(*
Author: Tobias Nipkow, Daniel Stüwe
*)
section \<open>AA Tree Implementation of Sets\<close>
theory AA_Set
imports
Isin2
Cmp
begin
type_synonym 'a aa_tree = "('a,nat) tree"
definition empty :: "'a aa_tree" where
"empty = Leaf"
fun lvl :: "'a aa_tree \<Rightarrow> nat" where
"lvl Leaf = 0" |
"lvl (Node _ _ lv _) = lv"
fun invar :: "'a aa_tree \<Rightarrow> bool" where
"invar Leaf = True" |
"invar (Node l a h r) =
(invar l \<and> invar r \<and>
h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = Node lr b h rr \<and> h = lvl rr + 1)))"
fun skew :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
"skew (Node (Node t1 b lvb t2) a lva t3) =
(if lva = lvb then Node t1 b lvb (Node t2 a lva t3) else Node (Node t1 b lvb t2) a lva t3)" |
"skew t = t"
fun split :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
"split (Node t1 a lva (Node t2 b lvb (Node t3 c lvc t4))) =
(if lva = lvb \<and> lvb = lvc \<comment> \<open>\<open>lva = lvc\<close> suffices\<close>
then Node (Node t1 a lva t2) b (lva+1) (Node t3 c lva t4)
else Node t1 a lva (Node t2 b lvb (Node t3 c lvc t4)))" |
"split t = t"
hide_const (open) insert
fun insert :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
"insert x Leaf = Node Leaf x 1 Leaf" |
"insert x (Node t1 a lv t2) =
(case cmp x a of
LT \<Rightarrow> split (skew (Node (insert x t1) a lv t2)) |
GT \<Rightarrow> split (skew (Node t1 a lv (insert x t2))) |
EQ \<Rightarrow> Node t1 x lv t2)"
fun sngl :: "'a aa_tree \<Rightarrow> bool" where
"sngl Leaf = False" |
"sngl (Node _ _ _ Leaf) = True" |
"sngl (Node _ _ lva (Node _ _ lvb _)) = (lva > lvb)"
definition adjust :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
"adjust t =
(case t of
Node l x lv r \<Rightarrow>
(if lvl l >= lv-1 \<and> lvl r >= lv-1 then t else
if lvl r < lv-1 \<and> sngl l then skew (Node l x (lv-1) r) else
if lvl r < lv-1
then case l of
Node t1 a lva (Node t2 b lvb t3)
\<Rightarrow> Node (Node t1 a lva t2) b (lvb+1) (Node t3 x (lv-1) r)
else
if lvl r < lv then split (Node l x (lv-1) r)
else
case r of
Node t1 b lvb t4 \<Rightarrow>
(case t1 of
Node t2 a lva t3
\<Rightarrow> Node (Node l x (lv-1) t2) a (lva+1)
(split (Node t3 b (if sngl t1 then lva else lva+1) t4)))))"
text\<open>In the paper, the last case of \<^const>\<open>adjust\<close> is expressed with the help of an
incorrect auxiliary function \texttt{nlvl}.
Function \<open>split_max\<close> below is called \texttt{dellrg} in the paper.
