rbtreeify_g.induct)
    case (1 n kvs)
    show ?case
    proof(cases "n ≤ 1")
      case True thus ?thesis using "1.prems"
        by(cases n kvs rule: nat.exhaust[case_product list.exhaust])
          (auto simp add: f0[unfolded f0_def] f1[unfolded f1_def])
    next
      case False 
      hence ns: "n ≠ 0" "n ≠ 1" by simp_all
      hence ge0: "n div 2 > 0" by simp
      note IH = "1.IH"[OF ns]
      show ?thesis
      proof(cases "n mod 2 = 0")
        case True note ge0 
        moreover from "1.prems" have n2: "n div 2 ≤ length kvs" by simp
        moreover from True n2 have "P (n div 2) kvs" by(rule IH)
        moreover from length_rbtreeify_f[OF n2] ge0 "1.prems" obtain t k v kvs' 
          where kvs': "rbtreeify_f (n div 2) kvs = (t, (k, v) # kvs')"
          by(cases "snd (rbtreeify_f (n div 2) kvs)")
            (auto simp add: snd_def split: prod.split_asm)
        moreover from "1.prems" length_rbtreeify_f[OF n2] ge0
        have n2': "n div 2 ≤ Suc (length kvs')" by(simp add: kvs')
        moreover from True kvs'[symmetric] refl refl n2'
        have "Q (n div 2) kvs'" by(rule IH)
        moreover note feven[unfolded feven_def]
          (* FIXME: why does by(rule feven[unfolded feven_def]) not work? *)
        ultimately have "P (2 * (n div 2)) kvs" by -
        thus ?thesis using True by (metis minus_mod_eq_div_mult [symmetric] minus_nat.diff_0 mult.commute)
      next
        case False note ge0
        moreover from "1.prems" have n2: "n div 2 ≤ length kvs" by simp
        moreover from False n2 have "P (n div 2) kvs" by(rule IH)
        moreover from length_rbtreeify_f[OF n2] ge0 "1.prems" obtain t k v kvs' 
          where kvs': "rbtreeify_f (n div 2) kvs = (t, (k, v) # kvs')"
          by(cases "snd (rbtreeify_f (n div 2) kvs)")
            (auto simp add: snd_def split: prod.split_asm)
        moreover from "1.prems" length_rbtreeify_f[OF n2] ge0 False
        have n2': "n div 2 ≤ length kvs'" by(simp add: kvs') arith
        moreover from False kvs'[symmetric] refl refl n2' have "P (n div 2) kvs'" by(rule IH)
        moreover note fodd[unfolded fodd_def]
        ultimately have "P (Suc (2 * (n div 2))) kvs" by -
        thus ?thesis using False 
          by simp (metis One_nat_def Suc_eq_plus1_left le_add_diff_inverse mod_less_eq_dividend minus_mod_eq_mult_div [symmetric])
      qed
    qed
  next
    case (2 n kvs)
    show ?case
    proof(cases "n ≤ 1")
      case True thus ?thesis using "2.prems"
        by(cases n kvs rule: nat.exhaust[case_product list.exhaust])
          (auto simp add: g0[unfolded g0_def] g1[unfolded g1_def])
    next
      case False 
      hence ns: "¬ (n = 0 ∨ n = 1)" by simp
      hence ge0: "n div 2 > 0" by simp
      note IH = "2.IH"[OF ns]
      show ?thesis
      proof(cases "n mod 2 = 0")
        case True note ge0
        moreover from "2.prems" have n2: "n div 2 ≤ Suc (length kvs)" by simp
        moreover from True n2 have "Q (n div 2) kvs" by(rule IH)
        moreover from length_rbtreeify_g[OF ge0 n2] ge0 "2.prems" obtain t k v kvs' 
          where kvs': "rbtreeify_g (n div 2) kvs = (t, (k, v) # kvs')"
          by(cases "snd (rbtreeify_g (n div 2) kvs)")
            (auto simp add: snd_def split: prod.split_asm)
        moreover from "2.prems" length_rbtreeify_g[OF ge0 n2] ge0
        have n2': "n div 2 ≤ Suc (length kvs')" by(simp add: kvs')
        moreover from True kvs'[symmetric] refl refl  n2'
        have "Q (n div 2) kvs'" by(rule IH)
        moreover note geven[unfolded geven_def]
        ultimately have "Q (2 * (n div 2)) kvs" by -
        thus ?thesis using True 
          by(metis minus_mod_eq_div_mult [symmetric] minus_nat.diff_0 mult.commute)
      next
        case False note ge0
        moreover from "2.prems" have n2: "n div 2 ≤ length kvs" by simp
        moreover from False n2 have "P (n div 2) kvs" by(rule IH)
        moreover from length_rbtreeify_f[OF n2] ge0 "2.prems" False obtain t k v kvs' 
          where kvs': "rbtreeify_f (n div 2) kvs = (t, (k, v) # kvs')"
          by(cases "snd (rbtreeify_f (n div 2) kvs)")
            (auto simp add: snd_def split: prod.split_asm, arith)
        moreover from "2.prems" length_rbtreeify_f[OF n2] ge0 False
        have n2': "n div 2 ≤ Suc (length kvs')" by(simp add: kvs') arith
        moreover from False kvs'[symmetric] refl refl n2'
        have "Q (n div 2) kvs'" by(rule IH)
        moreover note godd[unfolded godd_def]
        ultimately have "Q (Suc (2 * (n div 2))) kvs" by -
        thus ?thesis using False 
          by simp (metis One_nat_def Suc_eq_plus1_left le_add_diff_inverse mod_less_eq_dividend minus_mod_eq_mult_div [symmetric])
      qed
    qed
  qed
qed

lemma inv1_rbtreeify_f: "n ≤ length kvs 
  ⟹ inv1 (fst (rbtreeify_f n kvs))"
  and inv1_rbtreeify_g: "n ≤ Suc (length kvs)
  ⟹ inv1 (fst (rbtreeify_g n kvs))"
by(induct n kvs and n kvs rule: rbtreeify_induct) simp_all

fun plog2 :: "nat ⇒ nat" 
where "plog2 n = (if n ≤ 1 then 0 else plog2 (n div 2) + 1)"

declare plog2.simps [simp del]

lemma plog2_simps [simp]:
  "plog2 0 = 0" "plog2 (Suc 0) = 0"
  "0 < n ⟹ plog2 (2 * n) = 1 + plog2 n"
  "0 < n ⟹ plog2 (Suc (2 * n)) = 1 + plog2 n"
by(subst plog2.simps, simp add: Suc_double_half)+

lemma bheight_rbtreeify_f: "n ≤ length kvs
  ⟹ bheight (fst (rbtreeify_f n kvs)) = plog2 n"
  and bheight_rbtreeify_g: "n ≤ Suc (length kvs)
  ⟹ bheight (fst (rbtreeify_g n kvs)) = plog2 n"
by(induct n kvs and n kvs rule: rbtreeify_induct) simp_all

lemma bheight_rbtreeify_f_eq_plog2I:
  "⟦ rbtreeify_f n kvs = (t, kvs'); n ≤ length kvs ⟧ 
  ⟹ bheight t = plog2 n"
using bheight_rbtreeify_f[of n kvs] by simp

lemma bheight_rbtreeify_g_eq_plog2I: 
  "⟦ rbtreeify_g n kvs = (t, kvs'); n ≤ Suc (length kvs) ⟧
  ⟹ bheight t = plog2 n"
using bheight_rbtreeify_g[of n kvs] by simp

hide_const (open) plog2

lemma inv2_rbtreeify_f: "n ≤ length kvs
  ⟹ inv2 (fst (rbtreeify_f n kvs))"
  and inv2_rbtreeify_g: "n ≤ Suc (length kvs)
  ⟹ inv2 (fst (rbtreeify_g n kvs))"
by(induct n kvs and n kvs rule: rbtreeify_induct)
  (auto simp add: bheight_rbtreeify_f bheight_rbtreeify_g 
        intro: bheight_rbtreeify_f_eq_plog2I bheight_rbtreeify_g_eq_plog2I)

lemma "n ≤ length kvs ⟹ True"
  and color_of_rbtreeify_g:
  "⟦ n ≤ Suc (length kvs); 0 < n ⟧ 
  ⟹ color_of (fst (rbtreeify_g n kvs)) = B"
by(induct n kvs and n kvs rule: rbtreeify_induct) simp_all

lemma entries_rbtreeify_f_append:
  "n ≤ length kvs 
  ⟹ entries (fst (rbtreeify_f n kvs)) @ snd (rbtreeify_f n kvs) = kvs"
  and entries_rbtreeify_g_append: 
  "n ≤ Suc (length kvs) 
  ⟹ entries (fst (rbtreeify_g n kvs)) @ snd (rbtreeify_g n kvs) = kvs"
by(induction rule: rbtreeify_induct) simp_all

lemma length_entries_rbtreeify_f:
  "n ≤ length kvs ⟹ length (entries (fst (rbtreeify_f n kvs))) = n"
  and length_entries_rbtreeify_g: 
  "n ≤ Suc (length kvs) ⟹ length (entries (fst (rbtreeify_g n kvs))) = n - 1"
by(induct rule: rbtreeify_induct) simp_all

lemma rbtreeify_f_conv_drop: 
  "n ≤ length kvs ⟹ snd (rbtreeify_f n kvs) = drop n kvs"
using entries_rbtreeify_f_append[of n kvs]
by(simp add: append_eq_conv_conj length_entries_rbtreeify_f)

lemma rbtreeify_g_conv_drop: 
  "n ≤ Suc (length kvs) ⟹ snd (rbtreeify_g n kvs) = drop (n - 1) kvs"
using entries_rbtreeify_g_append[of n kvs]
by(simp add: append_eq_conv_conj length_entries_rbtreeify_g)

lemma entries_rbtreeify_f [simp]:
  "n ≤ length kvs ⟹ entries (fst (rbtreeify_f n kvs)) = take n kvs"
using entries_rbtreeify_f_append[of n kvs]
by(simp add: append_eq_conv_conj length_entries_rbtreeify_f)

lemma entries_rbtreeify_g [simp]:
  "n ≤ Suc (length kvs) ⟹ 
  entries (fst (rbtreeify_g n kvs)) = take (n - 1) kvs"
using entries_rbtreeify_g_append[of n kvs]
by(simp add: append_eq_conv_conj length_entries_rbtreeify_g)

lemma keys_rbtreeify_f [simp]: "n ≤ length kvs
  ⟹ keys (fst (rbtreeify_f n kvs)) = take n (map fst kvs)"
by(simp add: keys_def take_map)

lemma keys_rbtreeify_g [simp]: "n ≤ Suc (length kvs)
  ⟹ keys (fst (rbtreeify_g n kvs)) = take (n - 1) (map fst kvs)"
by(simp add: keys_def take_map)

lemma rbtreeify_fD: 
  "⟦ rbtreeify_f n kvs = (t, kvs'); n ≤ length kvs ⟧ 
  ⟹ entries t = take n kvs ∧ kvs' = drop n kvs"
using rbtreeify_f_conv_drop[of n kvs] entries_rbtreeify_f[of n kvs] by simp

lemma rbtreeify_gD: 
  "⟦ rbtreeify_g n kvs = (t, kvs'); n ≤ Suc (length kvs) ⟧
  ⟹ entries t = take (n - 1) kvs ∧ kvs' = drop (n - 1) kvs"
using rbtreeify_g_conv_drop[of n kvs] entries_rbtreeify_g[of n kvs] by simp

lemma entries_rbtreeify [simp]: "entries (rbtreeify kvs) = kvs"
by(simp add: rbtreeify_def entries_rbtreeify_g)

