(* Author: Tobias Nipkow *)
section "Height-Balanced Trees"
theory Height_Balanced_Tree
imports
Cmp
Isin2
begin
text \<open>Height-balanced trees (HBTs) can be seen as a generalization of AVL trees.
The code and the proofs were obtained by small modifications of the AVL theories.
This is an implementation of sets via HBTs.\<close>
type_synonym 'a tree_ht = "('a*nat) tree"
definition empty :: "'a tree_ht" where
"empty = Leaf"
text \<open>The maximal amount by which the height of two siblings may differ:\<close>
locale HBT =
fixes m :: nat
assumes [arith]: "m > 0"
begin
text \<open>Invariant:\<close>
fun hbt :: "'a tree_ht \ bool" where
"hbt Leaf = True" |
"hbt (Node l (a,n) r) =
(abs(int(height l) - int(height r)) \<le> int(m) \<and>
n = max (height l) (height r) + 1 \<and> hbt l \<and> hbt r)"
fun ht :: "'a tree_ht \ nat" where
"ht Leaf = 0" |
"ht (Node l (a,n) r) = n"
definition node :: "'a tree_ht \ 'a \ 'a tree_ht \ 'a tree_ht" where
"node l a r = Node l (a, max (ht l) (ht r) + 1) r"
definition balL :: "'a tree_ht \ 'a \ 'a tree_ht \ 'a tree_ht" where
"balL AB b C =
(if ht AB = ht C + m + 1 then
case AB of
Node A (a, _) B \<Rightarrow>
if ht A \<ge> ht B then node A a (node B b C)
else
case B of
Node B\<^sub>1 (ab, _) B\<^sub>2 \<Rightarrow> node (node A a B\<^sub>1) ab (node B\<^sub>2 b C)
else node AB b C)"
definition balR :: "'a tree_ht \ 'a \ 'a tree_ht \ 'a tree_ht" where
"balR A a BC =
(if ht BC = ht A + m + 1 then
case BC of
Node B (b, _) C \<Rightarrow>
if ht B \<le> ht C then node (node A a B) b C
else
case B of
Node B\<^sub>1 (ab, _) B\<^sub>2 \<Rightarrow> node (node A a B\<^sub>1) ab (node B\<^sub>2 b C)
else node A a BC)"
fun insert :: "'a::linorder \ 'a tree_ht \ 'a tree_ht" where
"insert x Leaf = Node Leaf (x, 1) Leaf" |
"insert x (Node l (a, n) r) = (case cmp x a of
EQ \<Rightarrow> Node l (a, n) r |
LT \<Rightarrow> balL (insert x l) a r |
GT \<Rightarrow> balR l a (insert x r))"
fun split_max :: "'a tree_ht \ 'a tree_ht * 'a" where
"split_max (Node l (a, _) r) =
(if r = Leaf then (l,a) else let (r',a') = split_max r in (balL l a r', a'))"
lemmas split_max_induct = split_max.induct[case_names Node Leaf]
fun delete :: "'a::linorder \ 'a tree_ht \ 'a tree_ht" where
"delete _ Leaf = Leaf" |
"delete x (Node l (a, n) r) =
(case cmp x a of
EQ \<Rightarrow> if l = Leaf then r
else let (l', a') = split_max l in balR l' a' r |
LT \<Rightarrow> balR (delete x l) a r |
GT \<Rightarrow> balL l a (delete x r))"
subsection \<open>Functional Correctness Proofs\<close>
subsubsection "Proofs for insert"
lemma inorder_balL:
"inorder (balL l a r) = inorder l @ a # inorder r"
by (auto simp: node_def balL_def split:tree.splits)
lemma inorder_balR:
"inorder (balR l a r) = inorder l @ a # inorder r"
by (auto simp: node_def balR_def split:tree.splits)
theorem inorder_insert:
"sorted(inorder t) \ inorder(insert x t) = ins_list x (inorder t)"
by (induct t)
(auto simp: ins_list_simps inorder_balL inorder_balR)
subsubsection "Proofs for delete"
lemma inorder_split_maxD:
"\ split_max t = (t',a); t \ Leaf \ \
inorder t' @ [a] = inorder t"
by(induction t arbitrary: t' rule: split_max.induct)
(auto simp: inorder_balL split: if_splits prod.splits tree.split)
theorem inorder_delete:
"sorted(inorder t) \ inorder (delete x t) = del_list x (inorder t)"
by(induction t)
(auto simp: del_list_simps inorder_balL inorder_balR inorder_split_maxD split: prod.splits)