The latter is incorrect for two reasons: \texttt{dellrg} is meant to delete the largest
element but recurses on the left instead of the right subtree; the invariant
is not restored.\<close>
fun split_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
"split_max (Node l a lv Leaf) = (l,a)" |
"split_max (Node l a lv r) = (let (r',b) = split_max r in (adjust(Node l a lv r'), b))"
fun delete :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
"delete _ Leaf = Leaf" |
"delete x (Node l a lv r) =
(case cmp x a of
LT \<Rightarrow> adjust (Node (delete x l) a lv r) |
GT \<Rightarrow> adjust (Node l a lv (delete x r)) |
EQ \<Rightarrow> (if l = Leaf then r
else let (l',b) = split_max l in adjust (Node l' b lv r)))"
fun pre_adjust where
"pre_adjust (Node l a lv r) = (invar l \<and> invar r \<and>
((lv = lvl l + 1 \<and> (lv = lvl r + 1 \<or> lv = lvl r + 2 \<or> lv = lvl r \<and> sngl r)) \<or>
(lv = lvl l + 2 \<and> (lv = lvl r + 1 \<or> lv = lvl r \<and> sngl r))))"
declare pre_adjust.simps [simp del]
subsection "Auxiliary Proofs"
lemma split_case: "split t = (case t of
Node t1 x lvx (Node t2 y lvy (Node t3 z lvz t4)) \<Rightarrow>
(if lvx = lvy \<and> lvy = lvz
then Node (Node t1 x lvx t2) y (lvx+1) (Node t3 z lvx t4)
else t)
| t \<Rightarrow> t)"
by(auto split: tree.split)
lemma skew_case: "skew t = (case t of
Node (Node t1 y lvy t2) x lvx t3 \<Rightarrow>
(if lvx = lvy then Node t1 y lvx (Node t2 x lvx t3) else t)
| t \<Rightarrow> t)"
by(auto split: tree.split)
lemma lvl_0_iff: "invar t \<Longrightarrow> lvl t = 0 \<longleftrightarrow> t = Leaf"
by(cases t) auto
lemma lvl_Suc_iff: "lvl t = Suc n \<longleftrightarrow> (\<exists> l a r. t = Node l a (Suc n) r)"
by(cases t) auto
lemma lvl_skew: "lvl (skew t) = lvl t"
by(cases t rule: skew.cases) auto
lemma lvl_split: "lvl (split t) = lvl t \<or> lvl (split t) = lvl t + 1 \<and> sngl (split t)"
by(cases t rule: split.cases) auto
lemma invar_2Nodes:"invar (Node l x lv (Node rl rx rlv rr)) =
(invar l \<and> invar \<langle>rl, rx, rlv, rr\<rangle> \<and> lv = Suc (lvl l) \<and>
(lv = Suc rlv \<or> rlv = lv \<and> lv = Suc (lvl rr)))"
by simp
lemma invar_NodeLeaf[simp]:
"invar (Node l x lv Leaf) = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)"
by simp
lemma sngl_if_invar: "invar (Node l a n r) \<Longrightarrow> n = lvl r \<Longrightarrow> sngl r"
by(cases r rule: sngl.cases) clarsimp+
subsection "Invariance"
subsubsection "Proofs for insert"
lemma lvl_insert_aux:
"lvl (insert x t) = lvl t \<or> lvl (insert x t) = lvl t + 1 \<and> sngl (insert x t)"
apply(induction t)
apply (auto simp: lvl_skew)
apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+
done
lemma lvl_insert: obtains
(Same) "lvl (insert x t) = lvl t" |
(Incr) "lvl (insert x t) = lvl t + 1" "sngl (insert x t)"
using lvl_insert_aux by blast
lemma lvl_insert_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(insert x t) = lvl t"
proof (induction t rule: insert.induct)
case (2 x t1 a lv t2)
consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a"
using less_linear by blast
thus ?case proof cases
case LT
thus ?thesis using 2 by (auto simp add: skew_case split_case split: tree.splits)
next
case GT
thus ?thesis using 2 proof (cases t1)
case Node
thus ?thesis using 2 GT
apply (auto simp add: skew_case split_case split: tree.splits)
by (metis less_not_refl2 lvl.simps(2) lvl_insert_aux n_not_Suc_n sngl.simps(3))+
qed (auto simp add: lvl_0_iff)
qed simp
qed simp
lemma skew_invar: "invar t \<Longrightarrow> skew t = t"
by(cases t rule: skew.cases) auto
lemma split_invar: "invar t \<Longrightarrow> split t = t"
by(cases t rule: split.cases) clarsimp+
