(* Author: Tobias Nipkow *)
section \<open>Leftist Heap\<close>
theory Leftist_Heap
imports
"HOL-Library.Pattern_Aliases"
Tree2
Priority_Queue_Specs
Complex_Main
begin
fun mset_tree :: "('a,'b) tree \<Rightarrow> 'a multiset" where
"mset_tree Leaf = {#}" |
"mset_tree (Node l a _ r) = {#a#} + mset_tree l + mset_tree r"
type_synonym 'a lheap = "('a,nat)tree"
fun rank :: "'a lheap \<Rightarrow> nat" where
"rank Leaf = 0" |
"rank (Node _ _ _ r) = rank r + 1"
fun rk :: "'a lheap \<Rightarrow> nat" where
"rk Leaf = 0" |
"rk (Node _ _ n _) = n"
text\<open>The invariants:\<close>
fun (in linorder) heap :: "('a,'b) tree \<Rightarrow> bool" where
"heap Leaf = True" |
"heap (Node l m _ r) =
(heap l \<and> heap r \<and> (\<forall>x \<in> set_mset(mset_tree l + mset_tree r). m \<le> x))"
fun ltree :: "'a lheap \<Rightarrow> bool" where
"ltree Leaf = True" |
"ltree (Node l a n r) =
(n = rank r + 1 \<and> rank l \<ge> rank r \<and> ltree l & ltree r)"
definition node :: "'a lheap \<Rightarrow> 'a \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
"node l a r =
(let rl = rk l; rr = rk r
in if rl \<ge> rr then Node l a (rr+1) r else Node r a (rl+1) l)"
fun get_min :: "'a lheap \<Rightarrow> 'a" where
"get_min(Node l a n r) = a"
text \<open>For function \<open>merge\<close>:\<close>
unbundle pattern_aliases
fun merge :: "'a::ord lheap \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
"merge Leaf t2 = t2" |
"merge t1 Leaf = t1" |
"merge (Node l1 a1 n1 r1 =: t1) (Node l2 a2 n2 r2 =: t2) =
(if a1 \<le> a2 then node l1 a1 (merge r1 t2)
else node l2 a2 (merge t1 r2))"
lemma merge_code: "merge t1 t2 = (case (t1,t2) of
(Leaf, _) \<Rightarrow> t2 |
(_, Leaf) \<Rightarrow> t1 |
(Node l1 a1 n1 r1, Node l2 a2 n2 r2) \<Rightarrow>
if a1 \<le> a2 then node l1 a1 (merge r1 t2) else node l2 a2 (merge t1 r2))"
by(induction t1 t2 rule: merge.induct) (simp_all split: tree.split)
hide_const (open) insert
definition insert :: "'a::ord \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
"insert x t = merge (Node Leaf x 1 Leaf) t"
fun del_min :: "'a::ord lheap \<Rightarrow> 'a lheap" where
"del_min Leaf = Leaf" |
"del_min (Node l x n r) = merge l r"
subsection "Lemmas"
lemma mset_tree_empty: "mset_tree t = {#} \<longleftrightarrow> t = Leaf"
by(cases t) auto
lemma rk_eq_rank[simp]: "ltree t \<Longrightarrow> rk t = rank t"
by(cases t) auto
lemma ltree_node: "ltree (node l a r) \<longleftrightarrow> ltree l \<and> ltree r"
by(auto simp add: node_def)
lemma heap_node: "heap (node l a r) \<longleftrightarrow>
heap l \<and> heap r \<and> (\<forall>x \<in> set_mset(mset_tree l + mset_tree r). a \<le> x)"
by(auto simp add: node_def)
subsection "Functional Correctness"
lemma mset_merge: "mset_tree (merge h1 h2) = mset_tree h1 + mset_tree h2"
by (induction h1 h2 rule: merge.induct) (auto simp add: node_def ac_simps)
lemma mset_insert: "mset_tree (insert x t) = mset_tree t + {#x#}"
by (auto simp add: insert_def mset_merge)
lemma get_min: "\<lbrakk> heap h; h \<noteq> Leaf \<rbrakk> \<Longrightarrow> get_min h = Min_mset (mset_tree h)"
by (induction h) (auto simp add: eq_Min_iff)
lemma mset_del_min: "mset_tree (del_min h) = mset_tree h - {# get_min h #}"
by (cases h) (auto simp: mset_merge)
lemma ltree_merge: "\<lbrakk> ltree l; ltree r \<rbrakk> \<Longrightarrow> ltree (merge l r)"
proof(induction l r rule: merge.induct)
case (3 l1 a1 n1 r1 l2 a2 n2 r2)
show ?case (is "ltree(merge ?t1 ?t2)")
proof cases
assume "a1 \<le> a2"
hence "ltree (merge ?t1 ?t2) = ltree (node l1 a1 (merge r1 ?t2))" by simp
also have "\<dots> = (ltree l1 \<and> ltree(merge r1 ?t2))"
by(simp add: ltree_node)
also have "..." using "3.prems" "3.IH"(1)[OF \<open>a1 \<le> a2\<close>] by (simp)
finally show ?thesis .
