(* Author: Peter Lammich
Tobias Nipkow (tuning)
*)
section \<open>Binomial Heap\<close>
theory Binomial_Heap
imports
"HOL-Library.Pattern_Aliases"
Complex_Main
Priority_Queue_Specs
Define_Time_Function
begin
text \<open>
We formalize the binomial heap presentation from Okasaki's book.
We show the functional correctness and complexity of all operations.
The presentation is engineered for simplicity, and most
proofs are straightforward and automatic.
\<close>
subsection \<open>Binomial Tree and Heap Datatype\<close>
datatype 'a tree = Node (rank: nat) (root: 'a) (children: "'a tree list")
type_synonym 'a trees = "'a tree list"
subsubsection \<open>Multiset of elements\<close>
fun mset_tree :: "'a::linorder tree \<Rightarrow> 'a multiset" where
"mset_tree (Node _ a ts) = {#a#} + (\<Sum>t\<in>#mset ts. mset_tree t)"
definition mset_trees :: "'a::linorder trees \<Rightarrow> 'a multiset" where
"mset_trees ts = (\<Sum>t\<in>#mset ts. mset_tree t)"
lemma mset_tree_simp_alt[simp]:
"mset_tree (Node r a ts) = {#a#} + mset_trees ts"
unfolding mset_trees_def by auto
declare mset_tree.simps[simp del]
lemma mset_tree_nonempty[simp]: "mset_tree t \<noteq> {#}"
by (cases t) auto
lemma mset_trees_Nil[simp]:
"mset_trees [] = {#}"
by (auto simp: mset_trees_def)
lemma mset_trees_Cons[simp]: "mset_trees (t#ts) = mset_tree t + mset_trees ts"
by (auto simp: mset_trees_def)
lemma mset_trees_empty_iff[simp]: "mset_trees ts = {#} \<longleftrightarrow> ts=[]"
by (auto simp: mset_trees_def)
lemma root_in_mset[simp]: "root t \<in># mset_tree t"
by (cases t) auto
lemma mset_trees_rev_eq[simp]: "mset_trees (rev ts) = mset_trees ts"
by (auto simp: mset_trees_def)
subsubsection \<open>Invariants\<close>
text \<open>Binomial tree\<close>
fun btree :: "'a::linorder tree \<Rightarrow> bool" where
"btree (Node r x ts) \<longleftrightarrow>
(\<forall>t\<in>set ts. btree t) \<and> map rank ts = rev [0..<r]"
text \<open>Heap invariant\<close>
fun heap :: "'a::linorder tree \<Rightarrow> bool" where
"heap (Node _ x ts) \<longleftrightarrow> (\<forall>t\<in>set ts. heap t \<and> x \<le> root t)"
definition "bheap t \<longleftrightarrow> btree t \<and> heap t"
text \<open>Binomial Heap invariant\<close>
definition "invar ts \<longleftrightarrow> (\<forall>t\<in>set ts. bheap t) \<and> (sorted_wrt (<) (map rank ts))"
text \<open>The children of a node are a valid heap\<close>
lemma invar_children:
"bheap (Node r v ts) \<Longrightarrow> invar (rev ts)"
by (auto simp: bheap_def invar_def rev_map[symmetric])
subsection \<open>Operations and Their Functional Correctness\<close>
subsubsection \<open>\<open>link\<close>\<close>
context
includes pattern_aliases
begin
fun link :: "('a::linorder) tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"link (Node r x\<^sub>1 ts\<^sub>1 =: t\<^sub>1) (Node r' x\<^sub>2 ts\<^sub>2 =: t\<^sub>2) =
(if x\<^sub>1\<le>x\<^sub>2 then Node (r+1) x\<^sub>1 (t\<^sub>2#ts\<^sub>1) else Node (r+1) x\<^sub>2 (t\<^sub>1#ts\<^sub>2))"
end
lemma invar_link:
assumes "bheap t\<^sub>1"
assumes "bheap t\<^sub>2"
assumes "rank t\<^sub>1 = rank t\<^sub>2"
shows "bheap (link t\<^sub>1 t\<^sub>2)"
using assms unfolding bheap_def
