(* Author: Tobias Nipkow *)
section "Join-Based BST Implementation of Sets"
theory Set2_BST_Join
imports
"HOL-Library.Tree"
Cmp
Set_Specs
begin
text \<open>This theory implements the set operations \<open>insert\<close>, \<open>delete\<close>,
\<open>union\<close>, \<open>inter\<close>section and \<open>diff\<close>erence. The implementation is based on binary search trees.
All operations are reduced to a single operation \<open>join l x r\<close> that joins two BSTs \<open>l\<close> and \<open>r\<close>
and an element \<open>x\<close> such that \<open>l < x < r\<close>.
This theory illustrates the idea but is not suitable for an efficient implementation where
\<open>join\<close> balances the tree in some manner because type @{typ "'a tree"} in theory @{theory Tree}
has no additional fields for recording balance information. See theory \<open>Set2_BST2_Join\<close> for that.\<close>
text \<open>Function \<open>isin\<close> can also be expressed via \<open>join\<close> but this is more direct:\<close>
fun isin :: "('a::linorder) tree \<Rightarrow> 'a \<Rightarrow> bool" where
"isin Leaf x = False" |
"isin (Node l a r) x =
(case cmp x a of
LT \<Rightarrow> isin l x |
EQ \<Rightarrow> True |
GT \<Rightarrow> isin r x)"
lemma isin_set: "bst t \<Longrightarrow> isin t x = (x \<in> set_tree t)"
by (induction t) (auto)
locale Set2_BST_Join =
fixes join :: "('a::linorder) tree \<Rightarrow> 'a \<Rightarrow> 'a tree \<Rightarrow> 'a tree"
assumes set_join: "set_tree (join t1 x t2) = Set.insert x (set_tree t1 \<union> set_tree t2)"
assumes bst_join:
"\<lbrakk> bst l; bst r; \<forall>x \<in> set_tree l. x < k; \<forall>y \<in> set_tree r. k < y \<rbrakk>
\<Longrightarrow> bst (join l k r)"
fixes inv :: "'a tree \<Rightarrow> bool"
assumes inv_Leaf: "inv \<langle>\<rangle>"
assumes inv_join: "\<lbrakk> inv l; inv r \<rbrakk> \<Longrightarrow> inv (join l k r)"
assumes inv_Node: "\<lbrakk> inv (Node l x r) \<rbrakk> \<Longrightarrow> inv l \<and> inv r"
begin
declare set_join [simp]
subsection "\<open>split_min\<close>"
fun split_min :: "'a tree \<Rightarrow> 'a \<times> 'a tree" where
"split_min (Node l x r) =
(if l = Leaf then (x,r) else let (m,l') = split_min l in (m, join l' x r))"
lemma split_min_set:
"\<lbrakk> split_min t = (x,t'); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
x \<in> set_tree t \<and> set_tree t = Set.insert x (set_tree t')"
proof(induction t arbitrary: t')
case Node thus ?case by(auto split: prod.splits if_splits)
next
case Leaf thus ?case by simp
qed
lemma split_min_bst:
"\<lbrakk> split_min t = (x,t'); bst t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> bst t' \<and> (\<forall>x' \<in> set_tree t'. x < x')"
proof(induction t arbitrary: t')
case Node thus ?case by(fastforce simp: split_min_set bst_join split: prod.splits if_splits)
next
case Leaf thus ?case by simp
qed
lemma split_min_inv:
"\<lbrakk> split_min t = (x,t'); inv t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> inv t'"
proof(induction t arbitrary: t')
case Node thus ?case by(auto simp: inv_join split: prod.splits if_splits dest: inv_Node)
next
case Leaf thus ?case by simp
qed
subsection "\<open>join2\<close>"
definition join2 :: "'a tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"join2 l r = (if r = Leaf then l else let (x,r') = split_min r in join l x r')"
