Theory RBT_Set

(* Author: Tobias Nipkow *)

section ‹Red-Black Tree Implementation of Sets›

theory RBT_Set
imports
  Complex_Main
  RBT
  Cmp
  Isin2
begin

definition empty :: "'a rbt" where
"empty = Leaf"

fun ins :: "'a::linorder ⇒ 'a rbt ⇒ 'a rbt" where
"ins x Leaf = R Leaf x Leaf" |
"ins x (B l a r) =
  (case cmp x a of
     LT ⇒ baliL (ins x l) a r |
     GT ⇒ baliR l a (ins x r) |
     EQ ⇒ B l a r)" |
"ins x (R l a r) =
  (case cmp x a of
    LT ⇒ R (ins x l) a r |
    GT ⇒ R l a (ins x r) |
    EQ ⇒ R l a r)"

definition insert :: "'a::linorder ⇒ 'a rbt ⇒ 'a rbt" where
"insert x t = paint Black (ins x t)"

fun color :: "'a rbt ⇒ color" where
"color Leaf = Black" |
"color (Node _ (_, c) _) = c"

fun del :: "'a::linorder ⇒ 'a rbt ⇒ 'a rbt" where
"del x Leaf = Leaf" |
"del x (Node l (a, _) r) =
  (case cmp x a of
     LT ⇒ if l ≠ Leaf ∧ color l = Black
           then baldL (del x l) a r else R (del x l) a r |
     GT ⇒ if r ≠ Leaf∧ color r = Black
           then baldR l a (del x r) else R l a (del x r) |
     EQ ⇒ join l r)"

definition delete :: "'a::linorder ⇒ 'a rbt ⇒ 'a rbt" where
"delete x t = paint Black (del x t)"


subsection "Functional Correctness Proofs"

lemma inorder_paint: "inorder(paint c t) = inorder t"
by(cases t) (auto)

lemma inorder_baliL:
  "inorder(baliL l a r) = inorder l @ a # inorder r"
by(cases "(l,a,r)" rule: baliL.cases) (auto)

lemma inorder_baliR:
  "inorder(baliR l a r) = inorder l @ a # inorder r"
by(cases "(l,a,r)" rule: baliR.cases) (auto)

lemma inorder_ins:
  "sorted(inorder t) ⟹ inorder(ins x t) = ins_list x (inorder t)"
by(induction x t rule: ins.induct)
  (auto simp: ins_list_simps inorder_baliL inorder_baliR)

lemma inorder_insert:
  "sorted(inorder t) ⟹ inorder(insert x t) = ins_list x (inorder t)"
by (simp add: insert_def inorder_ins inorder_paint)

lemma inorder_baldL:
  "inorder(baldL l a r) = inorder l @ a # inorder r"
by(cases "(l,a,r)" rule: baldL.cases)
  (auto simp:  inorder_baliL inorder_baliR inorder_paint)

lemma inorder_baldR:
  "inorder(baldR l a r) = inorder l @ a # inorder r"
by(cases "(l,a,r)" rule: baldR.cases)
  (auto simp:  inorder_baliL inorder_baliR inorder_paint)

lemma inorder_join:
  "inorder(join l r) = inorder l @ inorder r"
by(induction l r rule: join.induct)
  (auto simp: inorder_baldL inorder_baldR split: tree.split color.split)

lemma inorder_del:
 "sorted(inorder t) ⟹  inorder(del x t) = del_list x (inorder t)"
by(induction x t rule: del.induct)
  (auto simp: del_list_simps inorder_join inorder_baldL inorder_baldR)

lemma inorder_delete:
  "sorted(inorder t) ⟹ inorder(delete x t) = del_list x (inorder t)"
by (auto simp: delete_def inorder_del inorder_paint)


subsection ‹Structural invariants›

lemma neq_Black[simp]: "(c ≠ Black) = (c = Red)"
by (cases c) auto

text‹The proofs are due to Markus Reiter and Alexander Krauss.›

fun bheight :: "'a rbt ⇒ nat" where
"bheight Leaf = 0" |
"bheight (Node l (x, c) r) = (if c = Black then bheight l + 1 else bheight l)"

fun invc :: "'a rbt ⇒ bool" where
"invc Leaf = True" |
"invc (Node l (a,c) r) =
  ((c = Red ⟶ color l = Black ∧ color r = Black) ∧ invc l ∧ invc r)"

text ‹Weaker version:›
abbreviation invc2 :: "'a rbt ⇒ bool" where
"invc2 t ≡ invc(paint Black t)"

fun invh :: "'a rbt ⇒ bool" where
"invh Leaf = True" |
"invh (Node l (x, c) r) = (bheight l = bheight r ∧ invh l ∧ invh r)"

lemma invc2I: "invc t ⟹ invc2 t"
by (cases t rule: tree2_cases) simp+

definition rbt :: "'a rbt ⇒ bool" where
"rbt t = (invc t ∧ invh t ∧ color t = Black)"

