# Theory AA_Map

theory AA_Map
imports AA_Set Lookup2
```(* Author: Tobias Nipkow *)

section "AA Tree Implementation of Maps"

theory AA_Map
imports
AA_Set
Lookup2
begin

fun update :: "'a::linorder ⇒ 'b ⇒ ('a*'b) aa_tree ⇒ ('a*'b) aa_tree" where
"update x y Leaf = Node Leaf (x,y) 1 Leaf" |
"update x y (Node t1 (a,b) lv t2) =
(case cmp x a of
LT ⇒ split (skew (Node (update x y t1) (a,b) lv t2)) |
GT ⇒ split (skew (Node t1 (a,b) lv (update x y t2))) |
EQ ⇒ Node t1 (x,y) lv t2)"

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

subsection "Invariance"

subsubsection "Proofs for insert"

lemma lvl_update_aux:
"lvl (update x y t) = lvl t ∨ lvl (update x y t) = lvl t + 1 ∧ sngl (update x y t)"
apply(induction t)
apply (auto simp: lvl_skew)
apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+
done

lemma lvl_update: obtains
(Same) "lvl (update x y t) = lvl t" |
(Incr) "lvl (update x y t) = lvl t + 1" "sngl (update x y t)"
using lvl_update_aux by fastforce

declare invar.simps(2)[simp]

lemma lvl_update_sngl: "invar t ⟹ sngl t ⟹ lvl(update x y t) = lvl t"
proof (induction t rule: update.induct)
case (2 x y t1 a b 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_update_aux n_not_Suc_n sngl.simps(3))+
qed simp
qed simp

lemma lvl_update_incr_iff: "(lvl(update a b t) = lvl t + 1) ⟷
(∃l x r. update a b 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_update: "invar t ⟹ invar(update a b t)"
proof(induction t)
case N: (Node l xy 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]
obtain x y where [simp]: "xy = (x,y)" by fastforce
let ?t = "Node l xy n r"
have "a < x ∨ a = x ∨ x < a" by auto
moreover
have ?case if "a < x"
proof (cases rule: lvl_update[of a b 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]: "update a b 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_update_incr_iff tree.inject)
have "update a b ?t = split(skew(Node (update a b l) xy n r))"
also have "skew(Node (update a b l) xy n r) = Node t1 w n (Node t2 xy 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 "update a b ?t = split(skew(Node l xy n (update a b r)))"
using ‹a>x› by(auto)
also have "skew(Node l xy n (update a b r)) = Node l xy n (update a b r)"
using N.prems by(simp add: skew_case split: tree.split)
also have "invar(split …)"
proof -
from lvl_update_sngl[OF ir sngl_if_invar[OF ‹invar ?t› 0], of a b]
obtain t1 p t2 where iar: "update a b r = Node t1 p 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_update[of a b 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"

declare invar.simps(2)[simp del]

theorem post_delete: "invar t ⟹ post_del t (delete x t)"
proof (induction t)
case (Node l ab lv r)

obtain a b where [simp]: "ab = (a,b)" by fastforce

let ?l' = "delete x l" and ?r' = "delete x r"
let ?t = "Node l ab 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 Node.prems
qed
qed

subsection ‹Functional Correctness Proofs›

theorem inorder_update:
"sorted1(inorder t) ⟹ inorder(update x y t) = upd_list x y (inorder t)"
by (induct t) (auto simp: upd_list_simps inorder_split inorder_skew)

theorem inorder_delete:
"⟦invar t; sorted1(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 I: Map_by_Ordered
where empty = empty and lookup = lookup and update = update 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: lookup_map_of)
next
case 3 thus ?case by(simp add: inorder_update)
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_update)
next
case 7 thus ?case using post_delete by(auto simp: post_del_def)
qed

end
```