context linorder begin

lemma rbt_sorted_rbtreeify_f: 
  "⟦ n ≤ length kvs; sorted (map fst kvs); distinct (map fst kvs) ⟧ 
  ⟹ rbt_sorted (fst (rbtreeify_f n kvs))"
  and rbt_sorted_rbtreeify_g: 
  "⟦ n ≤ Suc (length kvs); sorted (map fst kvs); distinct (map fst kvs) ⟧
  ⟹ rbt_sorted (fst (rbtreeify_g n kvs))"
proof(induction n kvs and n kvs rule: rbtreeify_induct)
  case (f_even n kvs t k v kvs')
  from rbtreeify_fD[OF ‹rbtreeify_f n kvs = (t, (k, v) # kvs')› ‹n ≤ length kvs›]
  have "entries t = take n kvs"
    and kvs': "drop n kvs = (k, v) # kvs'" by simp_all
  hence unfold: "kvs = take n kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
  from ‹sorted (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take n kvs). x ≤ k) ∧ (∀(x, y) ∈ set kvs'. k ≤ x)"
    by(subst (asm) unfold)(auto simp add: sorted_append)
  moreover from ‹distinct (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take n kvs). x ≠ k) ∧ (∀(x, y) ∈ set kvs'. x ≠ k)"
    by(subst (asm) unfold)(auto intro: rev_image_eqI)
  ultimately have "(∀(x, y) ∈ set (take n kvs). x < k) ∧ (∀(x, y) ∈ set kvs'. k < x)"
    by fastforce
  hence "fst (rbtreeify_f n kvs) |« k" "k «| fst (rbtreeify_g n kvs')"
    using ‹n ≤ Suc (length kvs')› ‹n ≤ length kvs› set_take_subset[of "n - 1" kvs']
    by(auto simp add: ord.rbt_greater_prop ord.rbt_less_prop take_map split_def)
  moreover from ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
  have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule f_even.IH)
  moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
    using ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
    by(subst (asm) (1 2) unfold, simp add: sorted_append)+
  hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule f_even.IH)
  ultimately show ?case
    using ‹0 < n› ‹rbtreeify_f n kvs = (t, (k, v) # kvs')› by simp
next
  case (f_odd n kvs t k v kvs')
  from rbtreeify_fD[OF ‹rbtreeify_f n kvs = (t, (k, v) # kvs')› ‹n ≤ length kvs›]
  have "entries t = take n kvs" 
    and kvs': "drop n kvs = (k, v) # kvs'" by simp_all
  hence unfold: "kvs = take n kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
  from ‹sorted (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take n kvs). x ≤ k) ∧ (∀(x, y) ∈ set kvs'. k ≤ x)"
    by(subst (asm) unfold)(auto simp add: sorted_append)
  moreover from ‹distinct (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take n kvs). x ≠ k) ∧ (∀(x, y) ∈ set kvs'. x ≠ k)"
    by(subst (asm) unfold)(auto intro: rev_image_eqI)
  ultimately have "(∀(x, y) ∈ set (take n kvs). x < k) ∧ (∀(x, y) ∈ set kvs'. k < x)"
    by fastforce
  hence "fst (rbtreeify_f n kvs) |« k" "k «| fst (rbtreeify_f n kvs')"
    using ‹n ≤ length kvs'› ‹n ≤ length kvs› set_take_subset[of n kvs']
    by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)
  moreover from ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
  have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule f_odd.IH)
  moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
    using ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
    by(subst (asm) (1 2) unfold, simp add: sorted_append)+
  hence "rbt_sorted (fst (rbtreeify_f n kvs'))" by(rule f_odd.IH)
  ultimately show ?case 
    using ‹0 < n› ‹rbtreeify_f n kvs = (t, (k, v) # kvs')› by simp
next
  case (g_even n kvs t k v kvs')
  from rbtreeify_gD[OF ‹rbtreeify_g n kvs = (t, (k, v) # kvs')› ‹n ≤ Suc (length kvs)›]
  have t: "entries t = take (n - 1) kvs" 
    and kvs': "drop (n - 1) kvs = (k, v) # kvs'" by simp_all
  hence unfold: "kvs = take (n - 1) kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
  from ‹sorted (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take (n - 1) kvs). x ≤ k) ∧ (∀(x, y) ∈ set kvs'. k ≤ x)"
    by(subst (asm) unfold)(auto simp add: sorted_append)
  moreover from ‹distinct (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take (n - 1) kvs). x ≠ k) ∧ (∀(x, y) ∈ set kvs'. x ≠ k)"
    by(subst (asm) unfold)(auto intro: rev_image_eqI)
  ultimately have "(∀(x, y) ∈ set (take (n - 1) kvs). x < k) ∧ (∀(x, y) ∈ set kvs'. k < x)"
    by fastforce
  hence "fst (rbtreeify_g n kvs) |« k" "k «| fst (rbtreeify_g n kvs')"
    using ‹n ≤ Suc (length kvs')› ‹n ≤ Suc (length kvs)› set_take_subset[of "n - 1" kvs']
    by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)
  moreover from ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
  have "rbt_sorted (fst (rbtreeify_g n kvs))" by(rule g_even.IH)
  moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
    using ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
    by(subst (asm) (1 2) unfold, simp add: sorted_append)+
  hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule g_even.IH)
  ultimately show ?case using ‹0 < n› ‹rbtreeify_g n kvs = (t, (k, v) # kvs')› by simp
next
  case (g_odd n kvs t k v kvs')
  from rbtreeify_fD[OF ‹rbtreeify_f n kvs = (t, (k, v) # kvs')› ‹n ≤ length kvs›]
  have "entries t = take n kvs"
    and kvs': "drop n kvs = (k, v) # kvs'" by simp_all
  hence unfold: "kvs = take n kvs @ (k, v) # kvs'" by(metis append_take_drop_id)
  from ‹sorted (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take n kvs). x ≤ k) ∧ (∀(x, y) ∈ set kvs'. k ≤ x)"
    by(subst (asm) unfold)(auto simp add: sorted_append)
  moreover from ‹distinct (map fst kvs)› kvs'
  have "(∀(x, y) ∈ set (take n kvs). x ≠ k) ∧ (∀(x, y) ∈ set kvs'. x ≠ k)"
    by(subst (asm) unfold)(auto intro: rev_image_eqI)
  ultimately have "(∀(x, y) ∈ set (take n kvs). x < k) ∧ (∀(x, y) ∈ set kvs'. k < x)"
    by fastforce
  hence "fst (rbtreeify_f n kvs) |« k" "k «| fst (rbtreeify_g n kvs')"
    using ‹n ≤ Suc (length kvs')› ‹n ≤ length kvs› set_take_subset[of "n - 1" kvs']
    by(auto simp add: rbt_greater_prop rbt_less_prop take_map split_def)
  moreover from ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
  have "rbt_sorted (fst (rbtreeify_f n kvs))" by(rule g_odd.IH)
  moreover have "sorted (map fst kvs')" "distinct (map fst kvs')"
    using ‹sorted (map fst kvs)› ‹distinct (map fst kvs)›
    by(subst (asm) (1 2) unfold, simp add: sorted_append)+
  hence "rbt_sorted (fst (rbtreeify_g n kvs'))" by(rule g_odd.IH)
  ultimately show ?case
    using ‹0 < n› ‹rbtreeify_f n kvs = (t, (k, v) # kvs')› by simp
qed simp_all

lemma rbt_sorted_rbtreeify: 
  "⟦ sorted (map fst kvs); distinct (map fst kvs) ⟧ ⟹ rbt_sorted (rbtreeify kvs)"
by(simp add: rbtreeify_def rbt_sorted_rbtreeify_g)

lemma is_rbt_rbtreeify: 
  "⟦ sorted (map fst kvs); distinct (map fst kvs) ⟧
  ⟹ is_rbt (rbtreeify kvs)"
by(simp add: is_rbt_def rbtreeify_def inv1_rbtreeify_g inv2_rbtreeify_g rbt_sorted_rbtreeify_g color_of_rbtreeify_g)

lemma rbt_lookup_rbtreeify:
  "⟦ sorted (map fst kvs); distinct (map fst kvs) ⟧ ⟹ 
  rbt_lookup (rbtreeify kvs) = map_of kvs"
by(simp add: map_of_entries[symmetric] rbt_sorted_rbtreeify)

end

text ‹
  Functions to compare the height of two rbt trees, taken from 
  Andrew W. Appel, Efficient Verified Red-Black Trees (September 2011)
›

fun skip_red :: "('a, 'b) rbt ⇒ ('a, 'b) rbt"
where
  "skip_red (Branch color.R l k v r) = l"
| "skip_red t = t"

definition skip_black :: "('a, 'b) rbt ⇒ ('a, 'b) rbt"
where
  "skip_black t = (let t' = skip_red t in case t' of Branch color.B l k v r ⇒ l | _ ⇒ t')"

datatype compare = LT | GT | EQ

partial_function (tailrec) compare_height :: "('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ compare"
where
  "compare_height sx s t tx =
  (case (skip_red sx, skip_red s, skip_red t, skip_red tx) of
     (Branch _ sx' _ _ _, Branch _ s' _ _ _, Branch _ t' _ _ _, Branch _ tx' _ _ _) ⇒ 
       compare_height (skip_black sx') s' t' (skip_black tx')
   | (_, rbt.Empty, _, Branch _ _ _ _ _) ⇒ LT
   | (Branch _ _ _ _ _, _, rbt.Empty, _) ⇒ GT
   | (Branch _ sx' _ _ _, Branch _ s' _ _ _, Branch _ t' _ _ _, rbt.Empty) ⇒
       compare_height (skip_black sx') s' t' rbt.Empty
   | (rbt.Empty, Branch _ s' _ _ _, Branch _ t' _ _ _, Branch _ tx' _ _ _) ⇒
       compare_height rbt.Empty s' t' (skip_black tx')
   | _ ⇒ EQ)"

declare compare_height.simps [code]

hide_type (open) compare
hide_const (open)
  compare_height skip_black skip_red LT GT EQ case_compare rec_compare
  Abs_compare Rep_compare
hide_fact (open)
  Abs_compare_cases Abs_compare_induct Abs_compare_inject Abs_compare_inverse
  Rep_compare Rep_compare_cases Rep_compare_induct Rep_compare_inject Rep_compare_inverse
  compare.simps compare.exhaust compare.induct compare.rec compare.simps
  compare.size compare.case_cong compare.case_cong_weak compare.case
  compare.nchotomy compare.split compare.split_asm compare.eq.refl compare.eq.simps
  equal_compare_def
  skip_red.simps skip_red.cases skip_red.induct 
  skip_black_def
  compare_height.simps

subsection ‹union and intersection of sorted associative lists›

context ord begin

function sunion_with :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a × 'b) list ⇒ ('a × 'b) list ⇒ ('a × 'b) list" 
where
  "sunion_with f ((k, v) # as) ((k', v') # bs) =
   (if k > k' then (k', v') # sunion_with f ((k, v) # as) bs
    else if k < k' then (k, v) # sunion_with f as ((k', v') # bs)
    else (k, f k v v') # sunion_with f as bs)"
| "sunion_with f [] bs = bs"
| "sunion_with f as [] = as"
by pat_completeness auto
termination by lexicographic_order

function sinter_with :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a × 'b) list ⇒ ('a × 'b) list ⇒ ('a × 'b) list"
where
  "sinter_with f ((k, v) # as) ((k', v') # bs) =
  (if k > k' then sinter_with f ((k, v) # as) bs
   else if k < k' then sinter_with f as ((k', v') # bs)
   else (k, f k v v') # sinter_with f as bs)"
| "sinter_with f [] _ = []"
| "sinter_with f _ [] = []"
by pat_completeness auto
termination by lexicographic_order

end

declare ord.sunion_with.simps [code] ord.sinter_with.simps[code]

context linorder begin

lemma set_fst_sunion_with: 
  "set (map fst (sunion_with f xs ys)) = set (map fst xs) ∪ set (map fst ys)"
by(induct f xs ys rule: sunion_with.induct) auto

lemma sorted_sunion_with [simp]:
  "⟦ sorted (map fst xs); sorted (map fst ys) ⟧ 
  ⟹ sorted (map fst (sunion_with f xs ys))"
by(induct f xs ys rule: sunion_with.induct)
  (auto simp add: set_fst_sunion_with simp del: set_map)

lemma distinct_sunion_with [simp]:
  "⟦ distinct (map fst xs); distinct (map fst ys); sorted (map fst xs); sorted (map fst ys) ⟧
  ⟹ distinct (map fst (sunion_with f xs ys))"
proof(induct f xs ys rule: sunion_with.induct)
  case (1 f k v xs k' v' ys)
  have "⟦ ¬ k < k'; ¬ k' < k ⟧ ⟹ k = k'" by simp
  thus ?case using "1"
    by(auto simp add: set_fst_sunion_with simp del: set_map)
qed simp_all

lemma map_of_sunion_with: 
  "⟦ sorted (map fst xs); sorted (map fst ys) ⟧
  ⟹ map_of (sunion_with f xs ys) k = 
  (case map_of xs k of None ⇒ map_of ys k 
  | Some v ⇒ case map_of ys k of None ⇒ Some v 
              | Some w ⇒ Some (f k v w))"
by(induct f xs ys rule: sunion_with.induct)(auto split: option.split dest: map_of_SomeD bspec)

lemma set_fst_sinter_with [simp]:
  "⟦ sorted (map fst xs); sorted (map fst ys) ⟧
  ⟹ set (map fst (sinter_with f xs ys)) = set (map fst xs) ∩ set (map fst ys)"
by(induct f xs ys rule: sinter_with.induct)(auto simp del: set_map)

lemma set_fst_sinter_with_subset1:
  "set (map fst (sinter_with f xs ys)) ⊆ set (map fst xs)"
by(induct f xs ys rule: sinter_with.induct) auto

lemma set_fst_sinter_with_subset2:
  "set (map fst (sinter_with f xs ys)) ⊆ set (map fst ys)"
by(induct f xs ys rule: sinter_with.induct)(auto simp del: set_map)

lemma sorted_sinter_with [simp]:
  "⟦ sorted (map fst xs); sorted (map fst ys) ⟧
  ⟹ sorted (map fst (sinter_with f xs ys))"
by(induct f xs ys rule: sinter_with.induct)(auto simp del: set_map)

lemma distinct_sinter_with [simp]:
  "⟦ distinct (map fst xs); distinct (map fst ys) ⟧
  ⟹ distinct (map fst (sinter_with f xs ys))"
proof(induct f xs ys rule: sinter_with.induct)
  case (1 f k v as k' v' bs)
  have "⟦ ¬ k < k'; ¬ k' < k ⟧ ⟹ k = k'" by simp
  thus ?case using "1" set_fst_sinter_with_subset1[of f as bs]
    set_fst_sinter_with_subset2[of f as bs]
    by(auto simp del: set_map)
qed simp_all

lemma map_of_sinter_with:
  "⟦ sorted (map fst xs); sorted (map fst ys) ⟧
  ⟹ map_of (sinter_with f xs ys) k = 
  (case map_of xs k of None ⇒ None | Some v ⇒ map_option (f k v) (map_of ys k))"
apply(induct f xs ys rule: sinter_with.induct)
apply(auto simp add: map_option_case split: option.splits dest: map_of_SomeD bspec)
done

end

lemma distinct_map_of_rev: "distinct (map fst xs) ⟹ map_of (rev xs) = map_of xs"
by(induct xs)(auto 4 3 simp add: map_add_def intro!: ext split: option.split intro: rev_image_eqI)

lemma map_map_filter: 
  "map f (List.map_filter g xs) = List.map_filter (map_option f ∘ g) xs"
by(auto simp add: List.map_filter_def)

lemma map_filter_map_option_const: 
  "List.map_filter (λx. map_option (λy. f x) (g (f x))) xs = filter (λx. g x ≠ None) (map f xs)"
by(auto simp add: map_filter_def filter_map o_def)

lemma set_map_filter: "set (List.map_filter P xs) = the ` (P ` set xs - {None})"
by(auto simp add: List.map_filter_def intro: rev_image_eqI)

