(* Author: Bohua Zhan, with modifications by Tobias Nipkow *)
section \<open>Interval Trees\<close>
theory Interval_Tree
imports
"HOL-Data_Structures.Cmp"
"HOL-Data_Structures.List_Ins_Del"
"HOL-Data_Structures.Isin2"
"HOL-Data_Structures.Set_Specs"
begin
subsection \<open>Intervals\<close>
text \<open>The following definition of intervals uses the \<^bold>\<open>typedef\<close> command
to define the type of non-empty intervals as a subset of the type of pairs \<open>p\<close>
where @{prop "fst p \<le> snd p"}:\<close>
typedef (overloaded) 'a::linorder ivl =
"{p :: 'a \<times> 'a. fst p \<le> snd p}" by auto
text \<open>More precisely, @{typ "'a ivl"} is isomorphic with that subset via the function
@{const Rep_ivl}. Hence the basic interval properties are not immediate but
need simple proofs:\<close>
definition low :: "'a::linorder ivl \<Rightarrow> 'a" where
"low p = fst (Rep_ivl p)"
definition high :: "'a::linorder ivl \<Rightarrow> 'a" where
"high p = snd (Rep_ivl p)"
lemma ivl_is_interval: "low p \<le> high p"
by (metis Rep_ivl high_def low_def mem_Collect_eq)
lemma ivl_inj: "low p = low q \<Longrightarrow> high p = high q \<Longrightarrow> p = q"
by (metis Rep_ivl_inverse high_def low_def prod_eqI)
text \<open>Now we can forget how exactly intervals were defined.\<close>
instantiation ivl :: (linorder) linorder begin
definition int_less: "(x < y) = (low x < low y | (low x = low y \<and> high x < high y))"
definition int_less_eq: "(x \<le> y) = (low x < low y | (low x = low y \<and> high x \<le> high y))"
instance proof
fix x y z :: "'a ivl"
show a: "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
using int_less int_less_eq by force
show b: "x \<le> x"
by (simp add: int_less_eq)
show c: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
by (smt int_less_eq dual_order.trans less_trans)
show d: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
using int_less_eq a ivl_inj int_less by fastforce
show e: "x \<le> y \<or> y \<le> x"
by (meson int_less_eq leI not_less_iff_gr_or_eq)
qed end
definition overlap :: "('a::linorder) ivl \<Rightarrow> 'a ivl \<Rightarrow> bool" where
"overlap x y \<longleftrightarrow> (high x \<ge> low y \<and> high y \<ge> low x)"
definition has_overlap :: "('a::linorder) ivl set \<Rightarrow> 'a ivl \<Rightarrow> bool" where
"has_overlap S y \<longleftrightarrow> (\<exists>x\<in>S. overlap x y)"
subsection \<open>Interval Trees\<close>
type_synonym 'a ivl_tree = "('a ivl * 'a) tree"
fun max_hi :: "('a::order_bot) ivl_tree \<Rightarrow> 'a" where
"max_hi Leaf = bot" |
"max_hi (Node _ (_,m) _) = m"
definition max3 :: "('a::linorder) ivl \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"max3 a m n = max (high a) (max m n)"
fun inv_max_hi :: "('a::{linorder,order_bot}) ivl_tree \<Rightarrow> bool" where
"inv_max_hi Leaf \<longleftrightarrow> True" |
"inv_max_hi (Node l (a, m) r) \<longleftrightarrow> (inv_max_hi l \<and> inv_max_hi r \<and> m = max3 a (max_hi l) (max_hi r))"
lemma max_hi_is_max:
"inv_max_hi t \<Longrightarrow> a \<in> set_tree t \<Longrightarrow> high a \<le> max_hi t"
by (induct t, auto simp add: max3_def max_def)
lemma max_hi_exists:
"inv_max_hi t \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> \<exists>a\<in>set_tree t. high a = max_hi t"
proof (induction t rule: tree2_induct)
case Leaf