by (cases "(t\<^sub>1, t\<^sub>2)" rule: link.cases) auto
lemma rank_link[simp]: "rank (link t\<^sub>1 t\<^sub>2) = rank t\<^sub>1 + 1"
by (cases "(t\<^sub>1, t\<^sub>2)" rule: link.cases) simp
lemma mset_link[simp]: "mset_tree (link t\<^sub>1 t\<^sub>2) = mset_tree t\<^sub>1 + mset_tree t\<^sub>2"
by (cases "(t\<^sub>1, t\<^sub>2)" rule: link.cases) simp
subsubsection \<open>\<open>ins_tree\<close>\<close>
fun ins_tree :: "'a::linorder tree \<Rightarrow> 'a trees \<Rightarrow> 'a trees" where
"ins_tree t [] = [t]"
| "ins_tree t\<^sub>1 (t\<^sub>2#ts) =
(if rank t\<^sub>1 < rank t\<^sub>2 then t\<^sub>1#t\<^sub>2#ts else ins_tree (link t\<^sub>1 t\<^sub>2) ts)"
lemma bheap0[simp]: "bheap (Node 0 x [])"
unfolding bheap_def by auto
lemma invar_Cons[simp]:
"invar (t#ts)
\<longleftrightarrow> bheap t \<and> invar ts \<and> (\<forall>t'\<in>set ts. rank t < rank t')"
by (auto simp: invar_def)
lemma invar_ins_tree:
assumes "bheap t"
assumes "invar ts"
assumes "\<forall>t'\<in>set ts. rank t \<le> rank t'"
shows "invar (ins_tree t ts)"
using assms
by (induction t ts rule: ins_tree.induct) (auto simp: invar_link less_eq_Suc_le[symmetric])
lemma mset_trees_ins_tree[simp]:
"mset_trees (ins_tree t ts) = mset_tree t + mset_trees ts"
by (induction t ts rule: ins_tree.induct) auto
lemma ins_tree_rank_bound:
assumes "t' \<in> set (ins_tree t ts)"
assumes "\<forall>t'\<in>set ts. rank t\<^sub>0 < rank t'"
assumes "rank t\<^sub>0 < rank t"
shows "rank t\<^sub>0 < rank t'"
using assms
by (induction t ts rule: ins_tree.induct) (auto split: if_splits)
subsubsection \<open>\<open>insert\<close>\<close>
hide_const (open) insert
definition insert :: "'a::linorder \<Rightarrow> 'a trees \<Rightarrow> 'a trees" where
"insert x ts = ins_tree (Node 0 x []) ts"
lemma invar_insert[simp]: "invar t \<Longrightarrow> invar (insert x t)"
by (auto intro!: invar_ins_tree simp: insert_def)
lemma mset_trees_insert[simp]: "mset_trees (insert x t) = {#x#} + mset_trees t"
by(auto simp: insert_def)
subsubsection \<open>\<open>merge\<close>\<close>
context
includes pattern_aliases
begin
fun merge :: "'a::linorder trees \<Rightarrow> 'a trees \<Rightarrow> 'a trees" where
"merge ts\<^sub>1 [] = ts\<^sub>1"
| "merge [] ts\<^sub>2 = ts\<^sub>2"
| "merge (t\<^sub>1#ts\<^sub>1 =: h\<^sub>1) (t\<^sub>2#ts\<^sub>2 =: h\<^sub>2) = (
if rank t\<^sub>1 < rank t\<^sub>2 then t\<^sub>1 # merge ts\<^sub>1 h\<^sub>2 else
if rank t\<^sub>2 < rank t\<^sub>1 then t\<^sub>2 # merge h\<^sub>1 ts\<^sub>2
else ins_tree (link t\<^sub>1 t\<^sub>2) (merge ts\<^sub>1 ts\<^sub>2)
)"
end
lemma merge_simp2[simp]: "merge [] ts\<^sub>2 = ts\<^sub>2"
by (cases ts\<^sub>2) auto
lemma merge_rank_bound:
assumes "t' \<in> set (merge ts\<^sub>1 ts\<^sub>2)"
assumes "\<forall>t\<^sub>1\<in>set ts\<^sub>1. rank t < rank t\<^sub>1"
assumes "\<forall>t\<^sub>2\<in>set ts\<^sub>2. rank t < rank t\<^sub>2"
shows "rank t < rank t'"
using assms
by (induction ts\<^sub>1 ts\<^sub>2 arbitrary: t' rule: merge.induct)
(auto split: if_splits simp: ins_tree_rank_bound)
lemma invar_merge[simp]:
assumes "invar ts\<^sub>1"
assumes "invar ts\<^sub>2"
shows "invar (merge ts\<^sub>1 ts\<^sub>2)"
using assms