(* Split and Join *)

definition is_rbt_empty :: "('a, 'b) rbt ⇒ bool" where
  "is_rbt_empty t ⟷ (case t of RBT_Impl.Empty ⇒ True | _ ⇒ False)"

lemma is_rbt_empty_prop[simp]: "is_rbt_empty t ⟷ t = RBT_Impl.Empty"
  by (auto simp: is_rbt_empty_def split: RBT_Impl.rbt.splits)

definition small_rbt :: "('a, 'b) rbt ⇒ bool" where
  "small_rbt t ⟷ bheight t < 4"

definition flip_rbt :: "('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ bool" where
  "flip_rbt t1 t2 ⟷ bheight t2 < bheight t1"

abbreviation (input) MR where "MR l a b r ≡ Branch RBT_Impl.R l a b r"
abbreviation (input) MB where "MB l a b r ≡ Branch RBT_Impl.B l a b r"

fun rbt_baliL :: "('a, 'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_baliL (MR (MR t1 a b t2) a' b' t3) a'' b'' t4 = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
| "rbt_baliL (MR t1 a b (MR t2 a' b' t3)) a'' b'' t4 = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
| "rbt_baliL t1 a b t2 = MB t1 a b t2"

fun rbt_baliR :: "('a, 'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_baliR t1 a b (MR t2 a' b' (MR t3 a'' b'' t4)) = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
| "rbt_baliR t1 a b (MR (MR t2 a' b' t3) a'' b'' t4) = MR (MB t1 a b t2) a' b' (MB t3 a'' b'' t4)"
| "rbt_baliR t1 a b t2 = MB t1 a b t2"

fun rbt_baldL :: "('a, 'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_baldL (MR t1 a b t2) a' b' t3 = MR (MB t1 a b t2) a' b' t3"
| "rbt_baldL t1 a b (MB t2 a' b' t3) = rbt_baliR t1 a b (MR t2 a' b' t3)"
| "rbt_baldL t1 a b (MR (MB t2 a' b' t3) a'' b'' t4) =
  MR (MB t1 a b t2) a' b' (rbt_baliR t3 a'' b'' (paint RBT_Impl.R t4))"
| "rbt_baldL t1 a b t2 = MR t1 a b t2"

fun rbt_baldR :: "('a, 'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_baldR t1 a b (MR t2 a' b' t3) = MR t1 a b (MB t2 a' b' t3)"
| "rbt_baldR (MB t1 a b t2) a' b' t3 = rbt_baliL (MR t1 a b t2) a' b' t3"
| "rbt_baldR (MR t1 a b (MB t2 a' b' t3)) a'' b'' t4 =
  MR (rbt_baliL (paint RBT_Impl.R t1) a b t2) a' b' (MB t3 a'' b'' t4)"
| "rbt_baldR t1 a b t2 = MR t1 a b t2"

fun rbt_app :: "('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_app RBT_Impl.Empty t = t"
| "rbt_app t RBT_Impl.Empty = t"
| "rbt_app (MR t1 a b t2) (MR t3 a'' b'' t4) = (case rbt_app t2 t3 of
    MR u2 a' b' u3 ⇒ (MR (MR t1 a b u2) a' b' (MR u3 a'' b'' t4))
  | t23 ⇒ MR t1 a b (MR t23 a'' b'' t4))"
| "rbt_app (MB t1 a b t2) (MB t3 a'' b'' t4) = (case rbt_app t2 t3 of
    MR u2 a' b' u3 ⇒ MR (MB t1 a b u2) a' b' (MB u3 a'' b'' t4)
  | t23 ⇒ rbt_baldL t1 a b (MB t23 a'' b'' t4))"
| "rbt_app t1 (MR t2 a b t3) = MR (rbt_app t1 t2) a b t3"
| "rbt_app (MR t1 a b t2) t3 = MR t1 a b (rbt_app t2 t3)"

fun rbt_joinL :: "('a, 'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_joinL l a b r = (if bheight l ≥ bheight r then MR l a b r
    else case r of MB l' a' b' r' ⇒ rbt_baliL (rbt_joinL l a b l') a' b' r'
    | MR l' a' b' r' ⇒ MR (rbt_joinL l a b l') a' b' r')"

declare rbt_joinL.simps[simp del]

fun rbt_joinR :: "('a, 'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_joinR l a b r = (if bheight l ≤ bheight r then MR l a b r
    else case l of MB l' a' b' r' ⇒ rbt_baliR l' a' b' (rbt_joinR r' a b r)
    | MR l' a' b' r' ⇒ MR l' a' b' (rbt_joinR r' a b r))"

declare rbt_joinR.simps[simp del]

definition rbt_join :: "('a, 'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_join l a b r =
    (let bhl = bheight l; bhr = bheight r
    in if bhl > bhr
    then paint RBT_Impl.B (rbt_joinR l a b r)
    else if bhl < bhr
    then paint RBT_Impl.B (rbt_joinL l a b r)
    else MB l a b r)"

lemma size_paint[simp]: "size (paint c t) = size t"
  by (cases t) auto

lemma size_baliL[simp]: "size (rbt_baliL t1 a b t2) = Suc (size t1 + size t2)"
  by (induction t1 a b t2 rule: rbt_baliL.induct) auto

lemma size_baliR[simp]: "size (rbt_baliR t1 a b t2) = Suc (size t1 + size t2)"
  by (induction t1 a b t2 rule: rbt_baliR.induct) auto

lemma size_baldL[simp]: "size (rbt_baldL t1 a b t2) = Suc (size t1 + size t2)"
  by (induction t1 a b t2 rule: rbt_baldL.induct) auto

lemma size_baldR[simp]: "size (rbt_baldR t1 a b t2) = Suc (size t1 + size t2)"
  by (induction t1 a b t2 rule: rbt_baldR.induct) auto

lemma size_rbt_app[simp]: "size (rbt_app t1 t2) = size t1 + size t2"
  by (induction t1 t2 rule: rbt_app.induct)
     (auto split: RBT_Impl.rbt.splits RBT_Impl.color.splits)

lemma size_rbt_joinL[simp]: "size (rbt_joinL t1 a b t2) = Suc (size t1 + size t2)"
  by (induction t1 a b t2 rule: rbt_joinL.induct)
     (auto simp: rbt_joinL.simps split: RBT_Impl.rbt.splits RBT_Impl.color.splits)

lemma size_rbt_joinR[simp]: "size (rbt_joinR t1 a b t2) = Suc (size t1 + size t2)"
  by (induction t1 a b t2 rule: rbt_joinR.induct)
     (auto simp: rbt_joinR.simps split: RBT_Impl.rbt.splits RBT_Impl.color.splits)

lemma size_rbt_join[simp]: "size (rbt_join t1 a b t2) = Suc (size t1 + size t2)"
  by (auto simp: rbt_join_def Let_def)

definition "inv_12 t ⟷ inv1 t ∧ inv2 t"

lemma rbt_Node: "inv_12 (RBT_Impl.Branch c l a b r) ⟹ inv_12 l ∧ inv_12 r"
  by (auto simp: inv_12_def)

lemma paint2: "paint c2 (paint c1 t) = paint c2 t"
  by (cases t) auto

lemma inv1_rbt_baliL: "inv1l l ⟹ inv1 r ⟹ inv1 (rbt_baliL l a b r)"
  by (induct l a b r rule: rbt_baliL.induct) auto

lemma inv1_rbt_baliR: "inv1 l ⟹ inv1l r ⟹ inv1 (rbt_baliR l a b r)"
  by (induct l a b r rule: rbt_baliR.induct) auto

lemma rbt_bheight_rbt_baliL: "bheight l = bheight r ⟹ bheight (rbt_baliL l a b r) = Suc (bheight l)"
  by (induct l a b r rule: rbt_baliL.induct) auto

lemma rbt_bheight_rbt_baliR: "bheight l = bheight r ⟹ bheight (rbt_baliR l a b r) = Suc (bheight l)"
  by (induct l a b r rule: rbt_baliR.induct) auto

lemma inv2_rbt_baliL: "inv2 l ⟹ inv2 r ⟹ bheight l = bheight r ⟹ inv2 (rbt_baliL l a b r)"
  by (induct l a b r rule: rbt_baliL.induct) auto

lemma inv2_rbt_baliR: "inv2 l ⟹ inv2 r ⟹ bheight l = bheight r ⟹ inv2 (rbt_baliR l a b r)"
  by (induct l a b r rule: rbt_baliR.induct) auto

lemma inv_rbt_baliR: "inv2 l ⟹ inv2 r ⟹ inv1 l ⟹ inv1l r ⟹ bheight l = bheight r ⟹
  inv1 (rbt_baliR l a b r) ∧ inv2 (rbt_baliR l a b r) ∧ bheight (rbt_baliR l a b r) = Suc (bheight l)"
  by (induct l a b r rule: rbt_baliR.induct) auto

lemma inv_rbt_baliL: "inv2 l ⟹ inv2 r ⟹ inv1l l ⟹ inv1 r ⟹ bheight l = bheight r ⟹
  inv1 (rbt_baliL l a b r) ∧ inv2 (rbt_baliL l a b r) ∧ bheight (rbt_baliL l a b r) = Suc (bheight l)"
  by (induct l a b r rule: rbt_baliL.induct) auto

lemma inv2_rbt_baldL_inv1: "inv2 l ⟹ inv2 r ⟹ bheight l + 1 = bheight r ⟹ inv1 r ⟹
  inv2 (rbt_baldL l a b r) ∧ bheight (rbt_baldL l a b r) = bheight r"
  by (induct l a b r rule: rbt_baldL.induct) (auto simp: inv2_rbt_baliR rbt_bheight_rbt_baliR)

lemma inv2_rbt_baldL_B: "inv2 l ⟹ inv2 r ⟹ bheight l + 1 = bheight r ⟹ color_of r = RBT_Impl.B ⟹
  inv2 (rbt_baldL l a b r) ∧ bheight (rbt_baldL l a b r) = bheight r"
  by (induct l a b r rule: rbt_baldL.induct) (auto simp add: inv2_rbt_baliR rbt_bheight_rbt_baliR)

lemma inv1_rbt_baldL: "inv1l l ⟹ inv1 r ⟹ color_of r = RBT_Impl.B ⟹ inv1 (rbt_baldL l a b r)"
  by (induct l a b r rule: rbt_baldL.induct) (simp_all add: inv1_rbt_baliR)

lemma inv1lI: "inv1 t ⟹ inv1l t"
  by (cases t) auto

lemma neq_Black[simp]: "(c ≠ RBT_Impl.B) = (c = RBT_Impl.R)"
  by (cases c) auto

lemma inv1l_rbt_baldL: "inv1l l ⟹ inv1 r ⟹ inv1l (rbt_baldL l a b r)"
  by (induct l a b r rule: rbt_baldL.induct) (auto simp: inv1_rbt_baliR paint2)

lemma inv2_rbt_baldR_inv1: "inv2 l ⟹ inv2 r ⟹ bheight l = bheight r + 1 ⟹ inv1 l ⟹
  inv2 (rbt_baldR l a b r) ∧ bheight (rbt_baldR l a b r) = bheight l"
  by (induct l a b r rule: rbt_baldR.induct) (auto simp: inv2_rbt_baliL rbt_bheight_rbt_baliL)

lemma inv1_rbt_baldR: "inv1 l ⟹ inv1l r ⟹ color_of l = RBT_Impl.B ⟹ inv1 (rbt_baldR l a b r)"
  by (induct l a b r rule: rbt_baldR.induct) (simp_all add: inv1_rbt_baliL)

lemma inv1l_rbt_baldR: "inv1 l ⟹ inv1l r ⟹inv1l (rbt_baldR l a b r)"
  by (induct l a b r rule: rbt_baldR.induct) (auto simp: inv1_rbt_baliL paint2)

