# Theory AA_Set

theory AA_Set
imports Isin2
```(*
Author: Tobias Nipkow, Daniel StÃ¼we
*)

section ‹AA Tree Implementation of Sets›

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 ⇒ nat" where
"lvl Leaf = 0" |
"lvl (Node _ _ lv _) = lv"

fun invar :: "'a aa_tree ⇒ bool" where
"invar Leaf = True" |
"invar (Node l a h r) =
(invar l ∧ invar r ∧
h = lvl l + 1 ∧ (h = lvl r + 1 ∨ (∃lr b rr. r = Node lr b h rr ∧ h = lvl rr + 1)))"

fun skew :: "'a aa_tree ⇒ '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 ⇒ 'a aa_tree" where
"split (Node t1 a lva (Node t2 b lvb (Node t3 c lvc t4))) =
(if lva = lvb ∧ lvb = lvc ― ‹‹lva = lvc› suffices›
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 ⇒ 'a aa_tree ⇒ '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 ⇒ split (skew (Node (insert x t1) a lv t2)) |
GT ⇒ split (skew (Node t1 a lv (insert x t2))) |
EQ ⇒ Node t1 x lv t2)"

fun sngl :: "'a aa_tree ⇒ bool" where
"sngl Leaf = False" |
"sngl (Node _ _ _ Leaf) = True" |
"sngl (Node _ _ lva (Node _ _ lvb _)) = (lva > lvb)"

definition adjust :: "'a aa_tree ⇒ 'a aa_tree" where
(case t of
Node l x lv r ⇒
(if lvl l >= lv-1 ∧ lvl r >= lv-1 then t else
if lvl r < lv-1 ∧ 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)
⇒ 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 ⇒
(case t1 of
Node t2 a lva t3
⇒ Node (Node l x (lv-1) t2) a (lva+1)
(split (Node t3 b (if sngl t1 then lva else lva+1) t4)))))"

text‹In the paper, the last case of \<^const>‹adjust› is expressed with the help of an
incorrect auxiliary function \texttt{nlvl}.

Function ‹split_max› 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.›

fun split_max :: "'a aa_tree ⇒ '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 ⇒ 'a aa_tree ⇒ 'a aa_tree" where
"delete _ Leaf = Leaf" |
"delete x (Node l a lv r) =
(case cmp x a of
LT ⇒ adjust (Node (delete x l) a lv r) |
GT ⇒ adjust (Node l a lv (delete x r)) |
EQ ⇒ (if l = Leaf then r
else let (l',b) = split_max l in adjust (Node l' b lv r)))"

"pre_adjust (Node l a lv r) = (invar l ∧ invar r ∧
((lv = lvl l + 1 ∧ (lv = lvl r + 1 ∨ lv = lvl r + 2 ∨ lv = lvl r ∧ sngl r)) ∨
(lv = lvl l + 2 ∧ (lv = lvl r + 1 ∨ lv = lvl r ∧ sngl r))))"

subsection "Auxiliary Proofs"

lemma split_case: "split t = (case t of
Node t1 x lvx (Node t2 y lvy (Node t3 z lvz t4)) ⇒
(if lvx = lvy ∧ lvy = lvz
then Node (Node t1 x lvx t2) y (lvx+1) (Node t3 z lvx t4)
else t)
| t ⇒ t)"
by(auto split: tree.split)

lemma skew_case: "skew t = (case t of
Node (Node t1 y lvy t2) x lvx t3 ⇒
(if lvx = lvy then Node t1 y lvx (Node t2 x lvx t3) else t)
| t ⇒ t)"
by(auto split: tree.split)

lemma lvl_0_iff: "invar t ⟹ lvl t = 0 ⟷ t = Leaf"
by(cases t) auto

lemma lvl_Suc_iff: "lvl t = Suc n ⟷ (∃ 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 ∨ lvl (split t) = lvl t + 1 ∧ 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 ∧ invar ⟨rl, rx, rlv, rr⟩ ∧ lv = Suc (lvl l) ∧
(lv = Suc rlv ∨ rlv = lv ∧ lv = Suc (lvl rr)))"
by simp

lemma invar_NodeLeaf[simp]:
"invar (Node l x lv Leaf) = (invar l ∧ lv = Suc (lvl l) ∧ lv = Suc 0)"
by simp

lemma sngl_if_invar: "invar (Node l a n r) ⟹ n = lvl r ⟹ 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 ∨ lvl (insert x t) = lvl t + 1 ∧ 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 ⟹ sngl t ⟹ 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 simp
qed simp

lemma skew_invar: "invar t ⟹ skew t = t"
by(cases t rule: skew.cases) auto

lemma split_invar: "invar t ⟹ split t = t"
by(cases t rule: split.cases) clarsimp+

lemma invar_NodeL:
"⟦ invar(Node l x n r); invar l'; lvl l' = lvl l ⟧ ⟹ invar(Node l' x n r)"
by(auto)

lemma invar_NodeR:
"⟦ invar(Node l x n r); n = lvl r + 1; invar r'; lvl r' = lvl r ⟧ ⟹ invar(Node l x n r')"
by(auto)

lemma invar_NodeR2:
"⟦ invar(Node l x n r); sngl r'; n = lvl r + 1; invar r'; lvl r' = n ⟧ ⟹ 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) ⟷
(∃l x r. insert a t = Node l x (lvl t + 1) r ∧ 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 ⟹ 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 ∨ a = x ∨ x < a" by auto
moreover
have ?case if "a < x"
proof (cases rule: lvl_insert[of a l])
case (Same) thus ?thesis
using ‹a<x› 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))"
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 …)"
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 ≠ n" thus ?thesis using N(3) iil l12 by(auto)
qed
qed
finally show ?thesis .