lemma color_paint_Black: "color (paint Black t) = Black"
by (cases t) auto

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

lemma invh_paint: "invh t ⟹ invh (paint c t)"
by (cases t) auto

lemma invc_baliL:
  "⟦invc2 l; invc r⟧ ⟹ invc (baliL l a r)" 
by (induct l a r rule: baliL.induct) auto

lemma invc_baliR:
  "⟦invc l; invc2 r⟧ ⟹ invc (baliR l a r)" 
by (induct l a r rule: baliR.induct) auto

lemma bheight_baliL:
  "bheight l = bheight r ⟹ bheight (baliL l a r) = Suc (bheight l)"
by (induct l a r rule: baliL.induct) auto

lemma bheight_baliR:
  "bheight l = bheight r ⟹ bheight (baliR l a r) = Suc (bheight l)"
by (induct l a r rule: baliR.induct) auto

lemma invh_baliL: 
  "⟦ invh l; invh r; bheight l = bheight r ⟧ ⟹ invh (baliL l a r)"
by (induct l a r rule: baliL.induct) auto

lemma invh_baliR: 
  "⟦ invh l; invh r; bheight l = bheight r ⟧ ⟹ invh (baliR l a r)"
by (induct l a r rule: baliR.induct) auto

text ‹All in one:›

lemma inv_baliR: "⟦ invh l; invh r; invc l; invc2 r; bheight l = bheight r ⟧
 ⟹ invc (baliR l a r) ∧ invh (baliR l a r) ∧ bheight (baliR l a r) = Suc (bheight l)"
by (induct l a r rule: baliR.induct) auto

lemma inv_baliL: "⟦ invh l; invh r; invc2 l; invc r; bheight l = bheight r ⟧
 ⟹ invc (baliL l a r) ∧ invh (baliL l a r) ∧ bheight (baliL l a r) = Suc (bheight l)"
by (induct l a r rule: baliL.induct) auto

subsubsection ‹Insertion›

lemma invc_ins: "invc t ⟶ invc2 (ins x t) ∧ (color t = Black ⟶ invc (ins x t))"
by (induct x t rule: ins.induct) (auto simp: invc_baliL invc_baliR invc2I)

lemma invh_ins: "invh t ⟹ invh (ins x t) ∧ bheight (ins x t) = bheight t"
by(induct x t rule: ins.induct)
  (auto simp: invh_baliL invh_baliR bheight_baliL bheight_baliR)

theorem rbt_insert: "rbt t ⟹ rbt (insert x t)"
by (simp add: invc_ins invh_ins color_paint_Black invh_paint rbt_def insert_def)

text ‹All in one:›

lemma inv_ins: "⟦ invc t; invh t ⟧ ⟹
  invc2 (ins x t) ∧ (color t = Black ⟶ invc (ins x t)) ∧
  invh(ins x t) ∧ bheight (ins x t) = bheight t"
by (induct x t rule: ins.induct) (auto simp: inv_baliL inv_baliR invc2I)

theorem rbt_insert2: "rbt t ⟹ rbt (insert x t)"
by (simp add: inv_ins color_paint_Black invh_paint rbt_def insert_def)


subsubsection ‹Deletion›

lemma bheight_paint_Red:
  "color t = Black ⟹ bheight (paint Red t) = bheight t - 1"
by (cases t) auto

lemma invh_baldL_invc:
  "⟦ invh l;  invh r;  bheight l + 1 = bheight r;  invc r ⟧
   ⟹ invh (baldL l a r) ∧ bheight (baldL l a r) = bheight r"
by (induct l a r rule: baldL.induct)
   (auto simp: invh_baliR invh_paint bheight_baliR bheight_paint_Red)

lemma invh_baldL_Black: 
  "⟦ invh l;  invh r;  bheight l + 1 = bheight r;  color r = Black ⟧
   ⟹ invh (baldL l a r) ∧ bheight (baldL l a r) = bheight r"
by (induct l a r rule: baldL.induct) (auto simp add: invh_baliR bheight_baliR) 

lemma invc_baldL: "⟦invc2 l; invc r; color r = Black⟧ ⟹ invc (baldL l a r)"
by (induct l a r rule: baldL.induct) (simp_all add: invc_baliR)

lemma invc2_baldL: "⟦ invc2 l; invc r ⟧ ⟹ invc2 (baldL l a r)"
by (induct l a r rule: baldL.induct) (auto simp: invc_baliR paint2 invc2I)

lemma invh_baldR_invc:
  "⟦ invh l;  invh r;  bheight l = bheight r + 1;  invc l ⟧
  ⟹ invh (baldR l a r) ∧ bheight (baldR l a r) = bheight l"
by(induct l a r rule: baldR.induct)
  (auto simp: invh_baliL bheight_baliL invh_paint bheight_paint_Red)

lemma invc_baldR: "⟦invc l; invc2 r; color l = Black⟧ ⟹ invc (baldR l a r)"
by (induct l a r rule: baldR.induct) (simp_all add: invc_baliL)