lemma inv2_rbt_app: "inv2 l ⟹ inv2 r ⟹ bheight l = bheight r ⟹
  inv2 (rbt_app l r) ∧ bheight (rbt_app l r) = bheight l"
  by (induct l r rule: rbt_app.induct)
     (auto simp: inv2_rbt_baldL_B split: RBT_Impl.rbt.splits RBT_Impl.color.splits)

lemma inv1_rbt_app: "inv1 l ⟹ inv1 r ⟹ (color_of l = RBT_Impl.B ∧
  color_of r = RBT_Impl.B ⟶ inv1 (rbt_app l r)) ∧ inv1l (rbt_app l r)"
  by (induct l r rule: rbt_app.induct)
     (auto simp: inv1_rbt_baldL split: RBT_Impl.rbt.splits RBT_Impl.color.splits)

lemma inv_rbt_baldL: "inv2 l ⟹ inv2 r ⟹ bheight l + 1 = bheight r ⟹ inv1l l ⟹ inv1 r ⟹
  inv2 (rbt_baldL l a b r) ∧ bheight (rbt_baldL l a b r) = bheight r ∧
  inv1l (rbt_baldL l a b r) ∧ (color_of r = RBT_Impl.B ⟶ inv1 (rbt_baldL l a b r))"
  by (induct l a b r rule: rbt_baldL.induct) (auto simp: inv_rbt_baliR rbt_bheight_rbt_baliR paint2)

lemma inv_rbt_baldR: "inv2 l ⟹ inv2 r ⟹ bheight l = bheight r + 1 ⟹ inv1 l ⟹ inv1l r ⟹
  inv2 (rbt_baldR l a b r) ∧ bheight (rbt_baldR l a b r) = bheight l ∧
  inv1l (rbt_baldR l a b r) ∧ (color_of l = RBT_Impl.B ⟶ inv1 (rbt_baldR l a b r))"
  by (induct l a b r rule: rbt_baldR.induct) (auto simp: inv_rbt_baliL rbt_bheight_rbt_baliL paint2)

lemma inv_rbt_app: "inv2 l ⟹ inv2 r ⟹ bheight l = bheight r ⟹ inv1 l ⟹ inv1 r ⟹
  inv2 (rbt_app l r) ∧ bheight (rbt_app l r) = bheight l ∧
  inv1l (rbt_app l r) ∧ (color_of l = RBT_Impl.B ∧ color_of r = RBT_Impl.B ⟶ inv1 (rbt_app l r))"
  by (induct l r rule: rbt_app.induct)
     (auto simp: inv2_rbt_baldL_B inv_rbt_baldL split: RBT_Impl.rbt.splits RBT_Impl.color.splits)

lemma inv1l_rbt_joinL: "inv1 l ⟹ inv1 r ⟹ bheight l ≤ bheight r ⟹
  inv1l (rbt_joinL l a b r) ∧
  (bheight l ≠ bheight r ∧ color_of r = RBT_Impl.B ⟶ inv1 (rbt_joinL l a b r))"
proof (induct l a b r rule: rbt_joinL.induct)
  case (1 l a b r)
  then show ?case
    by (auto simp: inv1_rbt_baliL rbt_joinL.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma inv1l_rbt_joinR: "inv1 l ⟹ inv2 l ⟹ inv1 r ⟹ inv2 r ⟹ bheight l ≥ bheight r ⟹
  inv1l (rbt_joinR l a b r) ∧
  (bheight l ≠ bheight r ∧ color_of l = RBT_Impl.B ⟶ inv1 (rbt_joinR l a b r))"
proof (induct l a b r rule: rbt_joinR.induct)
  case (1 l a b r)
  then show ?case
    by (fastforce simp: inv1_rbt_baliR rbt_joinR.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma bheight_rbt_joinL: "inv2 l ⟹ inv2 r ⟹ bheight l ≤ bheight r ⟹
  bheight (rbt_joinL l a b r) = bheight r"
proof (induct l a b r rule: rbt_joinL.induct)
  case (1 l a b r)
  then show ?case
    by (auto simp: rbt_bheight_rbt_baliL rbt_joinL.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma inv2_rbt_joinL: "inv2 l ⟹ inv2 r ⟹ bheight l ≤ bheight r ⟹ inv2 (rbt_joinL l a b r)"
proof (induct l a b r rule: rbt_joinL.induct)
  case (1 l a b r)
  then show ?case
    by (auto simp: inv2_rbt_baliL bheight_rbt_joinL rbt_joinL.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma bheight_rbt_joinR: "inv2 l ⟹ inv2 r ⟹ bheight l ≥ bheight r ⟹
  bheight (rbt_joinR l a b r) = bheight l"
proof (induct l a b r rule: rbt_joinR.induct)
  case (1 l a b r)
  then show ?case
    by (fastforce simp: rbt_bheight_rbt_baliR rbt_joinR.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma inv2_rbt_joinR: "inv2 l ⟹ inv2 r ⟹ bheight l ≥ bheight r ⟹ inv2 (rbt_joinR l a b r)"
proof (induct l a b r rule: rbt_joinR.induct)
  case (1 l a b r)
  then show ?case
    by (fastforce simp: inv2_rbt_baliR bheight_rbt_joinR rbt_joinR.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma keys_paint[simp]: "RBT_Impl.keys (paint c t) = RBT_Impl.keys t"
  by (cases t) auto

lemma keys_rbt_baliL: "RBT_Impl.keys (rbt_baliL l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
  by (cases "(l,a,b,r)" rule: rbt_baliL.cases) auto

lemma keys_rbt_baliR: "RBT_Impl.keys (rbt_baliR l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
  by (cases "(l,a,b,r)" rule: rbt_baliR.cases) auto

lemma keys_rbt_baldL: "RBT_Impl.keys (rbt_baldL l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
  by (cases "(l,a,b,r)" rule: rbt_baldL.cases) (auto simp: keys_rbt_baliL keys_rbt_baliR)

lemma keys_rbt_baldR: "RBT_Impl.keys (rbt_baldR l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
  by (cases "(l,a,b,r)" rule: rbt_baldR.cases) (auto simp: keys_rbt_baliL keys_rbt_baliR)

lemma keys_rbt_app: "RBT_Impl.keys (rbt_app l r) = RBT_Impl.keys l @ RBT_Impl.keys r"
  by (induction l r rule: rbt_app.induct)
     (auto simp: keys_rbt_baldL keys_rbt_baldR split: RBT_Impl.rbt.splits RBT_Impl.color.splits)

lemma keys_rbt_joinL: "bheight l ≤ bheight r ⟹
  RBT_Impl.keys (rbt_joinL l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
proof (induction l a b r rule: rbt_joinL.induct)
  case (1 l a b r)
  thus ?case
    by (auto simp: keys_rbt_baliL rbt_joinL.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma keys_rbt_joinR: "RBT_Impl.keys (rbt_joinR l a b r) = RBT_Impl.keys l @ a # RBT_Impl.keys r"
proof (induction l a b r rule: rbt_joinR.induct)
  case (1 l a b r)
  thus ?case
    by (force simp: keys_rbt_baliR rbt_joinR.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma rbt_set_rbt_baliL: "set (RBT_Impl.keys (rbt_baliL l a b r)) =
  set (RBT_Impl.keys l) ∪ {a} ∪ set (RBT_Impl.keys r)"
  by (cases "(l,a,b,r)" rule: rbt_baliL.cases) auto

lemma set_rbt_joinL: "set (RBT_Impl.keys (rbt_joinL l a b r)) =
  set (RBT_Impl.keys l) ∪ {a} ∪ set (RBT_Impl.keys r)"
proof (induction l a b r rule: rbt_joinL.induct)
  case (1 l a b r)
  thus ?case
    by (auto simp: rbt_set_rbt_baliL rbt_joinL.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma rbt_set_rbt_baliR: "set (RBT_Impl.keys (rbt_baliR l a b r)) =
  set (RBT_Impl.keys l) ∪ {a} ∪ set (RBT_Impl.keys r)"
  by (cases "(l,a,b,r)" rule: rbt_baliR.cases) auto

lemma set_rbt_joinR: "set (RBT_Impl.keys (rbt_joinR l a b r)) =
  set (RBT_Impl.keys l) ∪ {a} ∪ set (RBT_Impl.keys r)"
proof (induction l a b r rule: rbt_joinR.induct)
  case (1 l a b r)
  thus ?case
    by (force simp: rbt_set_rbt_baliR rbt_joinR.simps[of l a b r]
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma set_keys_paint: "set (RBT_Impl.keys (paint c t)) = set (RBT_Impl.keys t)"
  by (cases t) auto

lemma set_rbt_join: "set (RBT_Impl.keys (rbt_join l a b r)) =
  set (RBT_Impl.keys l) ∪ {a} ∪ set (RBT_Impl.keys r)"
  by (simp add: set_rbt_joinL set_rbt_joinR set_keys_paint rbt_join_def Let_def)

lemma inv_rbt_join: "inv_12 l ⟹ inv_12 r ⟹ inv_12 (rbt_join l a b r)"
  by (auto simp: rbt_join_def Let_def inv1l_rbt_joinL inv1l_rbt_joinR
      inv2_rbt_joinL inv2_rbt_joinR inv_12_def)

fun rbt_recolor :: "('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_recolor (Branch RBT_Impl.R t1 k v t2) =
    (if color_of t1 = RBT_Impl.B ∧ color_of t2 = RBT_Impl.B then Branch RBT_Impl.B t1 k v t2
    else Branch RBT_Impl.R t1 k v t2)"
| "rbt_recolor t = t"

lemma rbt_recolor: "inv_12 t ⟹ inv_12 (rbt_recolor t)"
  by (induction t rule: rbt_recolor.induct) (auto simp: inv_12_def)

fun rbt_split_min :: "('a, 'b) rbt ⇒ 'a × 'b × ('a, 'b) rbt" where
  "rbt_split_min RBT_Impl.Empty = undefined"
| "rbt_split_min (RBT_Impl.Branch _ l a b r) =
    (if is_rbt_empty l then (a,b,r) else let (a',b',l') = rbt_split_min l in (a',b',rbt_join l' a b r))"

lemma rbt_split_min_set:
  "rbt_split_min t = (a,b,t') ⟹ t ≠ RBT_Impl.Empty ⟹
  a ∈ set (RBT_Impl.keys t) ∧ set (RBT_Impl.keys t) = {a} ∪ set (RBT_Impl.keys t')"
  by (induction t arbitrary: t') (auto simp: set_rbt_join split: prod.splits if_splits)

lemma rbt_split_min_inv: "rbt_split_min t = (a,b,t') ⟹ inv_12 t ⟹ t ≠ RBT_Impl.Empty ⟹ inv_12 t'"
  by (induction t arbitrary: t')
     (auto simp: inv_rbt_join split: if_splits prod.splits dest: rbt_Node)

definition rbt_join2 :: "('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_join2 l r = (if is_rbt_empty r then l else let (a,b,r') = rbt_split_min r in rbt_join l a b r')"

lemma set_rbt_join2[simp]: "set (RBT_Impl.keys (rbt_join2 l r)) =
  set (RBT_Impl.keys l) ∪ set (RBT_Impl.keys r)"
  by (simp add: rbt_join2_def rbt_split_min_set set_rbt_join split: prod.split)

lemma inv_rbt_join2: "inv_12 l ⟹ inv_12 r ⟹ inv_12 (rbt_join2 l r)"
  by (simp add: rbt_join2_def inv_rbt_join rbt_split_min_set rbt_split_min_inv split: prod.split)

context ord begin

fun rbt_split :: "('a, 'b) rbt ⇒ 'a ⇒ ('a, 'b) rbt × 'b option × ('a, 'b) rbt" where
  "rbt_split RBT_Impl.Empty k = (RBT_Impl.Empty, None, RBT_Impl.Empty)"
| "rbt_split (RBT_Impl.Branch _ l a b r) x =
  (if x < a then (case rbt_split l x of (l1, β, l2) ⇒ (l1, β, rbt_join l2 a b r))
  else if a < x then (case rbt_split r x of (r1, β, r2) ⇒ (rbt_join l a b r1, β, r2))
  else (l, Some b, r))"

lemma rbt_split: "rbt_split t x = (l,β,r) ⟹ inv_12 t ⟹ inv_12 l ∧ inv_12 r"
  by (induction t arbitrary: l r)
     (auto simp: set_rbt_join inv_rbt_join rbt_greater_prop rbt_less_prop
      split: if_splits prod.splits dest!: rbt_Node)

lemma rbt_split_size: "(l2,β,r2) = rbt_split t2 a ⟹ size l2 + size r2 ≤ size t2"
  by (induction t2 a arbitrary: l2 r2 rule: rbt_split.induct) (auto split: if_splits prod.splits)

function rbt_union_rec :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_union_rec f t1 t2 = (let (f, t2, t1) =
    if flip_rbt t2 t1 then (λk v v'. f k v' v, t1, t2) else (f, t2, t1) in
    if small_rbt t2 then RBT_Impl.fold (rbt_insert_with_key f) t2 t1
    else (case t1 of RBT_Impl.Empty ⇒ t2
      | RBT_Impl.Branch _ l1 a b r1 ⇒
        case rbt_split t2 a of (l2, β, r2) ⇒
          rbt_join (rbt_union_rec f l1 l2) a (case β of None ⇒ b | Some b' ⇒ f a b b') (rbt_union_rec f r1 r2)))"
  by pat_completeness auto
termination
  using rbt_split_size
  by (relation "measure (λ(f,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+