then show ?case by auto
next
case N: (Node l v m r)
then show ?case
proof (cases l rule: tree2_cases)
case Leaf
then show ?thesis
using N.prems(1) N.IH(2) by (cases r, auto simp add: max3_def max_def le_bot)
next
case Nl: Node
then show ?thesis
proof (cases r rule: tree2_cases)
case Leaf
then show ?thesis
using N.prems(1) N.IH(1) Nl by (auto simp add: max3_def max_def le_bot)
next
case Nr: Node
obtain p1 where p1: "p1 \<in> set_tree l" "high p1 = max_hi l"
using N.IH(1) N.prems(1) Nl by auto
obtain p2 where p2: "p2 \<in> set_tree r" "high p2 = max_hi r"
using N.IH(2) N.prems(1) Nr by auto
then show ?thesis
using p1 p2 N.prems(1) by (auto simp add: max3_def max_def)
qed
qed
qed
subsection \<open>Insertion and Deletion\<close>
definition node where
[simp]: "node l a r = Node l (a, max3 a (max_hi l) (max_hi r)) r"
fun insert :: "'a::{linorder,order_bot} ivl \<Rightarrow> 'a ivl_tree \<Rightarrow> 'a ivl_tree" where
"insert x Leaf = Node Leaf (x, high x) Leaf" |
"insert x (Node l (a, m) r) =
(case cmp x a of
EQ \<Rightarrow> Node l (a, m) r |
LT \<Rightarrow> node (insert x l) a r |
GT \<Rightarrow> node l a (insert x r))"
lemma inorder_insert:
"sorted (inorder t) \<Longrightarrow> inorder (insert x t) = ins_list x (inorder t)"
by (induct t rule: tree2_induct) (auto simp: ins_list_simps)
lemma inv_max_hi_insert:
"inv_max_hi t \<Longrightarrow> inv_max_hi (insert x t)"
by (induct t rule: tree2_induct) (auto simp add: max3_def)
fun split_min :: "'a::{linorder,order_bot} ivl_tree \<Rightarrow> 'a ivl \<times> 'a ivl_tree" where
"split_min (Node l (a, m) r) =
(if l = Leaf then (a, r)
else let (x,l') = split_min l in (x, node l' a r))"
fun delete :: "'a::{linorder,order_bot} ivl \<Rightarrow> 'a ivl_tree \<Rightarrow> 'a ivl_tree" where
"delete x Leaf = Leaf" |
"delete x (Node l (a, m) r) =
(case cmp x a of
LT \<Rightarrow> node (delete x l) a r |
GT \<Rightarrow> node l a (delete x r) |
EQ \<Rightarrow> if r = Leaf then l else
let (a', r') = split_min r in node l a' r')"
lemma split_minD:
"split_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> x # inorder t' = inorder t"
by (induct t arbitrary: t' rule: split_min.induct)
(auto simp: sorted_lems split: prod.splits if_splits)
lemma inorder_delete:
"sorted (inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
by (induct t)
(auto simp: del_list_simps split_minD Let_def split: prod.splits)
lemma inv_max_hi_split_min:
"\<lbrakk> t \<noteq> Leaf; inv_max_hi t \<rbrakk> \<Longrightarrow> inv_max_hi (snd (split_min t))"
by (induct t) (auto split: prod.splits)
lemma inv_max_hi_delete:
"inv_max_hi t \<Longrightarrow> inv_max_hi (delete x t)"
apply (induct t)
apply simp
using inv_max_hi_split_min by (fastforce simp add: Let_def split: prod.splits)
subsection \<open>Search\<close>
text \<open>Does interval \<open>x\<close> overlap with any interval in the tree?\<close>
fun search :: "'a::{linorder,order_bot} ivl_tree \<Rightarrow> 'a ivl \<Rightarrow> bool" where
"search Leaf x = False" |
"search (Node l (a, m) r) x =
(if overlap x a then True
else if l \<noteq> Leaf \<and> max_hi l \<ge> low x then search l x
else search r x)"
lemma search_correct:
"inv_max_hi t \<Longrightarrow> sorted (inorder t) \<Longrightarrow> search t x = has_overlap (set_tree t) x"
proof (induction t rule: tree2_induct)
case Leaf