lemma set_join2[simp]: "set_tree (join2 l r) = set_tree l \<union> set_tree r"
by(simp add: join2_def split_min_set split: prod.split)
lemma bst_join2: "\<lbrakk> bst l; bst r; \<forall>x \<in> set_tree l. \<forall>y \<in> set_tree r. x < y \<rbrakk>
\<Longrightarrow> bst (join2 l r)"
by(simp add: join2_def bst_join split_min_set split_min_bst split: prod.split)
lemma inv_join2: "\<lbrakk> inv l; inv r \<rbrakk> \<Longrightarrow> inv (join2 l r)"
by(simp add: join2_def inv_join split_min_set split_min_inv split: prod.split)
subsection "\<open>split\<close>"
fun split :: "'a tree \<Rightarrow> 'a \<Rightarrow> 'a tree \<times> bool \<times> 'a tree" where
"split Leaf k = (Leaf, False, Leaf)" |
"split (Node l a r) k =
(case cmp k a of
LT \<Rightarrow> let (l1,b,l2) = split l k in (l1, b, join l2 a r) |
GT \<Rightarrow> let (r1,b,r2) = split r k in (join l a r1, b, r2) |
EQ \<Rightarrow> (l, True, r))"
lemma split: "split t k = (l,kin,r) \<Longrightarrow> bst t \<Longrightarrow>
set_tree l = {x \<in> set_tree t. x < k} \<and> set_tree r = {x \<in> set_tree t. k < x}
\<and> (kin = (k \<in> set_tree t)) \<and> bst l \<and> bst r"
proof(induction t arbitrary: l kin r)
case Leaf thus ?case by simp
next
case Node thus ?case by(force split!: prod.splits if_splits intro!: bst_join)
qed
lemma split_inv: "split t k = (l,kin,r) \<Longrightarrow> inv t \<Longrightarrow> inv l \<and> inv r"
proof(induction t arbitrary: l kin r)
case Leaf thus ?case by simp
next
case Node
thus ?case by(force simp: inv_join split!: prod.splits if_splits dest!: inv_Node)
qed
declare split.simps[simp del]
subsection "\<open>insert\<close>"
definition insert :: "'a \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"insert k t = (let (l,_,r) = split t k in join l k r)"
lemma set_tree_insert: "bst t \<Longrightarrow> set_tree (insert x t) = Set.insert x (set_tree t)"
by(auto simp add: insert_def split split: prod.split)
lemma bst_insert: "bst t \<Longrightarrow> bst (insert x t)"
by(auto simp add: insert_def bst_join dest: split split: prod.split)
lemma inv_insert: "inv t \<Longrightarrow> inv (insert x t)"
by(force simp: insert_def inv_join dest: split_inv split: prod.split)
subsection "\<open>delete\<close>"
definition delete :: "'a \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"delete k t = (let (l,_,r) = split t k in join2 l r)"
lemma set_tree_delete: "bst t \<Longrightarrow> set_tree (delete k t) = set_tree t - {k}"
by(auto simp: delete_def split split: prod.split)
lemma bst_delete: "bst t \<Longrightarrow> bst (delete x t)"
by(force simp add: delete_def intro: bst_join2 dest: split split: prod.split)
lemma inv_delete: "inv t \<Longrightarrow> inv (delete x t)"
by(force simp: delete_def inv_join2 dest: split_inv split: prod.split)
subsection "\<open>union\<close>"
fun union :: "'a tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"union t1 t2 =
(if t1 = Leaf then t2 else
if t2 = Leaf then t1 else
case t1 of Node l1 k r1 \<Rightarrow>
let (l2,_ ,r2) = split t2 k;
l' = union l1 l2; r' = union r1 r2
in join l' k r')"
declare union.simps [simp del]
lemma set_tree_union: "bst t2 \<Longrightarrow> set_tree (union t1 t2) = set_tree t1 \<union> set_tree t2"
proof(induction t1 t2 rule: union.induct)
case (1 t1 t2)
then show ?case
by (auto simp: union.simps[of t1 t2] split split: tree.split prod.split)
qed