declare rbt_union_rec.simps[simp del]

function rbt_union_swap_rec :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ bool ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_union_swap_rec f γ t1 t2 = (let (γ, t2, t1) =
    if flip_rbt t2 t1 then (¬γ, t1, t2) else (γ, t2, t1);
    f' = (if γ then (λk v v'. f k v' v) else f) in
    if small_rbt t2 then RBT_Impl.fold (rbt_insert_with_key f') t2 t1
    else (case t1 of RBT_Impl.Empty ⇒ t2
      | RBT_Impl.Branch _ l1 a b r1 ⇒
        case rbt_split t2 a of (l2, β, r2) ⇒
          rbt_join (rbt_union_swap_rec f γ l1 l2) a (case β of None ⇒ b | Some b' ⇒ f' a b b') (rbt_union_swap_rec f γ r1 r2)))"
  by pat_completeness auto
termination
  using rbt_split_size
  by (relation "measure (λ(f,γ,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+

declare rbt_union_swap_rec.simps[simp del]

lemma rbt_union_swap_rec: "rbt_union_swap_rec f γ t1 t2 =
  rbt_union_rec (if γ then (λk v v'. f k v' v) else f) t1 t2"
proof (induction f γ t1 t2 rule: rbt_union_swap_rec.induct)
  case (1 f γ t1 t2)
  show ?case
    using 1[OF refl _ refl refl _ refl _ refl]
    unfolding rbt_union_swap_rec.simps[of _ _ t1] rbt_union_rec.simps[of _ t1]
    by (auto simp: Let_def split: rbt.splits prod.splits option.splits) (* slow *)
qed

lemma rbt_fold_rbt_insert:
  assumes "inv_12 t2"
  shows "inv_12 (RBT_Impl.fold (rbt_insert_with_key f) t1 t2)"
proof -
  define xs where "xs = RBT_Impl.entries t1"
  from assms show ?thesis
    unfolding RBT_Impl.fold_def xs_def[symmetric]
    by (induct xs rule: rev_induct)
       (auto simp: inv_12_def rbt_insert_with_key_def ins_inv1_inv2)
qed

lemma rbt_union_rec: "inv_12 t1 ⟹ inv_12 t2 ⟹ inv_12 (rbt_union_rec f t1 t2)"
proof (induction f t1 t2 rule: rbt_union_rec.induct)
  case (1 t1 t2)
  thus ?case
    by (auto simp: rbt_union_rec.simps[of t1 t2] inv_rbt_join rbt_split rbt_fold_rbt_insert
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits prod.split if_splits dest: rbt_Node)
qed

definition "map_filter_inter f t1 t2 = List.map_filter (λ(k, v).
  case rbt_lookup t1 k of None ⇒ None
  | Some v' ⇒ Some (k, f k v' v)) (RBT_Impl.entries t2)"

function rbt_inter_rec :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_inter_rec f t1 t2 = (let (f, t2, t1) =
    if flip_rbt t2 t1 then (λk v v'. f k v' v, t1, t2) else (f, t2, t1) in
    if small_rbt t2 then rbtreeify (map_filter_inter f t1 t2)
    else case t1 of RBT_Impl.Empty ⇒ RBT_Impl.Empty
    | RBT_Impl.Branch _ l1 a b r1 ⇒
      case rbt_split t2 a of (l2, β, r2) ⇒ let l' = rbt_inter_rec f l1 l2; r' = rbt_inter_rec f r1 r2 in
      (case β of None ⇒ rbt_join2 l' r' | Some b' ⇒ rbt_join l' a (f a b b') r'))"
  by pat_completeness auto
termination
  using rbt_split_size
  by (relation "measure (λ(f,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+

declare rbt_inter_rec.simps[simp del]

function rbt_inter_swap_rec :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ bool ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_inter_swap_rec f γ t1 t2 = (let (γ, t2, t1) =
    if flip_rbt t2 t1 then (¬γ, t1, t2) else (γ, t2, t1);
    f' = (if γ then (λk v v'. f k v' v) else f) in
    if small_rbt t2 then rbtreeify (map_filter_inter f' t1 t2)
    else case t1 of RBT_Impl.Empty ⇒ RBT_Impl.Empty
    | RBT_Impl.Branch _ l1 a b r1 ⇒
      case rbt_split t2 a of (l2, β, r2) ⇒ let l' = rbt_inter_swap_rec f γ l1 l2; r' = rbt_inter_swap_rec f γ r1 r2 in
      (case β of None ⇒ rbt_join2 l' r' | Some b' ⇒ rbt_join l' a (f' a b b') r'))"
  by pat_completeness auto
termination
  using rbt_split_size
  by (relation "measure (λ(f,γ,t1,t2). size t1 + size t2)") (fastforce split: if_splits)+

declare rbt_inter_swap_rec.simps[simp del]

lemma rbt_inter_swap_rec: "rbt_inter_swap_rec f γ t1 t2 =
  rbt_inter_rec (if γ then (λk v v'. f k v' v) else f) t1 t2"
proof (induction f γ t1 t2 rule: rbt_inter_swap_rec.induct)
  case (1 f γ t1 t2)
  show ?case
    using 1[OF refl _ refl refl _ refl _ refl]
    unfolding rbt_inter_swap_rec.simps[of _ _ t1] rbt_inter_rec.simps[of _ t1]
    by (auto simp add: Let_def split: rbt.splits prod.splits option.splits)
qed

lemma rbt_rbtreeify[simp]: "inv_12 (rbtreeify kvs)"
  by (simp add: inv_12_def rbtreeify_def inv1_rbtreeify_g inv2_rbtreeify_g)

lemma rbt_inter_rec: "inv_12 t1 ⟹ inv_12 t2 ⟹ inv_12 (rbt_inter_rec f t1 t2)"
proof(induction f t1 t2 rule: rbt_inter_rec.induct)
  case (1 t1 t2)
  thus ?case
    by (auto simp: rbt_inter_rec.simps[of t1 t2] inv_rbt_join inv_rbt_join2 rbt_split Let_def
        split!: RBT_Impl.rbt.splits RBT_Impl.color.splits prod.split if_splits
        option.splits dest!: rbt_Node)
qed

definition "filter_minus t1 t2 = filter (λ(k, _). rbt_lookup t2 k = None) (RBT_Impl.entries t1)"

fun rbt_minus_rec :: "('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt" where
  "rbt_minus_rec t1 t2 = (if small_rbt t2 then RBT_Impl.fold (λk _ t. rbt_delete k t) t2 t1
    else if small_rbt t1 then rbtreeify (filter_minus t1 t2)
    else case t2 of RBT_Impl.Empty ⇒ t1
      | RBT_Impl.Branch _ l2 a b r2 ⇒
        case rbt_split t1 a of (l1, _, r1) ⇒ rbt_join2 (rbt_minus_rec l1 l2) (rbt_minus_rec r1 r2))"

declare rbt_minus_rec.simps[simp del]

end

context linorder begin

lemma rbt_sorted_entries_right_unique:
  "⟦ (k, v) ∈ set (entries t); (k, v') ∈ set (entries t); 
     rbt_sorted t ⟧ ⟹ v = v'"
by(auto dest!: distinct_entries inj_onD[where x="(k, v)" and y="(k, v')"] simp add: distinct_map)

lemma rbt_sorted_fold_rbt_insertwk:
  "rbt_sorted t ⟹ rbt_sorted (List.fold (λ(k, v). rbt_insert_with_key f k v) xs t)"
by(induct xs rule: rev_induct)(auto simp add: rbt_insertwk_rbt_sorted)

lemma is_rbt_fold_rbt_insertwk:
  assumes "is_rbt t1"
  shows "is_rbt (fold (rbt_insert_with_key f) t2 t1)"
proof -
  define xs where "xs = entries t2"
  from assms show ?thesis unfolding fold_def xs_def[symmetric]
    by(induct xs rule: rev_induct)(auto simp add: rbt_insertwk_is_rbt)
qed

lemma rbt_delete: "inv_12 t ⟹ inv_12 (rbt_delete x t)"
  using rbt_del_inv1_inv2[of t x]
  by (auto simp: inv_12_def rbt_delete_def rbt_del_inv1_inv2)

lemma rbt_sorted_delete: "rbt_sorted t ⟹ rbt_sorted (rbt_delete x t)"
  by (auto simp: rbt_delete_def rbt_del_rbt_sorted)

lemma rbt_fold_rbt_delete:
  assumes "inv_12 t2"
  shows "inv_12 (RBT_Impl.fold (λk _ t. rbt_delete k t) t1 t2)"
proof -
  define xs where "xs = RBT_Impl.entries t1"
  from assms show ?thesis
    unfolding RBT_Impl.fold_def xs_def[symmetric]
    by (induct xs rule: rev_induct) (auto simp: rbt_delete)
qed

lemma rbt_minus_rec: "inv_12 t1 ⟹ inv_12 t2 ⟹ inv_12 (rbt_minus_rec t1 t2)"
proof(induction t1 t2 rule: rbt_minus_rec.induct)
  case (1 t1 t2)
  thus ?case
    by (auto simp: rbt_minus_rec.simps[of t1 t2] inv_rbt_join inv_rbt_join2 rbt_split
        rbt_fold_rbt_delete split!: RBT_Impl.rbt.splits RBT_Impl.color.splits prod.split if_splits
        dest: rbt_Node)
qed

end

context linorder begin

lemma rbt_sorted_rbt_baliL: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_sorted (rbt_baliL l a b r)"
  using rbt_greater_trans rbt_less_trans
  by (cases "(l,a,b,r)" rule: rbt_baliL.cases) fastforce+

lemma rbt_lookup_rbt_baliL: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_lookup (rbt_baliL l a b r) k =
  (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
  by (cases "(l,a,b,r)" rule: rbt_baliL.cases) (auto split!: if_splits)

lemma rbt_sorted_rbt_baliR: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_sorted (rbt_baliR l a b r)"
  using rbt_greater_trans rbt_less_trans
  by (cases "(l,a,b,r)" rule: rbt_baliR.cases) fastforce+

lemma rbt_lookup_rbt_baliR: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_lookup (rbt_baliR l a b r) k =
  (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
  by (cases "(l,a,b,r)" rule: rbt_baliR.cases) (auto split!: if_splits)

lemma rbt_sorted_rbt_joinL: "rbt_sorted (RBT_Impl.Branch c l a b r) ⟹ bheight l ≤ bheight r ⟹
  rbt_sorted (rbt_joinL l a b r)"
proof (induction l a b r arbitrary: c rule: rbt_joinL.induct)
  case (1 l a b r)
  thus ?case
    by (auto simp: rbt_set_rbt_baliL rbt_joinL.simps[of l a b r] set_rbt_joinL rbt_less_prop
        intro!: rbt_sorted_rbt_baliL split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma rbt_lookup_rbt_joinL: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_lookup (rbt_joinL l a b r) k =
  (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
proof (induction l a b r rule: rbt_joinL.induct)
  case (1 l a b r)
  have less_rbt_joinL:
    "rbt_sorted r1 ⟹ r1 |« x ⟹ a «| r1 ⟹ a < x ⟹ rbt_joinL l a b r1 |« x" for x r1
    using 1(5)
    by (auto simp: rbt_less_prop rbt_greater_prop set_rbt_joinL)
  show ?case
    using 1 less_rbt_joinL rbt_lookup_rbt_baliL[OF rbt_sorted_rbt_joinL[of _ l a b], where ?k=k]
    by (auto simp: rbt_joinL.simps[of l a b r] split!: if_splits rbt.splits color.splits)
qed

lemma rbt_sorted_rbt_joinR: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_sorted (rbt_joinR l a b r)"
proof (induction l a b r rule: rbt_joinR.induct)
  case (1 l a b r)
  thus ?case
    by (auto simp: rbt_set_rbt_baliR rbt_joinR.simps[of l a b r] set_rbt_joinR rbt_greater_prop
        intro!: rbt_sorted_rbt_baliR split!: RBT_Impl.rbt.splits RBT_Impl.color.splits)
qed

lemma rbt_lookup_rbt_joinR: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_lookup (rbt_joinR l a b r) k =
  (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
proof (induction l a b r rule: rbt_joinR.induct)
  case (1 l a b r)
  have less_rbt_joinR:
    "rbt_sorted l1 ⟹ x «| l1 ⟹ l1 |« a ⟹ x < a ⟹ x «| rbt_joinR l1 a b r" for x l1
    using 1(6)
    by (auto simp: rbt_less_prop rbt_greater_prop set_rbt_joinR)
  show ?case
    using 1 less_rbt_joinR rbt_lookup_rbt_baliR[OF _ rbt_sorted_rbt_joinR[of _ r a b], where ?k=k]
    by (auto simp: rbt_joinR.simps[of l a b r] split!: if_splits rbt.splits color.splits)
qed