subsection \<open>Invariant preservation\<close>
subsubsection \<open>Insertion maintains balance\<close>
declare Let_def [simp]
lemma ht_height[simp]: "hbt t \ ht t = height t"
by (cases t rule: tree2_cases) simp_all
text \<open>First, a fast but relatively manual proof with many lemmas:\<close>
lemma height_balL:
"\ hbt l; hbt r; height l = height r + m + 1 \ \
height (balL l a r) \<in> {height r + m + 1, height r + m + 2}"
by (auto simp:node_def balL_def split:tree.split)
lemma height_balR:
"\ hbt l; hbt r; height r = height l + m + 1 \ \
height (balR l a r) \<in> {height l + m + 1, height l + m + 2}"
by(auto simp add:node_def balR_def split:tree.split)
lemma height_node[simp]: "height(node l a r) = max (height l) (height r) + 1"
by (simp add: node_def)
lemma height_balL2:
"\ hbt l; hbt r; height l \ height r + m + 1 \ \
height (balL l a r) = 1 + max (height l) (height r)"
by (simp_all add: balL_def)
lemma height_balR2:
"\ hbt l; hbt r; height r \ height l + m + 1 \ \
height (balR l a r) = 1 + max (height l) (height r)"
by (simp_all add: balR_def)
lemma hbt_balL:
"\ hbt l; hbt r; height r - m \ height l \ height l \ height r + m + 1 \ \ hbt(balL l a r)"
by(auto simp: balL_def node_def max_def split!: if_splits tree.split)
lemma hbt_balR:
"\ hbt l; hbt r; height l - m \ height r \ height r \ height l + m + 1 \ \ hbt(balR l a r)"
by(auto simp: balR_def node_def max_def split!: if_splits tree.split)
text\<open>Insertion maintains @{const hbt}. Requires simultaneous proof.\<close>
theorem hbt_insert:
"hbt t \ hbt(insert x t)"
"hbt t \ height (insert x t) \ {height t, height t + 1}"
proof (induction t rule: tree2_induct)
case (Node l a _ r)
case 1
show ?case
proof(cases "x = a")
case True with Node 1 show ?thesis by simp
next
case False
show ?thesis
proof(cases "x)
case True with 1 Node(1,2) show ?thesis by (auto intro!: hbt_balL)
next
case False with 1 Node(3,4) \<open>x\<noteq>a\<close> show ?thesis by (auto intro!: hbt_balR)
qed
qed
case 2
show ?case
proof(cases "x = a")
case True with 2 show ?thesis by simp
next
case False
show ?thesis
proof(cases "x)
case True
show ?thesis
proof(cases "height (insert x l) = height r + m + 1")
case False with 2 Node(1,2) \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
next
case True
hence "(height (balL (insert x l) a r) = height r + m + 1) \
(height (balL (insert x l) a r) = height r + m + 2)" (is "?A \<or> ?B")
using 2 Node(1,2) height_balL[OF _ _ True] by simp
thus ?thesis
proof
assume ?A with 2 Node(2) True \<open>x < a\<close> show ?thesis by (auto)
next
assume ?B with 2 Node(2) True \<open>x < a\<close> show ?thesis by (simp) arith
qed
qed
next
case False
show ?thesis
proof(cases "height (insert x r) = height l + m + 1")
case False with 2 Node(3,4) \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
next
case True
hence "(height (balR l a (insert x r)) = height l + m + 1) \
(height (balR l a (insert x r)) = height l + m + 2)" (is "?A \<or> ?B")
using Node 2 height_balR[OF _ _ True] by simp
thus ?thesis
proof
assume ?A with 2 Node(4) True \<open>\<not>x < a\<close> show ?thesis by (auto)
next
assume ?B with 2 Node(4) True \<open>\<not>x < a\<close> show ?thesis by (simp) arith
qed
qed
qed
qed
qed simp_all
text \<open>Now an automatic proof without lemmas:\<close>
theorem hbt_insert_auto: "hbt t \
hbt(insert x t) \<and> height (insert x t) \<in> {height t, height t + 1}"