lemma bst_union: "\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> bst (union t1 t2)"
proof(induction t1 t2 rule: union.induct)
case (1 t1 t2)
thus ?case
by(fastforce simp: union.simps[of t1 t2] set_tree_union split intro!: bst_join
split: tree.split prod.split)
qed
lemma inv_union: "\<lbrakk> inv t1; inv t2 \<rbrakk> \<Longrightarrow> inv (union t1 t2)"
proof(induction t1 t2 rule: union.induct)
case (1 t1 t2)
thus ?case
by(auto simp:union.simps[of t1 t2] inv_join split_inv
split!: tree.split prod.split dest: inv_Node)
qed
subsection "\<open>inter\<close>"
fun inter :: "'a tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"inter t1 t2 =
(if t1 = Leaf then Leaf else
if t2 = Leaf then Leaf else
case t1 of Node l1 k r1 \<Rightarrow>
let (l2,kin,r2) = split t2 k;
l' = inter l1 l2; r' = inter r1 r2
in if kin then join l' k r' else join2 l' r')"
declare inter.simps [simp del]
lemma set_tree_inter:
"\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> set_tree (inter t1 t2) = set_tree t1 \<inter> set_tree t2"
proof(induction t1 t2 rule: inter.induct)
case (1 t1 t2)
show ?case
proof (cases t1)
case Leaf thus ?thesis by (simp add: inter.simps)
next
case [simp]: (Node l1 k r1)
show ?thesis
proof (cases "t2 = Leaf")
case True thus ?thesis by (simp add: inter.simps)
next
case False
let ?L1 = "set_tree l1" let ?R1 = "set_tree r1"
have *: "k \<notin> ?L1 \<union> ?R1" using \<open>bst t1\<close> by (fastforce)
obtain l2 kin r2 where sp: "split t2 k = (l2,kin,r2)" using prod_cases3 by blast
let ?L2 = "set_tree l2" let ?R2 = "set_tree r2" let ?K = "if kin then {k} else {}"
have t2: "set_tree t2 = ?L2 \<union> ?R2 \<union> ?K" and
**: "?L2 \<inter> ?R2 = {}" "k \<notin> ?L2 \<union> ?R2" "?L1 \<inter> ?R2 = {}" "?L2 \<inter> ?R1 = {}"
using split[OF sp] \<open>bst t1\<close> \<open>bst t2\<close> by (force, force, force, force, force)
have IHl: "set_tree (inter l1 l2) = set_tree l1 \<inter> set_tree l2"
using "1.IH"(1)[OF _ False _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp
have IHr: "set_tree (inter r1 r2) = set_tree r1 \<inter> set_tree r2"
using "1.IH"(2)[OF _ False _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp
have "set_tree t1 \<inter> set_tree t2 = (?L1 \<union> ?R1 \<union> {k}) \<inter> (?L2 \<union> ?R2 \<union> ?K)"
by(simp add: t2)
also have "\<dots> = (?L1 \<inter> ?L2) \<union> (?R1 \<inter> ?R2) \<union> ?K"
using * ** by auto
also have "\<dots> = set_tree (inter t1 t2)"
using IHl IHr sp inter.simps[of t1 t2] False by(simp)
finally show ?thesis by simp
qed
qed
qed
lemma bst_inter: "\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> bst (inter t1 t2)"
proof(induction t1 t2 rule: inter.induct)
case (1 t1 t2)
thus ?case
by(fastforce simp: inter.simps[of t1 t2] set_tree_inter split Let_def
intro!: bst_join bst_join2 split: tree.split prod.split)
qed
lemma inv_inter: "\<lbrakk> inv t1; inv t2 \<rbrakk> \<Longrightarrow> inv (inter t1 t2)"
proof(induction t1 t2 rule: inter.induct)
case (1 t1 t2)
thus ?case
by(auto simp: inter.simps[of t1 t2] inv_join inv_join2 split_inv Let_def
split!: tree.split prod.split dest: inv_Node)
qed
subsection "\<open>diff\<close>"
fun diff :: "'a tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"diff t1 t2 =
(if t1 = Leaf then Leaf else
if t2 = Leaf then t1 else
case t2 of Node l2 k r2 \<Rightarrow>
let (l1,_,r1) = split t1 k;