lemma rbt_sorted_paint: "rbt_sorted (paint c t) = rbt_sorted t"
  by (cases t) auto

lemma rbt_sorted_rbt_join: "rbt_sorted (RBT_Impl.Branch c l a b r) ⟹
  rbt_sorted (rbt_join l a b r)"
  by (auto simp: rbt_sorted_paint rbt_sorted_rbt_joinL rbt_sorted_rbt_joinR rbt_join_def Let_def)

lemma rbt_lookup_rbt_join: "rbt_sorted l ⟹ rbt_sorted r ⟹ l |« a ⟹ a «| r ⟹
  rbt_lookup (rbt_join l a b r) k =
  (if k < a then rbt_lookup l k else if k = a then Some b else rbt_lookup r k)"
  by (auto simp: rbt_join_def Let_def rbt_lookup_rbt_joinL rbt_lookup_rbt_joinR)

lemma rbt_split_min_rbt_sorted: "rbt_split_min t = (a,b,t') ⟹ rbt_sorted t ⟹ t ≠ RBT_Impl.Empty ⟹
  rbt_sorted t' ∧ (∀x ∈ set (RBT_Impl.keys t'). a < x)"
  by (induction t arbitrary: t')
     (fastforce simp: rbt_split_min_set rbt_sorted_rbt_join set_rbt_join rbt_less_prop rbt_greater_prop
      split: if_splits prod.splits)+

lemma rbt_split_min_rbt_lookup: "rbt_split_min t = (a,b,t') ⟹ rbt_sorted t ⟹ t ≠ RBT_Impl.Empty ⟹
  rbt_lookup t k = (if k < a then None else if k = a then Some b else rbt_lookup t' k)"
  apply (induction t arbitrary: a b t')
   apply(simp_all split: if_splits prod.splits)
     apply(auto simp: rbt_less_prop rbt_split_min_set rbt_lookup_rbt_join rbt_split_min_rbt_sorted)
  done

lemma rbt_sorted_rbt_join2: "rbt_sorted l ⟹ rbt_sorted r ⟹
  ∀x ∈ set (RBT_Impl.keys l). ∀y ∈ set (RBT_Impl.keys r). x < y ⟹ rbt_sorted (rbt_join2 l r)"
  by (simp add: rbt_join2_def rbt_sorted_rbt_join rbt_split_min_set rbt_split_min_rbt_sorted set_rbt_join
      rbt_greater_prop rbt_less_prop split: prod.split)

lemma rbt_lookup_rbt_join2: "rbt_sorted l ⟹ rbt_sorted r ⟹
  ∀x ∈ set (RBT_Impl.keys l). ∀y ∈ set (RBT_Impl.keys r). x < y ⟹
  rbt_lookup (rbt_join2 l r) k = (case rbt_lookup l k of None ⇒ rbt_lookup r k | Some v ⇒ Some v)"
  using rbt_lookup_keys
  by (fastforce simp: rbt_join2_def rbt_greater_prop rbt_less_prop rbt_lookup_rbt_join
      rbt_split_min_rbt_lookup rbt_split_min_rbt_sorted rbt_split_min_set split: option.splits prod.splits)

lemma rbt_split_props: "rbt_split t x = (l,β,r) ⟹ rbt_sorted t ⟹
  set (RBT_Impl.keys l) = {a ∈ set (RBT_Impl.keys t). a < x} ∧
  set (RBT_Impl.keys r) = {a ∈ set (RBT_Impl.keys t). x < a} ∧
  rbt_sorted l ∧ rbt_sorted r"
  apply (induction t arbitrary: l r)
   apply(simp_all split!: prod.splits if_splits)
    apply(force simp: set_rbt_join rbt_greater_prop rbt_less_prop
      intro: rbt_sorted_rbt_join)+
  done

lemma rbt_split_lookup: "rbt_split t x = (l,β,r) ⟹ rbt_sorted t ⟹
  rbt_lookup t k = (if k < x then rbt_lookup l k else if k = x then β else rbt_lookup r k)"
proof (induction t arbitrary: x l β r)
  case (Branch c t1 a b t2)
  have "rbt_sorted r1" "r1 |« a" if "rbt_split t1 x = (l, β, r1)" for r1
    using rbt_split_props Branch(4) that
    by (fastforce simp: rbt_less_prop)+
  moreover have "rbt_sorted l1" "a «| l1" if "rbt_split t2 x = (l1, β, r)" for l1
    using rbt_split_props Branch(4) that
    by (fastforce simp: rbt_greater_prop)+
  ultimately show ?case
    using Branch rbt_lookup_rbt_join[of t1 _ a b k] rbt_lookup_rbt_join[of _ t2 a b k]
    by (auto split!: if_splits prod.splits)
qed simp

lemma rbt_sorted_fold_insertwk: "rbt_sorted t ⟹
  rbt_sorted (RBT_Impl.fold (rbt_insert_with_key f) t' t)"
  by (induct t' arbitrary: t)
     (simp_all add: rbt_insertwk_rbt_sorted)

lemma rbt_lookup_iff_keys:
  "rbt_sorted t ⟹ set (RBT_Impl.keys t) = {k. ∃v. rbt_lookup t k = Some v}"
  "rbt_sorted t ⟹ rbt_lookup t k = None ⟷ k ∉ set (RBT_Impl.keys t)"
  "rbt_sorted t ⟹ (∃v. rbt_lookup t k = Some v) ⟷ k ∈ set (RBT_Impl.keys t)"
  using entry_in_tree_keys rbt_lookup_keys[of t]
  by force+

lemma rbt_lookup_fold_rbt_insertwk:
  assumes t1: "rbt_sorted t1" and t2: "rbt_sorted t2"
  shows "rbt_lookup (fold (rbt_insert_with_key f) t1 t2) k =
  (case rbt_lookup t1 k of None ⇒ rbt_lookup t2 k
   | Some v ⇒ case rbt_lookup t2 k of None ⇒ Some v
               | Some w ⇒ Some (f k w v))"
proof -
  define xs where "xs = entries t1"
  hence dt1: "distinct (map fst xs)" using t1 by(simp add: distinct_entries)
  with t2 show ?thesis
    unfolding fold_def map_of_entries[OF t1, symmetric]
      xs_def[symmetric] distinct_map_of_rev[OF dt1, symmetric]
    apply(induct xs rule: rev_induct)
    apply(auto simp add: rbt_lookup_rbt_insertwk rbt_sorted_fold_rbt_insertwk split: option.splits)
    apply(auto simp add: distinct_map_of_rev intro: rev_image_eqI)
    done
qed

lemma rbt_lookup_union_rec: "rbt_sorted t1 ⟹ rbt_sorted t2 ⟹
  rbt_sorted (rbt_union_rec f t1 t2) ∧ rbt_lookup (rbt_union_rec f t1 t2) k =
  (case rbt_lookup t1 k of None ⇒ rbt_lookup t2 k
  | Some v ⇒ (case rbt_lookup t2 k of None ⇒ Some v
              | Some w ⇒ Some (f k v w)))"
proof(induction f t1 t2 arbitrary: k rule: rbt_union_rec.induct)
  case (1 f t1 t2)
  obtain f' t1' t2' where flip: "(f', t2', t1') =
    (if flip_rbt t2 t1 then (λk v v'. f k v' v, t1, t2) else (f, t2, t1))"
    by fastforce
  have rbt_sorted': "rbt_sorted t1'" "rbt_sorted t2'"
    using 1(3,4) flip
    by (auto split: if_splits)
  show ?case
  proof (cases t1')
    case Empty
    show ?thesis
      unfolding rbt_union_rec.simps[of _ t1] flip[symmetric]
      using flip rbt_sorted' rbt_split_props[of t2]
      by (auto simp: Empty rbt_lookup_fold_rbt_insertwk
          intro!: rbt_sorted_fold_insertwk split: if_splits option.splits)
  next
    case (Branch c l1 a b r1)
    {
      assume not_small: "¬small_rbt t2'"
      obtain l2 β r2 where rbt_split_t2': "rbt_split t2' a = (l2, β, r2)"
        by (cases "rbt_split t2' a") auto
      have rbt_sort: "rbt_sorted l1" "rbt_sorted r1"
        using 1(3,4) flip
        by (auto simp: Branch split: if_splits)
      note rbt_split_t2'_props = rbt_split_props[OF rbt_split_t2' rbt_sorted'(2)]
      have union_l1_l2: "rbt_sorted (rbt_union_rec f' l1 l2)" "rbt_lookup (rbt_union_rec f' l1 l2) k =
        (case rbt_lookup l1 k of None ⇒ rbt_lookup l2 k
        | Some v ⇒ (case rbt_lookup l2 k of None ⇒ Some v | Some w ⇒ Some (f' k v w)))" for k
        using 1(1)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
        by (auto simp: not_small)
      have union_r1_r2: "rbt_sorted (rbt_union_rec f' r1 r2)" "rbt_lookup (rbt_union_rec f' r1 r2) k =
        (case rbt_lookup r1 k of None ⇒ rbt_lookup r2 k
        | Some v ⇒ (case rbt_lookup r2 k of None ⇒ Some v | Some w ⇒ Some (f' k v w)))" for k
        using 1(2)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
        by (auto simp: not_small)
      have union_l1_l2_keys: "set (RBT_Impl.keys (rbt_union_rec f' l1 l2)) =
       set (RBT_Impl.keys l1) ∪ set (RBT_Impl.keys l2)"
        using rbt_sorted'(1) rbt_split_t2'_props
        by (auto simp: Branch rbt_lookup_iff_keys(1) union_l1_l2 split: option.splits)
      have union_r1_r2_keys: "set (RBT_Impl.keys (rbt_union_rec f' r1 r2)) =
        set (RBT_Impl.keys r1) ∪ set (RBT_Impl.keys r2)"
        using rbt_sorted'(1) rbt_split_t2'_props
        by (auto simp: Branch rbt_lookup_iff_keys(1) union_r1_r2 split: option.splits)
      have union_l1_l2_less: "rbt_union_rec f' l1 l2 |« a"
        using rbt_sorted'(1) rbt_split_t2'_props
        by (auto simp: Branch rbt_less_prop union_l1_l2_keys)
      have union_r1_r2_greater: "a «| rbt_union_rec f' r1 r2"
        using rbt_sorted'(1) rbt_split_t2'_props
        by (auto simp: Branch rbt_greater_prop union_r1_r2_keys)
      have "rbt_lookup (rbt_union_rec f t1 t2) k =
       (case rbt_lookup t1' k of None ⇒ rbt_lookup t2' k
       | Some v ⇒ (case rbt_lookup t2' k of None ⇒ Some v | Some w ⇒ Some (f' k v w)))"
        using rbt_sorted' union_l1_l2 union_r1_r2 rbt_split_t2'_props
          union_l1_l2_less union_r1_r2_greater not_small
        by (auto simp: rbt_union_rec.simps[of _ t1] flip[symmetric] Branch
            rbt_split_t2' rbt_lookup_rbt_join rbt_split_lookup[OF rbt_split_t2'] split: option.splits)
      moreover have "rbt_sorted (rbt_union_rec f t1 t2)"
        using rbt_sorted' rbt_split_t2'_props not_small
        by (auto simp: rbt_union_rec.simps[of _ t1] flip[symmetric] Branch rbt_split_t2'
            union_l1_l2 union_r1_r2 union_l1_l2_keys union_r1_r2_keys rbt_less_prop
            rbt_greater_prop intro!: rbt_sorted_rbt_join)
      ultimately have ?thesis
        using flip
        by (auto split: if_splits option.splits)
    }
    then show ?thesis
      unfolding rbt_union_rec.simps[of _ t1] flip[symmetric]
      using rbt_sorted' flip
      by (auto simp: rbt_sorted_fold_insertwk rbt_lookup_fold_rbt_insertwk split: option.splits)
  qed
qed

lemma rbtreeify_map_filter_inter:
  fixes f :: "'a ⇒ 'b ⇒ 'b ⇒ 'b"
  assumes "rbt_sorted t2"
  shows "rbt_sorted (rbtreeify (map_filter_inter f t1 t2))"
    "rbt_lookup (rbtreeify (map_filter_inter f t1 t2)) k =
    (case rbt_lookup t1 k of None ⇒ None
    | Some v ⇒ (case rbt_lookup t2 k of None ⇒ None | Some w ⇒ Some (f k v w)))"
proof -
  have map_of_map_filter: "map_of (List.map_filter (λ(k, v).
    case rbt_lookup t1 k of None ⇒ None | Some v' ⇒ Some (k, f k v' v)) xs) k =
    (case rbt_lookup t1 k of None ⇒ None
    | Some v ⇒ (case map_of xs k of None ⇒ None | Some w ⇒ Some (f k v w)))" for xs k
    by (induction xs) (auto simp: List.map_filter_def split: option.splits) (* slow *)
  have map_fst_map_filter: "map fst (List.map_filter (λ(k, v).
    case rbt_lookup t1 k of None ⇒ None | Some v' ⇒ Some (k, f k v' v)) xs) =
    filter (λk. rbt_lookup t1 k ≠ None) (map fst xs)" for xs
    by (induction xs) (auto simp: List.map_filter_def split: option.splits)
  have "sorted (map fst (map_filter_inter f t1 t2))"
    using sorted_filter[of id] rbt_sorted_entries[OF assms]
    by (auto simp: map_filter_inter_def map_fst_map_filter)
  moreover have "distinct (map fst (map_filter_inter f t1 t2))"
    using distinct_filter distinct_entries[OF assms]
    by (auto simp: map_filter_inter_def map_fst_map_filter)
  ultimately show
    "rbt_sorted (rbtreeify (map_filter_inter f t1 t2))"
    "rbt_lookup (rbtreeify (map_filter_inter f t1 t2)) k =
      (case rbt_lookup t1 k of None ⇒ None
      | Some v ⇒ (case rbt_lookup t2 k of None ⇒ None | Some w ⇒ Some (f k v w)))"
    using rbt_sorted_rbtreeify
    by (auto simp: rbt_lookup_rbtreeify map_filter_inter_def map_of_map_filter
        map_of_entries[OF assms] split: option.splits)
qed

