(* Author: Tobias Nipkow *)
section \<open>Red-Black Tree Implementation of Sets\<close>
theory RBT_Set
imports
Complex_Main
RBT
Cmp
Isin2
begin
definition empty :: "'a rbt" where
"empty = Leaf"
fun ins :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"ins x Leaf = R Leaf x Leaf" |
"ins x (B l a r) =
(case cmp x a of
LT \<Rightarrow> baliL (ins x l) a r |
GT \<Rightarrow> baliR l a (ins x r) |
EQ \<Rightarrow> B l a r)" |
"ins x (R l a r) =
(case cmp x a of
LT \<Rightarrow> R (ins x l) a r |
GT \<Rightarrow> R l a (ins x r) |
EQ \<Rightarrow> R l a r)"
definition insert :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"insert x t = paint Black (ins x t)"
fun color :: "'a rbt \<Rightarrow> color" where
"color Leaf = Black" |
"color (Node _ (_, c) _) = c"
fun del :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"del x Leaf = Leaf" |
"del x (Node l (a, _) r) =
(case cmp x a of
LT \<Rightarrow> if l \<noteq> Leaf \<and> color l = Black
then baldL (del x l) a r else R (del x l) a r |
GT \<Rightarrow> if r \<noteq> Leaf\<and> color r = Black
then baldR l a (del x r) else R l a (del x r) |
EQ \<Rightarrow> combine l r)"
definition delete :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> '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) \<Longrightarrow> 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) \<Longrightarrow> 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_combine:
"inorder(combine l r) = inorder l @ inorder r"
by(induction l r rule: combine.induct)
(auto simp: inorder_baldL inorder_baldR split: tree.split color.split)
lemma inorder_del:
"sorted(inorder t) \<Longrightarrow> inorder(del x t) = del_list x (inorder t)"
by(induction x t rule: del.induct)
(auto simp: del_list_simps inorder_combine inorder_baldL inorder_baldR)
lemma inorder_delete:
"sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
by (auto simp: delete_def inorder_del inorder_paint)
subsection \<open>Structural invariants\<close>
text\<open>The proofs are due to Markus Reiter and Alexander Krauss.\<close>
fun bheight :: "'a rbt \<Rightarrow> 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 \<Rightarrow> bool" where
"invc Leaf = True" |
"invc (Node l (a,c) r) =
(invc l \<and> invc r \<and> (c = Red \<longrightarrow> color l = Black \<and> color r = Black))"
text \<open>Weaker version:\<close>
abbreviation invc2 :: "'a rbt \<Rightarrow> bool" where
"invc2 t \<equiv> invc(paint Black t)"
fun invh :: "'a rbt \<Rightarrow> bool" where
"invh Leaf = True" |
"invh (Node l (x, c) r) = (invh l \<and> invh r \<and> bheight l = bheight r)"
lemma invc2I: "invc t \<Longrightarrow> invc2 t"
by (cases t rule: tree2_cases) simp+
definition rbt :: "'a rbt \<Rightarrow> bool" where
"rbt t = (invc t \<and> invh t \<and> 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 \<Longrightarrow> invh (paint c t)"
by (cases t) auto
lemma invc_baliL:
"\<lbrakk>invc2 l; invc r\<rbrakk> \<Longrightarrow> invc (baliL l a r)"
by (induct l a r rule: baliL.induct) auto
lemma invc_baliR:
"\<lbrakk>invc l; invc2 r\<rbrakk> \<Longrightarrow> invc (baliR l a r)"
by (induct l a r rule: baliR.induct) auto
lemma bheight_baliL:
"bheight l = bheight r \<Longrightarrow> bheight (baliL l a r) = Suc (bheight l)"
by (induct l a r rule: baliL.induct) auto
lemma bheight_baliR:
"bheight l = bheight r \<Longrightarrow> bheight (baliR l a r) = Suc (bheight l)"
by (induct l a r rule: baliR.induct) auto
lemma invh_baliL:
"\<lbrakk> invh l; invh r; bheight l = bheight r \<rbrakk> \<Longrightarrow> invh (baliL l a r)"
by (induct l a r rule: baliL.induct) auto
lemma invh_baliR:
"\<lbrakk> invh l; invh r; bheight l = bheight r \<rbrakk> \<Longrightarrow> invh (baliR l a r)"
by (induct l a r rule: baliR.induct) auto
text \<open>All in one:\<close>
lemma inv_baliR: "\<lbrakk> invh l; invh r; invc l; invc2 r; bheight l = bheight r \<rbrakk>
\<Longrightarrow> invc (baliR l a r) \<and> invh (baliR l a r) \<and> bheight (baliR l a r) = Suc (bheight l)"
by (induct l a r rule: baliR.induct) auto
lemma inv_baliL: "\<lbrakk> invh l; invh r; invc2 l; invc r; bheight l = bheight r \<rbrakk>
\<Longrightarrow> invc (baliL l a r) \<and> invh (baliL l a r) \<and> bheight (baliL l a r) = Suc (bheight l)"
by (induct l a r rule: baliL.induct) auto
subsubsection \<open>Insertion\<close>
lemma invc_ins: "invc t \<longrightarrow> invc2 (ins x t) \<and> (color t = Black \<longrightarrow> invc (ins x t))"
by (induct x t rule: ins.induct) (auto simp: invc_baliL invc_baliR invc2I)
lemma invh_ins: "invh t \<Longrightarrow> invh (ins x t) \<and> 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 \<Longrightarrow> rbt (insert x t)"