l' = diff l1 l2; r' = diff r1 r2
in join2 l' r')"
declare diff.simps [simp del]
lemma set_tree_diff:
"\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> set_tree (diff t1 t2) = set_tree t1 - set_tree t2"
proof(induction t1 t2 rule: diff.induct)
case (1 t1 t2)
show ?case
proof (cases t2)
case Leaf thus ?thesis by (simp add: diff.simps)
next
case [simp]: (Node l2 k r2)
show ?thesis
proof (cases "t1 = Leaf")
case True thus ?thesis by (simp add: diff.simps)
next
case False
let ?L2 = "set_tree l2" let ?R2 = "set_tree r2"
obtain l1 kin r1 where sp: "split t1 k = (l1,kin,r1)" using prod_cases3 by blast
let ?L1 = "set_tree l1" let ?R1 = "set_tree r1" let ?K = "if kin then {k} else {}"
have t1: "set_tree t1 = ?L1 \<union> ?R1 \<union> ?K" and
**: "k \<notin> ?L1 \<union> ?R1" "?L1 \<inter> ?R2 = {}" "?L2 \<inter> ?R1 = {}"
using split[OF sp] \<open>bst t1\<close> \<open>bst t2\<close> by (force, force, force, force)
have IHl: "set_tree (diff l1 l2) = set_tree l1 - set_tree l2"
using "1.IH"(1)[OF False _ _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp
have IHr: "set_tree (diff r1 r2) = set_tree r1 - set_tree r2"
using "1.IH"(2)[OF False _ _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp
have "set_tree t1 - set_tree t2 = (?L1 \<union> ?R1) - (?L2 \<union> ?R2 \<union> {k})"
by(simp add: t1)
also have "\<dots> = (?L1 - ?L2) \<union> (?R1 - ?R2)"
using ** by auto
also have "\<dots> = set_tree (diff t1 t2)"
using IHl IHr sp diff.simps[of t1 t2] False by(simp)
finally show ?thesis by simp
qed
qed
qed
lemma bst_diff: "\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> bst (diff t1 t2)"
proof(induction t1 t2 rule: diff.induct)
case (1 t1 t2)
thus ?case
by(fastforce simp: diff.simps[of t1 t2] set_tree_diff split Let_def
intro!: bst_join bst_join2 split: tree.split prod.split)
qed
lemma inv_diff: "\<lbrakk> inv t1; inv t2 \<rbrakk> \<Longrightarrow> inv (diff t1 t2)"
proof(induction t1 t2 rule: diff.induct)
case (1 t1 t2)
thus ?case
by(auto simp: diff.simps[of t1 t2] inv_join inv_join2 split_inv Let_def
split!: tree.split prod.split dest: inv_Node)
qed
text \<open>Locale @{locale Set2_BST_Join} implements locale @{locale Set2}:\<close>
sublocale Set2
where empty = Leaf and isin = isin and insert = insert and delete = delete
and union = union and inter = inter and diff = diff
and set = set_tree and invar = "\<lambda>t. bst t \<and> inv t"
proof (standard, goal_cases)
case 1 show ?case by simp
next
case 2 thus ?case by (simp add: isin_set)
next
case 3 thus ?case by (simp add: set_tree_insert)
next
case 4 thus ?case by (simp add: set_tree_delete)
next
case 5 thus ?case by (simp add: inv_Leaf)
next
case 6 thus ?case by (simp add: inv_insert bst_insert)
next
case 7 thus ?case by (simp add: inv_delete bst_delete)
next
case 8 thus ?case by (simp add: set_tree_union)
next
case 9 thus ?case by (simp add: set_tree_inter)
next
case 10 thus ?case by (simp add: set_tree_diff)
next
case 11 thus ?case by (simp add: bst_union inv_union)
next
case 12 thus ?case by (simp add: bst_inter inv_inter)
next
case 13 thus ?case by (simp add: bst_diff inv_diff)
qed
end (* Set2_BST_Join *)
text \<open>Interpretation of @{locale Set2_BST_Join} with unbalanced binary trees:\<close>
interpretation Set2_BST_Join where join = Node and inv = "\<lambda>t. True"
proof (standard, goal_cases)
qed auto
end