then show ?case by (auto simp add: has_overlap_def)
next
case (Node l a m r)
have search_l: "search l x = has_overlap (set_tree l) x"
using Node.IH(1) Node.prems by (auto simp: sorted_wrt_append)
have search_r: "search r x = has_overlap (set_tree r) x"
using Node.IH(2) Node.prems by (auto simp: sorted_wrt_append)
show ?case
proof (cases "overlap a x")
case True
thus ?thesis by (auto simp: overlap_def has_overlap_def)
next
case a_disjoint: False
then show ?thesis
proof cases
assume [simp]: "l = Leaf"
have search_eval: "search (Node l (a, m) r) x = search r x"
using a_disjoint overlap_def by auto
show ?thesis
unfolding search_eval search_r
by (auto simp add: has_overlap_def a_disjoint)
next
assume "l \<noteq> Leaf"
then show ?thesis
proof (cases "max_hi l \<ge> low x")
case max_hi_l_ge: True
have "inv_max_hi l"
using Node.prems(1) by auto
then obtain p where p: "p \<in> set_tree l" "high p = max_hi l"
using \<open>l \<noteq> Leaf\<close> max_hi_exists by auto
have search_eval: "search (Node l (a, m) r) x = search l x"
using a_disjoint \<open>l \<noteq> Leaf\<close> max_hi_l_ge by (auto simp: overlap_def)
show ?thesis
proof (cases "low p \<le> high x")
case True
have "overlap p x"
unfolding overlap_def using True p(2) max_hi_l_ge by auto
then show ?thesis
unfolding search_eval search_l
using p(1) by(auto simp: has_overlap_def overlap_def)
next
case False
have "\<not>overlap x rp" if asm: "rp \<in> set_tree r" for rp
proof -
have "p \<in> set (inorder l)" using p(1) by (simp)
moreover have "rp \<in> set (inorder r)" using asm by (simp)
ultimately have "low p \<le> low rp"
using Node(4) by(fastforce simp: sorted_wrt_append int_less)
then show ?thesis
using False by (auto simp: overlap_def)
qed
then show ?thesis
unfolding search_eval search_l
using a_disjoint by (fastforce simp: has_overlap_def overlap_def)
qed
next
case False
have search_eval: "search (Node l (a, m) r) x = search r x"
using a_disjoint False by (auto simp: overlap_def)
have "\<not>overlap x lp" if asm: "lp \<in> set_tree l" for lp
using asm False Node.prems(1) max_hi_is_max
by (fastforce simp: overlap_def)
then show ?thesis
unfolding search_eval search_r
using a_disjoint by (fastforce simp: has_overlap_def overlap_def)
qed
qed
qed
qed
definition empty :: "'a ivl_tree" where
"empty = Leaf"
subsection \<open>Specification\<close>
locale Interval_Set = Set +
fixes has_overlap :: "'t \<Rightarrow> 'a::linorder ivl \<Rightarrow> bool"
assumes set_overlap: "invar s \<Longrightarrow> has_overlap s x = Interval_Tree.has_overlap (set s) x"
interpretation S: Interval_Set
where empty = Leaf and insert = insert and delete = delete
and has_overlap = search and isin = isin and set = set_tree
and invar = "\<lambda>t. inv_max_hi t \<and> sorted (inorder t)"
proof (standard, goal_cases)
case 1
then show ?case by auto
next
case 2
then show ?case by (simp add: isin_set_inorder)
next
case 3
then show ?case by(simp add: inorder_insert set_ins_list flip: set_inorder)
next
case 4
then show ?case by(simp add: inorder_delete set_del_list flip: set_inorder)
next
case 5
then show ?case by auto
next
case 6
then show ?case by (simp add: inorder_insert inv_max_hi_insert sorted_ins_list)
next
case 7
then show ?case by (simp add: inorder_delete inv_max_hi_delete sorted_del_list)
next
case 8
then show ?case by (simp add: search_correct)
qed
end