by (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct)
(auto 0 3 simp: Suc_le_eq intro!: invar_ins_tree invar_link elim!: merge_rank_bound)
text \<open>Longer, more explicit proof of @{thm [source] invar_merge},
to illustrate the application of the @{thm [source] merge_rank_bound} lemma.\<close>
lemma
assumes "invar ts\<^sub>1"
assumes "invar ts\<^sub>2"
shows "invar (merge ts\<^sub>1 ts\<^sub>2)"
using assms
proof (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct)
case (3 t\<^sub>1 ts\<^sub>1 t\<^sub>2 ts\<^sub>2)
\<comment> \<open>Invariants of the parts can be shown automatically\<close>
from "3.prems" have [simp]:
"bheap t\<^sub>1" "bheap t\<^sub>2"
(*"invar (merge (t\<^sub>1#ts\<^sub>1) ts\<^sub>2)"
"invar (merge ts\<^sub>1 (t\<^sub>2#ts\<^sub>2))"
"invar (merge ts\<^sub>1 ts\<^sub>2)"*)
by auto
\<comment> \<open>These are the three cases of the @{const merge} function\<close>
consider (LT) "rank t\<^sub>1 < rank t\<^sub>2"
| (GT) "rank t\<^sub>1 > rank t\<^sub>2"
| (EQ) "rank t\<^sub>1 = rank t\<^sub>2"
using antisym_conv3 by blast
then show ?case proof cases
case LT
\<comment> \<open>@{const merge} takes the first tree from the left heap\<close>
then have "merge (t\<^sub>1 # ts\<^sub>1) (t\<^sub>2 # ts\<^sub>2) = t\<^sub>1 # merge ts\<^sub>1 (t\<^sub>2 # ts\<^sub>2)" by simp
also have "invar \<dots>" proof (simp, intro conjI)
\<comment> \<open>Invariant follows from induction hypothesis\<close>
show "invar (merge ts\<^sub>1 (t\<^sub>2 # ts\<^sub>2))"
using LT "3.IH" "3.prems" by simp
\<comment> \<open>It remains to show that \<open>t\<^sub>1\<close> has smallest rank.\<close>
show "\<forall>t'\<in>set (merge ts\<^sub>1 (t\<^sub>2 # ts\<^sub>2)). rank t\<^sub>1 < rank t'"
\<comment> \<open>Which is done by auxiliary lemma @{thm [source] merge_rank_bound}\<close>
using LT "3.prems" by (force elim!: merge_rank_bound)
qed
finally show ?thesis .
next
\<comment> \<open>@{const merge} takes the first tree from the right heap\<close>
case GT
\<comment> \<open>The proof is anaologous to the \<open>LT\<close> case\<close>
then show ?thesis using "3.prems" "3.IH" by (force elim!: merge_rank_bound)
next
case [simp]: EQ
\<comment> \<open>@{const merge} links both first trees, and inserts them into the merged remaining heaps\<close>
have "merge (t\<^sub>1 # ts\<^sub>1) (t\<^sub>2 # ts\<^sub>2) = ins_tree (link t\<^sub>1 t\<^sub>2) (merge ts\<^sub>1 ts\<^sub>2)" by simp
also have "invar \<dots>" proof (intro invar_ins_tree invar_link)
\<comment> \<open>Invariant of merged remaining heaps follows by IH\<close>
show "invar (merge ts\<^sub>1 ts\<^sub>2)"
using EQ "3.prems" "3.IH" by auto
\<comment> \<open>For insertion, we have to show that the rank of the linked tree is \<open>\<le>\<close> the
ranks in the merged remaining heaps\<close>
show "\<forall>t'\<in>set (merge ts\<^sub>1 ts\<^sub>2). rank (link t\<^sub>1 t\<^sub>2) \<le> rank t'"
proof -
\<comment> \<open>Which is, again, done with the help of @{thm [source] merge_rank_bound}\<close>
have "rank (link t\<^sub>1 t\<^sub>2) = Suc (rank t\<^sub>2)" by simp
thus ?thesis using "3.prems" by (auto simp: Suc_le_eq elim!: merge_rank_bound)
qed
qed simp_all
finally show ?thesis .