lemma rbt_lookup_inter_rec: "rbt_sorted t1 ⟹ rbt_sorted t2 ⟹
  rbt_sorted (rbt_inter_rec f t1 t2) ∧ rbt_lookup (rbt_inter_rec f t1 t2) k =
  (case rbt_lookup t1 k of None ⇒ None
  | Some v ⇒ (case rbt_lookup t2 k of None ⇒ None | Some w ⇒ Some (f k v w)))"
proof(induction f t1 t2 arbitrary: k rule: rbt_inter_rec.induct)
  case (1 f t1 t2)
  obtain f' t1' t2' where flip: "(f', t2', t1') =
    (if flip_rbt t2 t1 then (λk v v'. f k v' v, t1, t2) else (f, t2, t1))"
    by fastforce
  have rbt_sorted': "rbt_sorted t1'" "rbt_sorted t2'"
    using 1(3,4) flip
    by (auto split: if_splits)
  show ?case
  proof (cases t1')
    case Empty
    show ?thesis
      unfolding rbt_inter_rec.simps[of _ t1] flip[symmetric]
      using flip rbt_sorted' rbt_split_props[of t2] rbtreeify_map_filter_inter[OF rbt_sorted'(2)]
      by (auto simp: Empty split: option.splits)
  next
    case (Branch c l1 a b r1)
    {
      assume not_small: "¬small_rbt t2'"
      obtain l2 β r2 where rbt_split_t2': "rbt_split t2' a = (l2, β, r2)"
        by (cases "rbt_split t2' a") auto
      note rbt_split_t2'_props = rbt_split_props[OF rbt_split_t2' rbt_sorted'(2)]
      have rbt_sort: "rbt_sorted l1" "rbt_sorted r1" "rbt_sorted l2" "rbt_sorted r2"
        using 1(3,4) flip
        by (auto simp: Branch rbt_split_t2'_props split: if_splits)
      have inter_l1_l2: "rbt_sorted (rbt_inter_rec f' l1 l2)" "rbt_lookup (rbt_inter_rec f' l1 l2) k =
        (case rbt_lookup l1 k of None ⇒ None
        | Some v ⇒ (case rbt_lookup l2 k of None ⇒ None | Some w ⇒ Some (f' k v w)))" for k
        using 1(1)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
        by (auto simp: not_small)
      have inter_r1_r2: "rbt_sorted (rbt_inter_rec f' r1 r2)" "rbt_lookup (rbt_inter_rec f' r1 r2) k =
        (case rbt_lookup r1 k of None ⇒ None
        | Some v ⇒ (case rbt_lookup r2 k of None ⇒ None | Some w ⇒ Some (f' k v w)))" for k
        using 1(2)[OF flip refl refl _ Branch rbt_split_t2'[symmetric]] rbt_sort rbt_split_t2'_props
        by (auto simp: not_small)
      have inter_l1_l2_keys: "set (RBT_Impl.keys (rbt_inter_rec f' l1 l2)) =
        set (RBT_Impl.keys l1) ∩ set (RBT_Impl.keys l2)"
        using inter_l1_l2(1)
        by (auto simp: rbt_lookup_iff_keys(1) inter_l1_l2(2) rbt_sort split: option.splits)
      have inter_r1_r2_keys: "set (RBT_Impl.keys (rbt_inter_rec f' r1 r2)) =
        set (RBT_Impl.keys r1) ∩ set (RBT_Impl.keys r2)"
        using inter_r1_r2(1)
        by (auto simp: rbt_lookup_iff_keys(1) inter_r1_r2(2) rbt_sort split: option.splits)
      have inter_l1_l2_less: "rbt_inter_rec f' l1 l2 |« a"
        using rbt_sorted'(1) rbt_split_t2'_props
        by (auto simp: Branch rbt_less_prop inter_l1_l2_keys)
      have inter_r1_r2_greater: "a «| rbt_inter_rec f' r1 r2"
        using rbt_sorted'(1) rbt_split_t2'_props
        by (auto simp: Branch rbt_greater_prop inter_r1_r2_keys)
      have rbt_lookup_join2: "rbt_lookup (rbt_join2 (rbt_inter_rec f' l1 l2) (rbt_inter_rec f' r1 r2)) k =
        (case rbt_lookup (rbt_inter_rec f' l1 l2) k of None ⇒ rbt_lookup (rbt_inter_rec f' r1 r2) k
        | Some v ⇒ Some v)" for k
        using rbt_lookup_rbt_join2[OF inter_l1_l2(1) inter_r1_r2(1)] rbt_sorted'(1)
        by (fastforce simp: Branch inter_l1_l2_keys inter_r1_r2_keys rbt_less_prop rbt_greater_prop)
      have rbt_lookup_l1_k: "rbt_lookup l1 k = Some v ⟹ k < a" for k v
        using rbt_sorted'(1) rbt_lookup_iff_keys(3)
        by (auto simp: Branch rbt_less_prop)
      have rbt_lookup_r1_k: "rbt_lookup r1 k = Some v ⟹ a < k" for k v
        using rbt_sorted'(1) rbt_lookup_iff_keys(3)
        by (auto simp: Branch rbt_greater_prop)
      have "rbt_lookup (rbt_inter_rec f t1 t2) k =
        (case rbt_lookup t1' k of None ⇒ None
        | Some v ⇒ (case rbt_lookup t2' k of None ⇒ None | Some w ⇒ Some (f' k v w)))"
        by (auto simp: Let_def rbt_inter_rec.simps[of _ t1] flip[symmetric] not_small Branch
            rbt_split_t2' rbt_lookup_join2 rbt_lookup_rbt_join inter_l1_l2_less inter_r1_r2_greater
            rbt_split_lookup[OF rbt_split_t2' rbt_sorted'(2)] inter_l1_l2 inter_r1_r2
            split!: if_splits option.splits dest: rbt_lookup_l1_k rbt_lookup_r1_k)
      moreover have "rbt_sorted (rbt_inter_rec f t1 t2)"
        using rbt_sorted' inter_l1_l2 inter_r1_r2 rbt_split_t2'_props not_small
        by (auto simp: Let_def rbt_inter_rec.simps[of _ t1] flip[symmetric] Branch rbt_split_t2'
            rbt_less_prop rbt_greater_prop inter_l1_l2_less inter_r1_r2_greater
            inter_l1_l2_keys inter_r1_r2_keys intro!: rbt_sorted_rbt_join rbt_sorted_rbt_join2
            split: if_splits option.splits dest!: bspec)
      ultimately have ?thesis
        using flip
        by (auto split: if_splits split: option.splits)
    }
    then show ?thesis
      unfolding rbt_inter_rec.simps[of _ t1] flip[symmetric]
      using rbt_sorted' flip rbtreeify_map_filter_inter[OF rbt_sorted'(2)]
      by (auto split: option.splits)
  qed
qed

lemma rbt_lookup_delete:
  assumes "inv_12 t" "rbt_sorted t"
  shows "rbt_lookup (rbt_delete x t) k = (if x = k then None else rbt_lookup t k)"
proof -
  note rbt_sorted_del = rbt_del_rbt_sorted[OF assms(2), of x]
  show ?thesis
    using assms rbt_sorted_del rbt_del_in_tree rbt_lookup_from_in_tree[OF assms(2) rbt_sorted_del]
    by (fastforce simp: inv_12_def rbt_delete_def rbt_lookup_iff_keys(2) keys_entries)
qed

lemma fold_rbt_delete:
  assumes "inv_12 t1" "rbt_sorted t1" "rbt_sorted t2"
  shows "inv_12 (RBT_Impl.fold (λk _ t. rbt_delete k t) t2 t1) ∧
    rbt_sorted (RBT_Impl.fold (λk _ t. rbt_delete k t) t2 t1) ∧
    rbt_lookup (RBT_Impl.fold (λk _ t. rbt_delete k t) t2 t1) k =
    (case rbt_lookup t1 k of None ⇒ None
    | Some v ⇒ (case rbt_lookup t2 k of None ⇒ Some v | _ ⇒ None))"
proof -
  define xs where "xs = RBT_Impl.entries t2"
  show "inv_12 (RBT_Impl.fold (λk _ t. rbt_delete k t) t2 t1) ∧
    rbt_sorted (RBT_Impl.fold (λk _ t. rbt_delete k t) t2 t1) ∧
    rbt_lookup (RBT_Impl.fold (λk _ t. rbt_delete k t) t2 t1) k =
    (case rbt_lookup t1 k of None ⇒ None
    | Some v ⇒ (case rbt_lookup t2 k of None ⇒ Some v | _ ⇒ None))"
    using assms(1,2)
    unfolding map_of_entries[OF assms(3), symmetric] RBT_Impl.fold_def xs_def[symmetric]
    by (induction xs arbitrary: t1 rule: rev_induct)
       (auto simp: rbt_delete rbt_sorted_delete rbt_lookup_delete split!: option.splits)
qed

lemma rbtreeify_filter_minus:
  assumes "rbt_sorted t1"
  shows "rbt_sorted (rbtreeify (filter_minus t1 t2)) ∧
    rbt_lookup (rbtreeify (filter_minus t1 t2)) k =
    (case rbt_lookup t1 k of None ⇒ None
    | Some v ⇒ (case rbt_lookup t2 k of None ⇒ Some v | _ ⇒ None))"
proof -
  have map_of_filter: "map_of (filter (λ(k, _). rbt_lookup t2 k = None) xs) k =
    (case map_of xs k of None ⇒ None
    | Some v ⇒ (case rbt_lookup t2 k of None ⇒ Some v | Some x ⇒ Map.empty x))"
      for xs :: "('a × 'b) list"
    by (induction xs) (auto split: option.splits)
  have map_fst_filter_minus: "map fst (filter_minus t1 t2) =
    filter (λk. rbt_lookup t2 k = None) (map fst (RBT_Impl.entries t1))"
    by (auto simp: filter_minus_def filter_map comp_def case_prod_unfold)
  have "sorted (map fst (filter_minus t1 t2))" "distinct (map fst (filter_minus t1 t2))"
    using distinct_filter distinct_entries[OF assms]
      sorted_filter[of id] rbt_sorted_entries[OF assms]
    by (auto simp: map_fst_filter_minus intro!: rbt_sorted_rbtreeify)
  then show ?thesis
    by (auto simp: rbt_lookup_rbtreeify filter_minus_def map_of_filter map_of_entries[OF assms]
        intro!: rbt_sorted_rbtreeify)
qed

