--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Interval_Tree.thy Mon Feb 03 16:49:44 2020 +0100
@@ -0,0 +1,300 @@
+(* 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
--- a/src/HOL/ROOT Sat Feb 01 19:10:40 2020 +0100
+++ b/src/HOL/ROOT Mon Feb 03 16:49:44 2020 +0100
@@ -236,6 +236,7 @@
Sorting
Balance
Tree_Map
+ Interval_Tree
AVL_Map
RBT_Set2
RBT_Map