qed
qed auto
lemma mset_trees_merge[simp]:
"mset_trees (merge ts\<^sub>1 ts\<^sub>2) = mset_trees ts\<^sub>1 + mset_trees ts\<^sub>2"
by (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct) auto
subsubsection \<open>\<open>get_min\<close>\<close>
fun get_min :: "'a::linorder trees \<Rightarrow> 'a" where
"get_min [t] = root t"
| "get_min (t#ts) = min (root t) (get_min ts)"
lemma bheap_root_min:
assumes "bheap t"
assumes "x \<in># mset_tree t"
shows "root t \<le> x"
using assms unfolding bheap_def
by (induction t arbitrary: x rule: mset_tree.induct) (fastforce simp: mset_trees_def)
lemma get_min_mset:
assumes "ts\<noteq>[]"
assumes "invar ts"
assumes "x \<in># mset_trees ts"
shows "get_min ts \<le> x"
using assms
apply (induction ts arbitrary: x rule: get_min.induct)
apply (auto
simp: bheap_root_min min_def intro: order_trans;
meson linear order_trans bheap_root_min
)+
done
lemma get_min_member:
"ts\<noteq>[] \<Longrightarrow> get_min ts \<in># mset_trees ts"
by (induction ts rule: get_min.induct) (auto simp: min_def)
lemma get_min:
assumes "mset_trees ts \<noteq> {#}"
assumes "invar ts"
shows "get_min ts = Min_mset (mset_trees ts)"
using assms get_min_member get_min_mset
by (auto simp: eq_Min_iff)
subsubsection \<open>\<open>get_min_rest\<close>\<close>
fun get_min_rest :: "'a::linorder trees \<Rightarrow> 'a tree \<times> 'a trees" where
"get_min_rest [t] = (t,[])"
| "get_min_rest (t#ts) = (let (t',ts') = get_min_rest ts
in if root t \<le> root t' then (t,ts) else (t',t#ts'))"
lemma get_min_rest_get_min_same_root:
assumes "ts\<noteq>[]"
assumes "get_min_rest ts = (t',ts')"
shows "root t' = get_min ts"
using assms
by (induction ts arbitrary: t' ts' rule: get_min.induct) (auto simp: min_def split: prod.splits)
lemma mset_get_min_rest:
assumes "get_min_rest ts = (t',ts')"
assumes "ts\<noteq>[]"
shows "mset ts = {#t'#} + mset ts'"
using assms
by (induction ts arbitrary: t' ts' rule: get_min.induct) (auto split: prod.splits if_splits)
lemma set_get_min_rest:
assumes "get_min_rest ts = (t', ts')"
assumes "ts\<noteq>[]"
shows "set ts = Set.insert t' (set ts')"
using mset_get_min_rest[OF assms, THEN arg_cong[where f=set_mset]]
by auto
lemma invar_get_min_rest:
assumes "get_min_rest ts = (t',ts')"
assumes "ts\<noteq>[]"
assumes "invar ts"
shows "bheap t'" and "invar ts'"
proof -
have "bheap t' \<and> invar ts'"
using assms
proof (induction ts arbitrary: t' ts' rule: get_min.induct)
case (2 t v va)
then show ?case
apply (clarsimp split: prod.splits if_splits)
apply (drule set_get_min_rest; fastforce)
done
qed auto
thus "bheap t'" and "invar ts'" by auto
qed
subsubsection \<open>\<open>del_min\<close>\<close>
definition del_min :: "'a::linorder trees \<Rightarrow> 'a::linorder trees" where
"del_min ts = (case get_min_rest ts of
(Node r x ts\<^sub>1, ts\<^sub>2) \<Rightarrow> merge (rev ts\<^sub>1) ts\<^sub>2)"
lemma invar_del_min[simp]:
assumes "ts \<noteq> []"
assumes "invar ts"
shows "invar (del_min ts)"
using assms
unfolding del_min_def
by (auto
split: prod.split tree.split