next (* analogous but automatic *)
assume "\<not> a1 \<le> a2"
thus ?thesis using 3 by(simp)(auto simp: ltree_node)
qed
qed simp_all
lemma heap_merge: "\<lbrakk> heap l; heap r \<rbrakk> \<Longrightarrow> heap (merge l r)"
proof(induction l r rule: merge.induct)
case 3 thus ?case by(auto simp: heap_node mset_merge ball_Un)
qed simp_all
lemma ltree_insert: "ltree t \<Longrightarrow> ltree(insert x t)"
by(simp add: insert_def ltree_merge del: merge.simps split: tree.split)
lemma heap_insert: "heap t \<Longrightarrow> heap(insert x t)"
by(simp add: insert_def heap_merge del: merge.simps split: tree.split)
lemma ltree_del_min: "ltree t \<Longrightarrow> ltree(del_min t)"
by(cases t)(auto simp add: ltree_merge simp del: merge.simps)
lemma heap_del_min: "heap t \<Longrightarrow> heap(del_min t)"
by(cases t)(auto simp add: heap_merge simp del: merge.simps)
text \<open>Last step of functional correctness proof: combine all the above lemmas
to show that leftist heaps satisfy the specification of priority queues with merge.\<close>
interpretation lheap: Priority_Queue_Merge
where empty = Leaf and is_empty = "\<lambda>h. h = Leaf"
and insert = insert and del_min = del_min
and get_min = get_min and merge = merge
and invar = "\<lambda>h. heap h \<and> ltree h" and mset = mset_tree
proof(standard, goal_cases)
case 1 show ?case by simp
next
case (2 q) show ?case by (cases q) auto
next
case 3 show ?case by(rule mset_insert)
next
case 4 show ?case by(rule mset_del_min)
next
case 5 thus ?case by(simp add: get_min mset_tree_empty)
next
case 6 thus ?case by(simp)
next
case 7 thus ?case by(simp add: heap_insert ltree_insert)
next
case 8 thus ?case by(simp add: heap_del_min ltree_del_min)
next
case 9 thus ?case by (simp add: mset_merge)
next
case 10 thus ?case by (simp add: heap_merge ltree_merge)
qed
subsection "Complexity"
lemma pow2_rank_size1: "ltree t \<Longrightarrow> 2 ^ rank t \<le> size1 t"
proof(induction t)
case Leaf show ?case by simp
next
case (Node l a n r)
hence "rank r \<le> rank l" by simp
hence *: "(2::nat) ^ rank r \<le> 2 ^ rank l" by simp
have "(2::nat) ^ rank \<langle>l, a, n, r\<rangle> = 2 ^ rank r + 2 ^ rank r"
by(simp add: mult_2)
also have "\<dots> \<le> size1 l + size1 r"
using Node * by (simp del: power_increasing_iff)
also have "\<dots> = size1 \<langle>l, a, n, r\<rangle>" by simp
finally show ?case .
qed
text\<open>Explicit termination argument: sum of sizes\<close>
fun t_merge :: "'a::ord lheap \<Rightarrow> 'a lheap \<Rightarrow> nat" where
"t_merge Leaf t2 = 1" |
"t_merge t2 Leaf = 1" |
"t_merge (Node l1 a1 n1 r1 =: t1) (Node l2 a2 n2 r2 =: t2) =
(if a1 \<le> a2 then 1 + t_merge r1 t2
else 1 + t_merge t1 r2)"
definition t_insert :: "'a::ord \<Rightarrow> 'a lheap \<Rightarrow> nat" where
"t_insert x t = t_merge (Node Leaf x 1 Leaf) t"
fun t_del_min :: "'a::ord lheap \<Rightarrow> nat" where
"t_del_min Leaf = 1" |
"t_del_min (Node l a n r) = t_merge l r"
lemma t_merge_rank: "t_merge l r \<le> rank l + rank r + 1"
proof(induction l r rule: merge.induct)
case 3 thus ?case by(simp)
qed simp_all
corollary t_merge_log: assumes "ltree l" "ltree r"
shows "t_merge l r \<le> log 2 (size1 l) + log 2 (size1 r) + 1"
using le_log2_of_power[OF pow2_rank_size1[OF assms(1)]]
le_log2_of_power[OF pow2_rank_size1[OF assms(2)]] t_merge_rank[of l r]
by linarith
corollary t_insert_log: "ltree t \<Longrightarrow> t_insert x t \<le> log 2 (size1 t) + 2"
using t_merge_log[of "Node Leaf x 1 Leaf" t]
by(simp add: t_insert_def split: tree.split)
(* FIXME mv ? *)
lemma ld_ld_1_less:
assumes "x > 0" "y > 0" shows "log 2 x + log 2 y + 1 < 2 * log 2 (x+y)"
proof -
have "2 powr (log 2 x + log 2 y + 1) = 2*x*y"
using assms by(simp add: powr_add)
also have "\<dots> < (x+y)^2" using assms
by(simp add: numeral_eq_Suc algebra_simps add_pos_pos)
also have "\<dots> = 2 powr (2 * log 2 (x+y))"
using assms by(simp add: powr_add log_powr[symmetric])
finally show ?thesis by simp
qed
corollary t_del_min_log: assumes "ltree t"
shows "t_del_min t \<le> 2 * log 2 (size1 t) + 1"
proof(cases t)
case Leaf thus ?thesis using assms by simp
next
case [simp]: (Node t1 _ _ t2)
have "t_del_min t = t_merge t1 t2" by simp
also have "\<dots> \<le> log 2 (size1 t1) + log 2 (size1 t2) + 1"
using \<open>ltree t\<close> by (auto simp: t_merge_log simp del: t_merge.simps)
also have "\<dots> \<le> 2 * log 2 (size1 t) + 1"
using ld_ld_1_less[of "size1 t1" "size1 t2"] by (simp)
finally show ?thesis .
qed
end