by (simp add: invc_ins invh_ins color_paint_Black invh_paint rbt_def insert_def)
text \<open>All in one variant:\<close>
lemma inv_ins: "\<lbrakk> invc t; invh t \<rbrakk> \<Longrightarrow>
invc2 (ins x t) \<and> (color t = Black \<longrightarrow> invc (ins x t)) \<and>
invh(ins x t) \<and> 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 \<Longrightarrow> rbt (insert x t)"
by (simp add: inv_ins color_paint_Black invh_paint rbt_def insert_def)
subsubsection \<open>Deletion\<close>
lemma bheight_paint_Red:
"color t = Black \<Longrightarrow> bheight (paint Red t) = bheight t - 1"
by (cases t) auto
lemma invh_baldL_invc:
"\<lbrakk> invh l; invh r; bheight l + 1 = bheight r; invc r \<rbrakk>
\<Longrightarrow> invh (baldL l a r) \<and> bheight (baldL l a r) = bheight l + 1"
by (induct l a r rule: baldL.induct)
(auto simp: invh_baliR invh_paint bheight_baliR bheight_paint_Red)
lemma invh_baldL_Black:
"\<lbrakk> invh l; invh r; bheight l + 1 = bheight r; color r = Black \<rbrakk>
\<Longrightarrow> invh (baldL l a r) \<and> 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: "\<lbrakk>invc2 l; invc r; color r = Black\<rbrakk> \<Longrightarrow> invc (baldL l a r)"
by (induct l a r rule: baldL.induct) (simp_all add: invc_baliR)
lemma invc2_baldL: "\<lbrakk> invc2 l; invc r \<rbrakk> \<Longrightarrow> invc2 (baldL l a r)"
by (induct l a r rule: baldL.induct) (auto simp: invc_baliR paint2 invc2I)
lemma invh_baldR_invc:
"\<lbrakk> invh l; invh r; bheight l = bheight r + 1; invc l \<rbrakk>
\<Longrightarrow> invh (baldR l a r) \<and> 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: "\<lbrakk>invc l; invc2 r; color l = Black\<rbrakk> \<Longrightarrow> invc (baldR l a r)"
by (induct l a r rule: baldR.induct) (simp_all add: invc_baliL)
lemma invc2_baldR: "\<lbrakk> invc l; invc2 r \<rbrakk> \<Longrightarrow>invc2 (baldR l a r)"
by (induct l a r rule: baldR.induct) (auto simp: invc_baliL paint2 invc2I)
lemma invh_combine:
"\<lbrakk> invh l; invh r; bheight l = bheight r \<rbrakk>
\<Longrightarrow> invh (combine l r) \<and> bheight (combine l r) = bheight l"
by (induct l r rule: combine.induct)
(auto simp: invh_baldL_Black split: tree.splits color.splits)
lemma invc_combine:
"\<lbrakk> invc l; invc r \<rbrakk> \<Longrightarrow>
(color l = Black \<and> color r = Black \<longrightarrow> invc (combine l r)) \<and> invc2 (combine l r)"
by (induct l r rule: combine.induct)
(auto simp: invc_baldL invc2I split: tree.splits color.splits)
lemma neq_LeafD: "t \<noteq> Leaf \<Longrightarrow> \<exists>l x c r. t = Node l (x,c) r"
by(cases t rule: tree2_cases) auto
lemma del_invc_invh: "invh t \<Longrightarrow> invc t \<Longrightarrow> invh (del x t) \<and>
(color t = Red \<and> bheight (del x t) = bheight t \<and> invc (del x t) \<or>
color t = Black \<and> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
proof (induct x t rule: del.induct)
case (2 x _ y c)
have "x = y \<or> x < y \<or> x > y" by auto
thus ?case proof (elim disjE)
assume "x = y"
with 2 show ?thesis
by (cases c) (simp_all add: invh_combine invc_combine)
next
assume "x < y"
with 2 show ?thesis
by(cases c)
(auto simp: invh_baldL_invc invc_baldL invc2_baldL dest: neq_LeafD)
next
assume "y < x"
with 2 show ?thesis
by(cases c)
(auto simp: invh_baldR_invc invc_baldR invc2_baldR dest: neq_LeafD)
qed
qed auto
theorem rbt_delete: "rbt t \<Longrightarrow> rbt (delete x t)"
by (metis delete_def rbt_def color_paint_Black del_invc_invh invc2I invh_paint)
text \<open>Overall correctness:\<close>
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 \<open>Height-Size Relation\<close>
lemma neq_Black[simp]: "(c \<noteq> Black) = (c = Red)"
by (cases c) auto
lemma rbt_height_bheight_if: "invc t \<Longrightarrow> invh t \<Longrightarrow>
height t \<le> 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 \<Longrightarrow> height t / 2 \<le> bheight t "
by(auto simp: rbt_def dest: rbt_height_bheight_if)
lemma bheight_size_bound: "invc t \<Longrightarrow> invh t \<Longrightarrow> 2 ^ (bheight t) \<le> size1 t"
by (induction t) auto
lemma rbt_height_le: assumes "rbt t" shows "height t \<le> 2 * log 2 (size1 t)"
proof -
have "2 powr (height t / 2) \<le> 2 powr bheight t"
using rbt_height_bheight[OF assms] by (simp)
also have "\<dots> \<le> size1 t" using assms
by (simp add: powr_realpow bheight_size_bound rbt_def)
finally have "2 powr (height t / 2) \<le> size1 t" .
hence "height t / 2 \<le> log 2 (size1 t)"
by (simp add: le_log_iff size1_size del: divide_le_eq_numeral1(1))
thus ?thesis by simp
qed
end