(* Author: Tobias Nipkow *)
section "AVL Tree Implementation of Maps"
theory AVL_Map
imports
AVL_Set
Lookup2
begin
fun update :: "'a::linorder \ 'b \ ('a*'b) tree_ht \ ('a*'b) tree_ht" where
"update x y Leaf = Node Leaf ((x,y), 1) Leaf" |
"update x y (Node l ((a,b), h) r) = (case cmp x a of
EQ \<Rightarrow> Node l ((x,y), h) r |
LT \<Rightarrow> balL (update x y l) (a,b) r |
GT \<Rightarrow> balR l (a,b) (update x y r))"
fun delete :: "'a::linorder \ ('a*'b) tree_ht \ ('a*'b) tree_ht" where
"delete _ Leaf = Leaf" |
"delete x (Node l ((a,b), h) r) = (case cmp x a of
EQ \<Rightarrow> if l = Leaf then r
else let (l', ab') = split_max l in balR l' ab' r |
LT \<Rightarrow> balR (delete x l) (a,b) r |
GT \<Rightarrow> balL l (a,b) (delete x r))"
subsection \<open>Functional Correctness\<close>
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_balL inorder_balR)
theorem inorder_delete:
"sorted1(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>AVL invariants\<close>
subsubsection \<open>Insertion maintains AVL balance\<close>
theorem avl_update:
assumes "avl t"
shows "avl(update x y t)"
"(height (update x y t) = height t \ height (update x y t) = height t + 1)"
using assms
proof (induction x y t rule: update.induct)
case eq2: (2 x y l a b h r)
case 1
show ?case
proof(cases "x = a")
case True with eq2 1 show ?thesis by simp
next
case False
with eq2 1 show ?thesis
proof(cases "x)
case True with eq2 1 show ?thesis by (auto intro!: avl_balL)
next
case False with eq2 1 \<open>x\<noteq>a\<close> show ?thesis by (auto intro!: avl_balR)
qed
qed
case 2
show ?case
proof(cases "x = a")
case True with eq2 1 show ?thesis by simp
next
case False
show ?thesis
proof(cases "x)
case True
show ?thesis
proof(cases "height (update x y l) = height r + 2")
case False with eq2 2 \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
next
case True
hence "(height (balL (update x y l) (a,b) r) = height r + 2) \
(height (balL (update x y l) (a,b) r) = height r + 3)" (is "?A \<or> ?B")
using eq2 2 \<open>x<a\<close> height_balL[OF _ _ True] by simp
thus ?thesis
proof
assume ?A with 2 \<open>x < a\<close> show ?thesis by (auto)
next
assume ?B with True 1 eq2(2) \<open>x < a\<close> show ?thesis by (simp) arith
qed
qed
next
case False
show ?thesis
proof(cases "height (update x y r) = height l + 2")
case False with eq2 2 \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
next
case True
hence "(height (balR l (a,b) (update x y r)) = height l + 2) \
(height (balR l (a,b) (update x y r)) = height l + 3)" (is "?A \<or> ?B")
using eq2 2 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> height_balR[OF _ _ True] by simp
thus ?thesis
proof
assume ?A with 2 \<open>\<not>x < a\<close> show ?thesis by (auto)
next
assume ?B with True 1 eq2(4) \<open>\<not>x < a\<close> show ?thesis by (simp) arith
qed
qed
qed
qed
qed simp_all
subsubsection \<open>Deletion maintains AVL balance\<close>
theorem avl_delete:
assumes "avl t"
shows "avl(delete x t)" and "height t = (height (delete x t)) \ height t = height (delete x t) + 1"
using assms
proof (induct t rule: tree2_induct)
case (Node l ab h r)
obtain a b where [simp]: "ab = (a,b)" by fastforce
case 1
show ?case
proof(cases "x = a")
case True with Node 1 show ?thesis
using avl_split_max[of l] by (auto intro!: avl_balR split: prod.split)
next
case False
show ?thesis
proof(cases "x)
case True with Node 1 show ?thesis by (auto intro!: avl_balR)
next
case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto intro!: avl_balL)
qed
qed
case 2
show ?case
proof(cases "x = a")
case True then show ?thesis using 1 avl_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) + 2")
case False with Node 1 \<open>x < a\<close> show ?thesis by(auto simp: balR_def)
next
case True
thus ?thesis using height_balR[OF _ _ True, of ab] 2 Node(1,2) \<open>x < a\<close> by simp linarith
qed
next
case False
show ?thesis
proof(cases "height l = height (delete x r) + 2")
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
thus ?thesis
using height_balL[OF _ _ True, of ab] 2 Node(3,4) \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> by auto
qed
qed
qed
qed simp_all
interpretation M: Map_by_Ordered
where empty = empty and lookup = lookup and update = update and delete = delete
and inorder = inorder and inv = avl
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 show ?case by (simp add: empty_def)
next
case 6 thus ?case by(simp add: avl_update(1))
next
case 7 thus ?case by(simp add: avl_delete(1))
qed
end
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