lemma rbt_lookup_minus_rec: "inv_12 t1 ⟹ rbt_sorted t1 ⟹ rbt_sorted t2 ⟹
  rbt_sorted (rbt_minus_rec t1 t2) ∧ rbt_lookup (rbt_minus_rec t1 t2) k =
  (case rbt_lookup t1 k of None ⇒ None
  | Some v ⇒ (case rbt_lookup t2 k of None ⇒ Some v | _ ⇒ None))"
proof(induction t1 t2 arbitrary: k rule: rbt_minus_rec.induct)
  case (1 t1 t2)
  show ?case
  proof (cases t2)
    case Empty
    show ?thesis
      using rbtreeify_filter_minus[OF 1(4)] 1(4)
      by (auto simp: rbt_minus_rec.simps[of t1] Empty split: option.splits)
  next
    case (Branch c l2 a b r2)
    {
      assume not_small: "¬small_rbt t2" "¬small_rbt t1"
      obtain l1 β r1 where rbt_split_t1: "rbt_split t1 a = (l1, β, r1)"
        by (cases "rbt_split t1 a") auto
      note rbt_split_t1_props = rbt_split_props[OF rbt_split_t1 1(4)]
      have minus_l1_l2: "rbt_sorted (rbt_minus_rec l1 l2)"
        "rbt_lookup (rbt_minus_rec l1 l2) k =
        (case rbt_lookup l1 k of None ⇒ None
        | Some v ⇒ (case rbt_lookup l2 k of None ⇒ Some v | Some x ⇒ None))" for k
        using 1(1)[OF not_small Branch rbt_split_t1[symmetric] refl] 1(5) rbt_split_t1_props
          rbt_split[OF rbt_split_t1 1(3)]
        by (auto simp: Branch)
      have minus_r1_r2: "rbt_sorted (rbt_minus_rec r1 r2)"
        "rbt_lookup (rbt_minus_rec r1 r2) k =
        (case rbt_lookup r1 k of None ⇒ None
        | Some v ⇒ (case rbt_lookup r2 k of None ⇒ Some v | Some x ⇒ None))" for k
        using 1(2)[OF not_small Branch rbt_split_t1[symmetric] refl] 1(5) rbt_split_t1_props
          rbt_split[OF rbt_split_t1 1(3)]
        by (auto simp: Branch)
      have minus_l1_l2_keys: "set (RBT_Impl.keys (rbt_minus_rec l1 l2)) =
        set (RBT_Impl.keys l1) - set (RBT_Impl.keys l2)"
        using minus_l1_l2(1) 1(5) rbt_lookup_iff_keys(3) rbt_split_t1_props
        by (auto simp: Branch rbt_lookup_iff_keys(1) minus_l1_l2(2) split: option.splits)
      have minus_r1_r2_keys: "set (RBT_Impl.keys (rbt_minus_rec r1 r2)) =
        set (RBT_Impl.keys r1) - set (RBT_Impl.keys r2)"
        using minus_r1_r2(1) 1(5) rbt_lookup_iff_keys(3) rbt_split_t1_props
        by (auto simp: Branch rbt_lookup_iff_keys(1) minus_r1_r2(2) split: option.splits)
      have rbt_lookup_join2: "rbt_lookup (rbt_join2 (rbt_minus_rec l1 l2) (rbt_minus_rec r1 r2)) k =
        (case rbt_lookup (rbt_minus_rec l1 l2) k of None ⇒ rbt_lookup (rbt_minus_rec r1 r2) k
        | Some v ⇒ Some v)" for k
        using rbt_lookup_rbt_join2[OF minus_l1_l2(1) minus_r1_r2(1)] rbt_split_t1_props
        by (fastforce simp: minus_l1_l2_keys minus_r1_r2_keys)
      have lookup_l1_r1_a: "rbt_lookup l1 a = None" "rbt_lookup r1 a = None"
        using rbt_split_t1_props
        by (auto simp: rbt_lookup_iff_keys(2))
      have "rbt_lookup (rbt_minus_rec t1 t2) k =
        (case rbt_lookup t1 k of None ⇒ None
        | Some v ⇒ (case rbt_lookup t2 k of None ⇒ Some v | _ ⇒ None))"
        using not_small rbt_lookup_iff_keys(2)[of l1] rbt_lookup_iff_keys(3)[of l1]
          rbt_lookup_iff_keys(3)[of r1] rbt_split_t1_props
        using [[simp_depth_limit = 2]]
        by (auto simp: rbt_minus_rec.simps[of t1] Branch rbt_split_t1 rbt_lookup_join2
            minus_l1_l2(2) minus_r1_r2(2) rbt_split_lookup[OF rbt_split_t1 1(4)] lookup_l1_r1_a
            split: option.splits)
      moreover have "rbt_sorted (rbt_minus_rec t1 t2)"
        using not_small minus_l1_l2(1) minus_r1_r2(1) rbt_split_t1_props rbt_sorted_rbt_join2
        by (fastforce simp: rbt_minus_rec.simps[of t1] Branch rbt_split_t1 minus_l1_l2_keys minus_r1_r2_keys)
      ultimately have ?thesis
        by (auto split: if_splits split: option.splits)
    }
    then show ?thesis
      using fold_rbt_delete[OF 1(3,4,5)] rbtreeify_filter_minus[OF 1(4)]
      by (auto simp: rbt_minus_rec.simps[of t1])
  qed
qed

end

context ord begin

definition rbt_union_with_key :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt"
where
  "rbt_union_with_key f t1 t2 = paint B (rbt_union_swap_rec f False t1 t2)"

definition rbt_union_with where
  "rbt_union_with f = rbt_union_with_key (λ_. f)"

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

definition rbt_inter_with_key :: "('a ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt ⇒ ('a, 'b) rbt"
where
  "rbt_inter_with_key f t1 t2 = paint B (rbt_inter_swap_rec f False t1 t2)"

definition rbt_inter_with where
  "rbt_inter_with f = rbt_inter_with_key (λ_. f)"

definition rbt_inter where
  "rbt_inter = rbt_inter_with_key (λ_ _ rv. rv)"

definition rbt_minus where
  "rbt_minus t1 t2 = paint B (rbt_minus_rec t1 t2)"

end

context linorder begin

lemma is_rbt_rbt_unionwk [simp]:
  "⟦ is_rbt t1; is_rbt t2 ⟧ ⟹ is_rbt (rbt_union_with_key f t1 t2)"
  using rbt_union_rec rbt_lookup_union_rec
  by (fastforce simp: rbt_union_with_key_def rbt_union_swap_rec is_rbt_def inv_12_def)

lemma rbt_lookup_rbt_unionwk:
  "⟦ rbt_sorted t1; rbt_sorted t2 ⟧ 
  ⟹ rbt_lookup (rbt_union_with_key f t1 t2) k = 
  (case rbt_lookup t1 k of None ⇒ rbt_lookup t2 k 
   | Some v ⇒ case rbt_lookup t2 k of None ⇒ Some v 
              | Some w ⇒ Some (f k v w))"
  using rbt_lookup_union_rec
  by (auto simp: rbt_union_with_key_def rbt_union_swap_rec)

lemma rbt_unionw_is_rbt: "⟦ is_rbt lt; is_rbt rt ⟧ ⟹ is_rbt (rbt_union_with f lt rt)"
by(simp add: rbt_union_with_def)

lemma rbt_union_is_rbt: "⟦ is_rbt lt; is_rbt rt ⟧ ⟹ is_rbt (rbt_union lt rt)"
by(simp add: rbt_union_def)

lemma rbt_lookup_rbt_union:
  "⟦ rbt_sorted s; rbt_sorted t ⟧ ⟹
  rbt_lookup (rbt_union s t) = rbt_lookup s ++ rbt_lookup t"
by(rule ext)(simp add: rbt_lookup_rbt_unionwk rbt_union_def map_add_def split: option.split)

lemma rbt_interwk_is_rbt [simp]:
  "⟦ is_rbt t1; is_rbt t2 ⟧ ⟹ is_rbt (rbt_inter_with_key f t1 t2)"
  using rbt_inter_rec rbt_lookup_inter_rec
  by (fastforce simp: rbt_inter_with_key_def rbt_inter_swap_rec is_rbt_def inv_12_def rbt_sorted_paint)

lemma rbt_interw_is_rbt:
  "⟦ is_rbt t1; is_rbt t2 ⟧ ⟹ is_rbt (rbt_inter_with f t1 t2)"
by(simp add: rbt_inter_with_def)

lemma rbt_inter_is_rbt:
  "⟦ is_rbt t1; is_rbt t2 ⟧ ⟹ is_rbt (rbt_inter t1 t2)"
by(simp add: rbt_inter_def)

lemma rbt_lookup_rbt_interwk:
  "⟦ rbt_sorted t1; rbt_sorted t2 ⟧
  ⟹ rbt_lookup (rbt_inter_with_key f t1 t2) k =
  (case rbt_lookup t1 k of None ⇒ None 
   | Some v ⇒ case rbt_lookup t2 k of None ⇒ None
               | Some w ⇒ Some (f k v w))"
  using rbt_lookup_inter_rec
  by (auto simp: rbt_inter_with_key_def rbt_inter_swap_rec)

lemma rbt_lookup_rbt_inter:
  "⟦ rbt_sorted t1; rbt_sorted t2 ⟧
  ⟹ rbt_lookup (rbt_inter t1 t2) = rbt_lookup t2 |` dom (rbt_lookup t1)"
by(auto simp add: rbt_inter_def rbt_lookup_rbt_interwk restrict_map_def split: option.split)

lemma rbt_minus_is_rbt:
  "⟦ is_rbt t1; is_rbt t2 ⟧ ⟹ is_rbt (rbt_minus t1 t2)"
  using rbt_minus_rec[of t1 t2] rbt_lookup_minus_rec[of t1 t2]
  by (auto simp: rbt_minus_def is_rbt_def inv_12_def)

lemma rbt_lookup_rbt_minus:
  "⟦ is_rbt t1; is_rbt t2 ⟧
  ⟹ rbt_lookup (rbt_minus t1 t2) = rbt_lookup t1 |` (- dom (rbt_lookup t2))"
  by (rule ext)
     (auto simp: rbt_minus_def is_rbt_def inv_12_def restrict_map_def rbt_lookup_minus_rec
      split: option.splits)

end


subsection ‹Code generator setup›

lemmas [code] =
  ord.rbt_less_prop
  ord.rbt_greater_prop
  ord.rbt_sorted.simps
  ord.rbt_lookup.simps
  ord.is_rbt_def
  ord.rbt_ins.simps
  ord.rbt_insert_with_key_def
  ord.rbt_insertw_def
  ord.rbt_insert_def
  ord.rbt_del_from_left.simps
  ord.rbt_del_from_right.simps
  ord.rbt_del.simps
  ord.rbt_delete_def
  ord.rbt_split.simps
  ord.rbt_union_swap_rec.simps
  ord.map_filter_inter_def
  ord.rbt_inter_swap_rec.simps
  ord.filter_minus_def
  ord.rbt_minus_rec.simps
  ord.rbt_union_with_key_def
  ord.rbt_union_with_def
  ord.rbt_union_def
  ord.rbt_inter_with_key_def
  ord.rbt_inter_with_def
  ord.rbt_inter_def
  ord.rbt_minus_def
  ord.rbt_map_entry.simps
  ord.rbt_bulkload_def

text ‹More efficient implementations for \<^term>‹entries› and \<^term>‹keys››

definition gen_entries :: 
  "(('a × 'b) × ('a, 'b) rbt) list ⇒ ('a, 'b) rbt ⇒ ('a × 'b) list"
where
  "gen_entries kvts t = entries t @ concat (map (λ(kv, t). kv # entries t) kvts)"

lemma gen_entries_simps [simp, code]:
  "gen_entries [] Empty = []"
  "gen_entries ((kv, t) # kvts) Empty = kv # gen_entries kvts t"
  "gen_entries kvts (Branch c l k v r) = gen_entries (((k, v), r) # kvts) l"
by(simp_all add: gen_entries_def)

lemma entries_code [code]:
  "entries = gen_entries []"
by(simp add: gen_entries_def fun_eq_iff)

definition gen_keys :: "('a × ('a, 'b) rbt) list ⇒ ('a, 'b) rbt ⇒ 'a list"
where "gen_keys kts t = RBT_Impl.keys t @ concat (List.map (λ(k, t). k # keys t) kts)"

lemma gen_keys_simps [simp, code]:
  "gen_keys [] Empty = []"
  "gen_keys ((k, t) # kts) Empty = k # gen_keys kts t"
  "gen_keys kts (Branch c l k v r) = gen_keys ((k, r) # kts) l"
by(simp_all add: gen_keys_def)

lemma keys_code [code]:
  "keys = gen_keys []"
by(simp add: gen_keys_def fun_eq_iff)

text ‹Restore original type constraints for constants›
setup ‹
  fold Sign.add_const_constraint
    [(\<^const_name>‹rbt_less›, SOME \<^typ>‹('a :: order) ⇒ ('a, 'b) rbt ⇒ bool›),
     (\<^const_name>‹rbt_greater›, SOME \<^typ>‹('a :: order) ⇒ ('a, 'b) rbt ⇒ bool›),
     (\<^const_name>‹rbt_sorted›, SOME \<^typ>‹('a :: linorder, 'b) rbt ⇒ bool›),
     (\<^const_name>‹rbt_lookup›, SOME \<^typ>‹('a :: linorder, 'b) rbt ⇒ 'a ⇀ 'b›),
     (\<^const_name>‹is_rbt›, SOME \<^typ>‹('a :: linorder, 'b) rbt ⇒ bool›),
     (\<^const_name>‹rbt_ins›, SOME \<^typ>‹('a::linorder ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_insert_with_key›, SOME \<^typ>‹('a::linorder ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_insert_with›, SOME \<^typ>‹('b ⇒ 'b ⇒ 'b) ⇒ ('a :: linorder) ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_insert›, SOME \<^typ>‹('a :: linorder) ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_del_from_left›, SOME \<^typ>‹('a::linorder) ⇒ ('a,'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_del_from_right›, SOME \<^typ>‹('a::linorder) ⇒ ('a,'b) rbt ⇒ 'a ⇒ 'b ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_del›, SOME \<^typ>‹('a::linorder) ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_delete›, SOME \<^typ>‹('a::linorder) ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_union_with_key›, SOME \<^typ>‹('a::linorder ⇒ 'b ⇒ 'b ⇒ 'b) ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_union_with›, SOME \<^typ>‹('b ⇒ 'b ⇒ 'b) ⇒ ('a::linorder,'b) rbt ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_union›, SOME \<^typ>‹('a::linorder,'b) rbt ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_map_entry›, SOME \<^typ>‹'a::linorder ⇒ ('b ⇒ 'b) ⇒ ('a,'b) rbt ⇒ ('a,'b) rbt›),
     (\<^const_name>‹rbt_bulkload›, SOME \<^typ>‹('a × 'b) list ⇒ ('a::linorder,'b) rbt›)]
›

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

end