apply (induction t rule: tree2_induct)
(* if you want to save a few secs: apply (auto split!: if_split) *)
apply (auto simp: balL_def balR_def node_def max_absorb1 max_absorb2 split!: if_split tree.split)
done
subsubsection \<open>Deletion maintains balance\<close>
lemma hbt_split_max:
"\ hbt t; t \ Leaf \ \
hbt (fst (split_max t)) \<and>
height t \<in> {height(fst (split_max t)), height(fst (split_max t)) + 1}"
by(induct t rule: split_max_induct)
(auto simp: balL_def node_def max_absorb2 split!: prod.split if_split tree.split)
text\<open>Deletion maintains @{const hbt}:\<close>
theorem hbt_delete:
"hbt t \ hbt(delete x t)"
"hbt t \ height t \ {height (delete x t), height (delete x t) + 1}"
proof (induct t rule: tree2_induct)
case (Node l a n r)
case 1
thus ?case
using Node hbt_split_max[of l] by (auto intro!: hbt_balL hbt_balR split: prod.split)
case 2
show ?case
proof(cases "x = a")
case True then show ?thesis using 1 hbt_split_max[of l]
by(auto simp: balR_def max_absorb2 split!: if_splits prod.split tree.split)
next
case False
show ?thesis
proof(cases "x)
case True
show ?thesis
proof(cases "height r = height (delete x l) + m + 1")
case False with Node 1 \<open>x < a\<close> show ?thesis by(auto simp: balR_def)
next
case True
hence "(height (balR (delete x l) a r) = height (delete x l) + m + 1) \
height (balR (delete x l) a r) = height (delete x l) + m + 2" (is "?A \<or> ?B")
using Node 2height_balR[OF _ _ True] by simp
thus ?thesis
proof
assume ?A with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def split!: if_splits)
next
assume ?B with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def split!: if_splits)
qed
qed
next
case False
show ?thesis
proof(cases "height l = height (delete x r) + m + 1")
case False with Node 1 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> show ?thesis by(auto simp: balL_def)
next
case True
hence "(height (balL l a (delete x r)) = height (delete x r) + m + 1) \
height (balL l a (delete x r)) = height (delete x r) + m + 2" (is "?A \<or> ?B")
using Node 2 height_balL[OF _ _ True] by simp
thus ?thesis
proof
assume ?A with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def split: if_splits)
next
assume ?B with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def split: if_splits)
qed
qed
qed
qed
qed simp_all
text \<open>A more automatic proof.
Complete automation as for insertion seems hard due to resource requirements.\<close>
theorem hbt_delete_auto:
"hbt t \ hbt(delete x t)"
"hbt t \ height t \ {height (delete x t), height (delete x t) + 1}"
proof (induct t rule: tree2_induct)
case (Node l a n r)
case 1
thus ?case
using Node hbt_split_max[of l] by (auto intro!: hbt_balL hbt_balR split: prod.split)
case 2
show ?case
proof(cases "x = a")
case True thus ?thesis
using 2 hbt_split_max[of l]
by(auto simp: balR_def max_absorb2 split!: if_splits prod.split tree.split)
next
case False thus ?thesis
using height_balL[of l "delete x r" a] height_balR[of "delete x l" r a] 2 Node
by(auto simp: balL_def balR_def split!: if_split)
qed
qed simp_all
subsection "Overall correctness"
interpretation S: Set_by_Ordered
where empty = empty and isin = isin and insert = insert and delete = delete
and inorder = inorder and inv = hbt
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: hbt_insert(1))
next
case 7 thus ?case by (simp add: hbt_delete(1))
qed
end
end
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