intro!: invar_merge invar_children
dest: invar_get_min_rest
)
lemma mset_trees_del_min:
assumes "ts \<noteq> []"
shows "mset_trees ts = mset_trees (del_min ts) + {# get_min ts #}"
using assms
unfolding del_min_def
apply (clarsimp split: tree.split prod.split)
apply (frule (1) get_min_rest_get_min_same_root)
apply (frule (1) mset_get_min_rest)
apply (auto simp: mset_trees_def)
done
subsubsection \<open>Instantiating the Priority Queue Locale\<close>
text \<open>Last step of functional correctness proof: combine all the above lemmas
to show that binomial heaps satisfy the specification of priority queues with merge.\<close>
interpretation bheaps: Priority_Queue_Merge
where empty = "[]" and is_empty = "(=) []" and insert = insert
and get_min = get_min and del_min = del_min and merge = merge
and invar = invar and mset = mset_trees
proof (unfold_locales, goal_cases)
case 1 thus ?case by simp
next
case 2 thus ?case by auto
next
case 3 thus ?case by auto
next
case (4 q)
thus ?case using mset_trees_del_min[of q] get_min[OF _ \<open>invar q\<close>]
by (auto simp: union_single_eq_diff)
next
case (5 q) thus ?case using get_min[of q] by auto
next
case 6 thus ?case by (auto simp add: invar_def)
next
case 7 thus ?case by simp
next
case 8 thus ?case by simp
next
case 9 thus ?case by simp
next
case 10 thus ?case by simp
qed
subsection \<open>Complexity\<close>
text \<open>The size of a binomial tree is determined by its rank\<close>
lemma size_mset_btree:
assumes "btree t"
shows "size (mset_tree t) = 2^rank t"
using assms
proof (induction t)
case (Node r v ts)
hence IH: "size (mset_tree t) = 2^rank t" if "t \<in> set ts" for t
using that by auto
from Node have COMPL: "map rank ts = rev [0..<r]" by auto
have "size (mset_trees ts) = (\<Sum>t\<leftarrow>ts. size (mset_tree t))"
by (induction ts) auto
also have "\<dots> = (\<Sum>t\<leftarrow>ts. 2^rank t)" using IH
by (auto cong: map_cong)
also have "\<dots> = (\<Sum>r\<leftarrow>map rank ts. 2^r)"
by (induction ts) auto
also have "\<dots> = (\<Sum>i\<in>{0..<r}. 2^i)"
unfolding COMPL
by (auto simp: rev_map[symmetric] interv_sum_list_conv_sum_set_nat)
also have "\<dots> = 2^r - 1"
by (induction r) auto
finally show ?case
by (simp)
qed
lemma size_mset_tree:
assumes "bheap t"
shows "size (mset_tree t) = 2^rank t"
using assms unfolding bheap_def
by (simp add: size_mset_btree)
text \<open>The length of a binomial heap is bounded by the number of its elements\<close>
lemma size_mset_trees:
assumes "invar ts"
shows "length ts \<le> log 2 (size (mset_trees ts) + 1)"
proof -
from \<open>invar ts\<close> have
ASC: "sorted_wrt (<) (map rank ts)" and
TINV: "\<forall>t\<in>set ts. bheap t"
unfolding invar_def by auto
have "(2::nat)^length ts = (\<Sum>i\<in>{0..<length ts}. 2^i) + 1"
by (simp add: sum_power2)
also have "\<dots> = (\<Sum>i\<leftarrow>[0..<length ts]. 2^i) + 1" (is "_ = ?S + 1")
by (simp add: interv_sum_list_conv_sum_set_nat)
also have "?S \<le> (\<Sum>t\<leftarrow>ts. 2^rank t)" (is "_ \<le> ?T")
using sorted_wrt_less_idx[OF ASC] by(simp add: sum_list_mono2)