qed
moreover
have ?case if "x < a"
proof -
from ‹invar ?t› have "n = lvl r ∨ 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 ‹a>x› 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 …)"
proof -
from lvl_insert_sngl[OF ir sngl_if_invar[OF ‹invar ?t› 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 ‹x<a› 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 ‹invar ?t› Incr(2) 1 iir] 1 ‹x < a›
by (auto simp add: skew_invar split_invar split: if_splits)
qed
qed
qed
moreover
have "a = x ⟹ ?case" using N.prems by auto
ultimately show ?case by blast
qed simp

subsubsection "Proofs for delete"

lemma invarL: "ASSUMPTION(invar ⟨l, a, lv, r⟩) ⟹ invar l"

lemma invarR: "ASSUMPTION(invar ⟨lv, l, a, r⟩) ⟹ invar r"

lemma sngl_NodeI:
"sngl (Node l a lv r) ⟹ 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 ∧ invar r ∧
lv = Suc (lvl r) ∧ lvl l = lvl r" |
(tDouble) "invar l ∧ invar r ∧
lv = lvl r ∧ Suc (lvl l) = lvl r ∧ sngl r " |
(rDown) "invar l ∧ invar r ∧
lv = Suc (Suc (lvl r)) ∧  lv = Suc (lvl l)" |
(lDown_tSngl) "invar l ∧ invar r ∧
lv = Suc (lvl r) ∧ lv = Suc (Suc (lvl l))" |
(lDown_tDouble) "invar l ∧ invar r ∧
lv = lvl r ∧ lv = Suc (Suc (lvl l)) ∧ sngl r"
by auto

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)

assumes "pre_adjust (Node l a lv r)"
shows "lv = lvl (adjust(Node l a lv r)) ∨ 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 "⟨l, a, lvl r, r⟩"] 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)

"sngl ⟨l, a, lv, r⟩" "lv = lvl (adjust ⟨l, a, lv, r⟩)"
shows "sngl (adjust ⟨l, a, lv, r⟩)"
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' ∧
(lvl t' = lvl t ∨ lvl t' + 1 = lvl t) ∧
(lvl t' = lvl t ∧ sngl t ⟶ sngl t')"

"invar⟨lv, l, a, r⟩ ⟹ post_del r r' ⟹ pre_adjust ⟨lv, l, a, r'⟩"
by(cases "sngl r")
(auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)

"invar⟨l, a, lv, r⟩ ⟹ post_del l l' ⟹ pre_adjust ⟨l', b, lv, r⟩"
by(cases "sngl r")
(auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)

"⟦ invar⟨l, a, lv, r⟩; pre_adjust ⟨l', b, lv, r⟩ ⟧
⟹ post_del ⟨l, a, lv, r⟩ (adjust ⟨l', b, lv, r⟩)"
unfolding post_del_def

assumes "invar⟨lv, l, a, r⟩" "pre_adjust ⟨lv, l, a, r'⟩" "post_del r r'"
shows "post_del ⟨lv, l, a, r⟩ (adjust ⟨lv, l, a, r'⟩)"
proof(unfold post_del_def, safe del: disjCI)
let ?t = "⟨lv, l, a, r⟩"
let ?t' = "adjust ⟨lv, l, a, r'⟩"
show "invar ?t'" by(rule invar_adjust[OF assms(2)])
show "lvl ?t' = lvl ?t ∨ lvl ?t' + 1 = lvl ?t"
show "sngl ?t'" if as: "lvl ?t' = lvl ?t" "sngl ?t"
proof -
have s: "sngl ⟨lv, l, a, r'⟩"
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:
"⟦ invar t; (t', x) = split_max t; t ≠ Leaf ⟧ ⟹ post_del t t'"
proof (induction t arbitrary: t' rule: split_max.induct)
case (2 lv l a lvr rl ra rr)
let ?r =  "⟨lvr, rl, ra, rr⟩"
let ?t = "⟨lv, l, a, ?r⟩"
from "2.prems"(2) obtain r' where r': "(r', x) = split_max ?r"
and [simp]: "t' = adjust ⟨lv, l, a, r'⟩" by auto
from  "2.IH"[OF _ r'] ‹invar ?t› have post: "post_del ?r r'" by simp
note preR = pre_adj_if_postR[OF ‹invar ?t› post]
qed (auto simp: post_del_def)

theorem post_delete: "invar t ⟹ 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
next
case greater
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 ≠ Leaf" thus ?thesis using equal
qed
qed

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) ⟹ 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"

by(cases t)
split: tree.splits)

lemma split_maxD:
"⟦ split_max t = (t',x); t ≠ Leaf; invar t ⟧ ⟹ inorder t' @ [x] = inorder t"
by(induction t arbitrary: t' rule: split_max.induct)

lemma inorder_delete:
"invar t ⟹ sorted(inorder t) ⟹ inorder(delete x t) = del_list x (inorder t)"
by(induction t)
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
```