lemma invar_NodeL:
"\<lbrakk> invar(Node l x n r); invar l'; lvl l' = lvl l \<rbrakk> \<Longrightarrow> invar(Node l' x n r)"
by(auto)
lemma invar_NodeR:
"\<lbrakk> invar(Node l x n r); n = lvl r + 1; invar r'; lvl r' = lvl r \<rbrakk> \<Longrightarrow> invar(Node l x n r')"
by(auto)
lemma invar_NodeR2:
"\<lbrakk> invar(Node l x n r); sngl r'; n = lvl r + 1; invar r'; lvl r' = n \<rbrakk> \<Longrightarrow> invar(Node l x n r')"
by(cases r' rule: sngl.cases) clarsimp+
lemma lvl_insert_incr_iff: "(lvl(insert a t) = lvl t + 1) \<longleftrightarrow>
(\<exists>l x r. insert a t = Node l x (lvl t + 1) r \<and> lvl l = lvl r)"
apply(cases t)
apply(auto simp add: skew_case split_case split: if_splits)
apply(auto split: tree.splits if_splits)
done
lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)"
proof(induction t)
case N: (Node l x n r)
hence il: "invar l" and ir: "invar r" by auto
note iil = N.IH(1)[OF il]
note iir = N.IH(2)[OF ir]
let ?t = "Node l x n r"
have "a < x \<or> a = x \<or> x < a" by auto
moreover
have ?case if "a < x"
proof (cases rule: lvl_insert[of a l])
case (Same) thus ?thesis
using \<open>a<x\<close> invar_NodeL[OF N.prems iil Same]
by (simp add: skew_invar split_invar del: invar.simps)
next
case (Incr)
then obtain t1 w t2 where ial[simp]: "insert a l = Node t1 w n t2"
using N.prems by (auto simp: lvl_Suc_iff)
have l12: "lvl t1 = lvl t2"
by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
have "insert a ?t = split(skew(Node (insert a l) x n r))"
by(simp add: \<open>a<x\<close>)
also have "skew(Node (insert a l) x n r) = Node t1 w n (Node t2 x n r)"
by(simp)
also have "invar(split \<dots>)"
proof (cases r)
case Leaf
hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
thus ?thesis using Leaf ial by simp
next
case [simp]: (Node t3 y m t4)
show ?thesis (*using N(3) iil l12 by(auto)*)
proof cases
assume "m = n" thus ?thesis using N(3) iil by(auto)
next
assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
qed
qed
finally show ?thesis .
qed
moreover
have ?case if "x < a"
proof -
from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
thus ?case
proof
assume 0: "n = lvl r"
have "insert a ?t = split(skew(Node l x n (insert a r)))"
using \<open>a>x\<close> by(auto)
also have "skew(Node l x n (insert a r)) = Node l x n (insert a r)"
using N.prems by(simp add: skew_case split: tree.split)
also have "invar(split \<dots>)"
proof -
from lvl_insert_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a]
obtain t1 y t2 where iar: "insert a r = Node t1 y n t2"
using N.prems 0 by (auto simp: lvl_Suc_iff)
from N.prems iar 0 iir
show ?thesis by (auto simp: split_case split: tree.splits)
qed
finally show ?thesis .
next
assume 1: "n = lvl r + 1"
hence "sngl ?t" by(cases r) auto
show ?thesis
proof (cases rule: lvl_insert[of a r])
case (Same)
show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF N.prems 1 iir Same]
by (auto simp add: skew_invar split_invar)
next
case (Incr)
thus ?thesis using invar_NodeR2[OF \<open>invar ?t\<close> Incr(2) 1 iir] 1 \<open>x < a\<close>
by (auto simp add: skew_invar split_invar split: if_splits)
qed
qed
qed
moreover
have "a = x \<Longrightarrow> ?case" using N.prems by auto
ultimately show ?case by blast
qed simp
subsubsection "Proofs for delete"
lemma invarL: "ASSUMPTION(invar \<langle>l, a, lv, r\<rangle>) \<Longrightarrow> invar l"
by(simp add: ASSUMPTION_def)
lemma invarR: "ASSUMPTION(invar \<langle>lv, l, a, r\<rangle>) \<Longrightarrow> invar r"
by(simp add: ASSUMPTION_def)
lemma sngl_NodeI:
"sngl (Node l a lv r) \<Longrightarrow> sngl (Node l' a' lv r)"
by(cases r) (simp_all)
declare invarL[simp] invarR[simp]
lemma pre_cases:
assumes "pre_adjust (Node l x lv r)"
obtains
(tSngl) "invar l \<and> invar r \<and>
lv = Suc (lvl r) \<and> lvl l = lvl r" |
(tDouble) "invar l \<and> invar r \<and>
lv = lvl r \<and> Suc (lvl l) = lvl r \<and> sngl r " |
(rDown) "invar l \<and> invar r \<and>
lv = Suc (Suc (lvl r)) \<and> lv = Suc (lvl l)" |
(lDown_tSngl) "invar l \<and> invar r \<and>
lv = Suc (lvl r) \<and> lv = Suc (Suc (lvl l))" |
(lDown_tDouble) "invar l \<and> invar r \<and>
lv = lvl r \<and> lv = Suc (Suc (lvl l)) \<and> sngl r"
using assms unfolding pre_adjust.simps
by auto
declare invar.simps(2)[simp del] invar_2Nodes[simp add]
lemma invar_adjust:
assumes pre: "pre_adjust (Node l a lv r)"
shows "invar(adjust (Node l a lv r))"
using pre proof (cases rule: pre_cases)
case (tDouble) thus ?thesis unfolding adjust_def by (cases r) (auto simp: invar.simps(2))
next
case (rDown)
from rDown obtain llv ll la lr where l: "l = Node ll la llv lr" by (cases l) auto
from rDown show ?thesis unfolding adjust_def by (auto simp: l invar.simps(2) split: tree.splits)
next
case (lDown_tDouble)
from lDown_tDouble obtain rlv rr ra rl where r: "r = Node rl ra rlv rr" by (cases r) auto
from lDown_tDouble and r obtain rrlv rrr rra rrl where
rr :"rr = Node rrr rra rrlv rrl" by (cases rr) auto
from lDown_tDouble show ?thesis unfolding adjust_def r rr
apply (cases rl) apply (auto simp add: invar.simps(2) split!: if_split)
using lDown_tDouble by (auto simp: split_case lvl_0_iff elim:lvl.elims split: tree.split)
qed (auto simp: split_case invar.simps(2) adjust_def split: tree.splits)
lemma lvl_adjust:
assumes "pre_adjust (Node l a lv r)"
shows "lv = lvl (adjust(Node l a lv r)) \<or> lv = lvl (adjust(Node l a lv r)) + 1"
using assms(1) proof(cases rule: pre_cases)
case lDown_tSngl thus ?thesis
using lvl_split[of "\<langle>l, a, lvl r, r\<rangle>"] by (auto simp: adjust_def)
next
case lDown_tDouble thus ?thesis
by (auto simp: adjust_def invar.simps(2) split: tree.split)
qed (auto simp: adjust_def split: tree.splits)
lemma sngl_adjust: assumes "pre_adjust (Node l a lv r)"
"sngl \<langle>l, a, lv, r\<rangle>" "lv = lvl (adjust \<langle>l, a, lv, r\<rangle>)"
shows "sngl (adjust \<langle>l, a, lv, r\<rangle>)"
using assms proof (cases rule: pre_cases)
case rDown
thus ?thesis using assms(2,3) unfolding adjust_def
by (auto simp add: skew_case) (auto split: tree.split)
qed (auto simp: adjust_def skew_case split_case split: tree.split)
definition "post_del t t' ==
invar t' \<and>
(lvl t' = lvl t \<or> lvl t' + 1 = lvl t) \<and>
(lvl t' = lvl t \<and> sngl t \<longrightarrow> sngl t')"
lemma pre_adj_if_postR:
"invar\<langle>lv, l, a, r\<rangle> \<Longrightarrow> post_del r r' \<Longrightarrow> pre_adjust \<langle>lv, l, a, r'\<rangle>"
by(cases "sngl r")
(auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
lemma pre_adj_if_postL:
"invar\<langle>l, a, lv, r\<rangle> \<Longrightarrow> post_del l l' \<Longrightarrow> pre_adjust \<langle>l', b, lv, r\<rangle>"
by(cases "sngl r")
(auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
lemma post_del_adjL:
"\<lbrakk> invar\<langle>l, a, lv, r\<rangle>; pre_adjust \<langle>l', b, lv, r\<rangle> \<rbrakk>
\<Longrightarrow> post_del \<langle>l, a, lv, r\<rangle> (adjust \<langle>l', b, lv, r\<rangle>)"
unfolding post_del_def
by (metis invar_adjust lvl_adjust sngl_NodeI sngl_adjust lvl.simps(2))
lemma post_del_adjR:
assumes "invar\<langle>lv, l, a, r\<rangle>" "pre_adjust \<langle>lv, l, a, r'\<rangle>" "post_del r r'"
shows "post_del \<langle>lv, l, a, r\<rangle> (adjust \<langle>lv, l, a, r'\<rangle>)"
proof(unfold post_del_def, safe del: disjCI)
let ?t = "\<langle>lv, l, a, r\<rangle>"