lemma invc2_baldR: "⟦ invc l; invc2 r ⟧ ⟹invc2 (baldR l a r)"
by (induct l a r rule: baldR.induct) (auto simp: invc_baliL paint2 invc2I)

lemma invh_join:
  "⟦ invh l; invh r; bheight l = bheight r ⟧
  ⟹ invh (join l r) ∧ bheight (join l r) = bheight l"
by (induct l r rule: join.induct) 
   (auto simp: invh_baldL_Black split: tree.splits color.splits)

lemma invc_join: 
  "⟦ invc l; invc r ⟧ ⟹
  (color l = Black ∧ color r = Black ⟶ invc (join l r)) ∧ invc2 (join l r)"
by (induct l r rule: join.induct)
   (auto simp: invc_baldL invc2I split: tree.splits color.splits)

text ‹All in one:›

lemma inv_baldL:
  "⟦ invh l;  invh r;  bheight l + 1 = bheight r; invc2 l; invc r ⟧
   ⟹ invh (baldL l a r) ∧ bheight (baldL l a r) = bheight r
  ∧ invc2 (baldL l a r) ∧ (color r = Black ⟶ invc (baldL l a r))"
by (induct l a r rule: baldL.induct)
   (auto simp: inv_baliR invh_paint bheight_baliR bheight_paint_Red paint2 invc2I)

lemma inv_baldR:
  "⟦ invh l;  invh r;  bheight l = bheight r + 1; invc l; invc2 r ⟧
   ⟹ invh (baldR l a r) ∧ bheight (baldR l a r) = bheight l
  ∧ invc2 (baldR l a r) ∧ (color l = Black ⟶ invc (baldR l a r))"
by (induct l a r rule: baldR.induct)
   (auto simp: inv_baliL invh_paint bheight_baliL bheight_paint_Red paint2 invc2I)

lemma inv_join:
  "⟦ invh l; invh r; bheight l = bheight r; invc l; invc r ⟧
  ⟹ invh (join l r) ∧ bheight (join l r) = bheight l
  ∧ invc2 (join l r) ∧ (color l = Black ∧ color r = Black ⟶ invc (join l r))"
by (induct l r rule: join.induct) 
   (auto simp: invh_baldL_Black inv_baldL invc2I split: tree.splits color.splits)

lemma neq_LeafD: "t ≠ Leaf ⟹ ∃l x c r. t = Node l (x,c) r"
by(cases t rule: tree2_cases) auto

lemma inv_del: "⟦ invh t; invc t ⟧ ⟹
   invh (del x t) ∧
   (color t = Red ⟶ bheight (del x t) = bheight t ∧ invc (del x t)) ∧
   (color t = Black ⟶ bheight (del x t) = bheight t - 1 ∧ invc2 (del x t))"
by(induct x t rule: del.induct)
  (auto simp: inv_baldL inv_baldR inv_join dest!: neq_LeafD)

theorem rbt_delete: "rbt t ⟹ rbt (delete x t)"
by (metis delete_def rbt_def color_paint_Black inv_del invh_paint)

text ‹Overall correctness:›

interpretation S: Set_by_Ordered
where empty = empty and isin = isin and insert = insert and delete = delete
and inorder = inorder and inv = rbt
proof (standard, goal_cases)
  case 1 show ?case by (simp add: empty_def)
next
  case 2 thus ?case by(simp add: isin_set_inorder)
next
  case 3 thus ?case by(simp add: inorder_insert)
next
  case 4 thus ?case by(simp add: inorder_delete)
next
  case 5 thus ?case by (simp add: rbt_def empty_def) 
next
  case 6 thus ?case by (simp add: rbt_insert) 
next
  case 7 thus ?case by (simp add: rbt_delete) 
qed


subsection ‹Height-Size Relation›

lemma rbt_height_bheight_if: "invc t ⟹ invh t ⟹
  height t ≤ 2 * bheight t + (if color t = Black then 0 else 1)"
by(induction t) (auto split: if_split_asm)

lemma rbt_height_bheight: "rbt t ⟹ height t / 2 ≤ bheight t "
by(auto simp: rbt_def dest: rbt_height_bheight_if)

lemma bheight_size_bound:  "invc t ⟹ invh t ⟹ 2 ^ (bheight t) ≤ size1 t"
by (induction t) auto

lemma rbt_height_le: assumes "rbt t" shows "height t ≤ 2 * log 2 (size1 t)"
proof -
  have "2 powr (height t / 2) ≤ 2 powr bheight t"
    using rbt_height_bheight[OF assms] by (simp)
  also have "… ≤ size1 t" using assms
    by (simp add: powr_realpow bheight_size_bound rbt_def)
  finally have "2 powr (height t / 2) ≤ size1 t" .
  hence "height t / 2 ≤ log 2 (size1 t)"
    by (simp add: le_log_iff size1_size del: divide_le_eq_numeral1(1))
  thus ?thesis by simp
qed

end