let ?t' = "adjust \<langle>lv, l, a, r'\<rangle>"
show "invar ?t'" by(rule invar_adjust[OF assms(2)])
show "lvl ?t' = lvl ?t \<or> lvl ?t' + 1 = lvl ?t"
using lvl_adjust[OF assms(2)] by auto
show "sngl ?t'" if as: "lvl ?t' = lvl ?t" "sngl ?t"
proof -
have s: "sngl \<langle>lv, l, a, r'\<rangle>"
proof(cases r')
case Leaf thus ?thesis by simp
next
case Node thus ?thesis using as(2) assms(1,3)
by (cases r) (auto simp: post_del_def)
qed
show ?thesis using as(1) sngl_adjust[OF assms(2) s] by simp
qed
qed
declare prod.splits[split]
theorem post_split_max:
"\<lbrakk> invar t; (t', x) = split_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'"
proof (induction t arbitrary: t' rule: split_max.induct)
case (2 lv l a lvr rl ra rr)
let ?r = "\<langle>lvr, rl, ra, rr\<rangle>"
let ?t = "\<langle>lv, l, a, ?r\<rangle>"
from "2.prems"(2) obtain r' where r': "(r', x) = split_max ?r"
and [simp]: "t' = adjust \<langle>lv, l, a, r'\<rangle>" by auto
from "2.IH"[OF _ r'] \<open>invar ?t\<close> have post: "post_del ?r r'" by simp
note preR = pre_adj_if_postR[OF \<open>invar ?t\<close> post]
show ?case by (simp add: post_del_adjR[OF "2.prems"(1) preR post])
qed (auto simp: post_del_def)
theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
proof (induction t)
case (Node l a lv r)
let ?l' = "delete x l" and ?r' = "delete x r"
let ?t = "Node l a lv r" let ?t' = "delete x ?t"
from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto)
note post_l' = Node.IH(1)[OF inv_l]
note preL = pre_adj_if_postL[OF Node.prems post_l']
note post_r' = Node.IH(2)[OF inv_r]
note preR = pre_adj_if_postR[OF Node.prems post_r']
show ?case
proof (cases rule: linorder_cases[of x a])
case less
thus ?thesis using Node.prems by (simp add: post_del_adjL preL)
next
case greater
thus ?thesis using Node.prems by (simp add: post_del_adjR preR post_r')
next
case equal
show ?thesis
proof cases
assume "l = Leaf" thus ?thesis using equal Node.prems
by(auto simp: post_del_def invar.simps(2))
next
assume "l \<noteq> Leaf" thus ?thesis using equal
by simp (metis Node.prems inv_l post_del_adjL post_split_max pre_adj_if_postL)
qed
qed
qed (simp add: post_del_def)
declare invar_2Nodes[simp del]
subsection "Functional Correctness"
subsubsection "Proofs for insert"
lemma inorder_split: "inorder(split t) = inorder t"
by(cases t rule: split.cases) (auto)
lemma inorder_skew: "inorder(skew t) = inorder t"
by(cases t rule: skew.cases) (auto)
lemma inorder_insert:
"sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
by(induction t) (auto simp: ins_list_simps inorder_split inorder_skew)
subsubsection "Proofs for delete"
lemma inorder_adjust: "t \<noteq> Leaf \<Longrightarrow> pre_adjust t \<Longrightarrow> inorder(adjust t) = inorder t"
by(cases t)
(auto simp: adjust_def inorder_skew inorder_split invar.simps(2) pre_adjust.simps
split: tree.splits)
lemma split_maxD:
"\<lbrakk> split_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t"
by(induction t arbitrary: t' rule: split_max.induct)
(auto simp: sorted_lems inorder_adjust pre_adj_if_postR post_split_max split: prod.splits)
lemma inorder_delete:
"invar t \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
by(induction t)
(auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR
post_split_max post_delete split_maxD split: prod.splits)
interpretation S: Set_by_Ordered
where empty = empty and isin = isin and insert = insert and delete = delete
and inorder = inorder and inv = invar
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: empty_def)
next
case 6 thus ?case by(simp add: invar_insert)
next
case 7 thus ?case using post_delete by(auto simp: post_del_def)
qed
end