also have "?T + 1 \<le> (\<Sum>t\<leftarrow>ts. size (mset_tree t)) + 1" using TINV
by (auto cong: map_cong simp: size_mset_tree)
also have "\<dots> = size (mset_trees ts) + 1"
unfolding mset_trees_def by (induction ts) auto
finally have "2^length ts \<le> size (mset_trees ts) + 1" by simp
then show ?thesis using le_log2_of_power by blast
qed
subsubsection \<open>Timing Functions\<close>
time_fun link
lemma T_link[simp]: "T_link t\<^sub>1 t\<^sub>2 = 0"
by(cases t\<^sub>1; cases t\<^sub>2, auto)
time_fun rank
lemma T_rank[simp]: "T_rank t = 0"
by(cases t, auto)
time_fun ins_tree
time_fun insert
lemma T_ins_tree_simple_bound: "T_ins_tree t ts \<le> length ts + 1"
by (induction t ts rule: T_ins_tree.induct) auto
subsubsection \<open>\<open>T_insert\<close>\<close>
lemma T_insert_bound:
assumes "invar ts"
shows "T_insert x ts \<le> log 2 (size (mset_trees ts) + 1) + 1"
proof -
have "real (T_insert x ts) \<le> real (length ts) + 1"
unfolding T_insert.simps using T_ins_tree_simple_bound
by (metis of_nat_1 of_nat_add of_nat_mono)
also note size_mset_trees[OF \<open>invar ts\<close>]
finally show ?thesis by simp
qed
subsubsection \<open>\<open>T_merge\<close>\<close>
time_fun merge
(* Warning: \<open>T_merge.induct\<close> is less convenient than the equivalent \<open>merge.induct\<close>,
apparently because of the \<open>let\<close> clauses introduced by pattern_aliases; should be investigated.
*)
text \<open>A crucial idea is to estimate the time in correlation with the
result length, as each carry reduces the length of the result.\<close>
lemma T_ins_tree_length:
"T_ins_tree t ts + length (ins_tree t ts) = 2 + length ts"
by (induction t ts rule: ins_tree.induct) auto
lemma T_merge_length:
"T_merge ts\<^sub>1 ts\<^sub>2 + length (merge ts\<^sub>1 ts\<^sub>2) \<le> 2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1"
by (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct)
(auto simp: T_ins_tree_length algebra_simps)
text \<open>Finally, we get the desired logarithmic bound\<close>
lemma T_merge_bound:
fixes ts\<^sub>1 ts\<^sub>2
defines "n\<^sub>1 \<equiv> size (mset_trees ts\<^sub>1)"
defines "n\<^sub>2 \<equiv> size (mset_trees ts\<^sub>2)"
assumes "invar ts\<^sub>1" "invar ts\<^sub>2"
shows "T_merge ts\<^sub>1 ts\<^sub>2 \<le> 4*log 2 (n\<^sub>1 + n\<^sub>2 + 1) + 1"
proof -
note n_defs = assms(1,2)
have "T_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * real (length ts\<^sub>1) + 2 * real (length ts\<^sub>2) + 1"
using T_merge_length[of ts\<^sub>1 ts\<^sub>2] by simp
also note size_mset_trees[OF \<open>invar ts\<^sub>1\<close>]
also note size_mset_trees[OF \<open>invar ts\<^sub>2\<close>]
finally have "T_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * log 2 (n\<^sub>1 + 1) + 2 * log 2 (n\<^sub>2 + 1) + 1"
unfolding n_defs by (simp add: algebra_simps)
also have "log 2 (n\<^sub>1 + 1) \<le> log 2 (n\<^sub>1 + n\<^sub>2 + 1)"
unfolding n_defs by (simp add: algebra_simps)
also have "log 2 (n\<^sub>2 + 1) \<le> log 2 (n\<^sub>1 + n\<^sub>2 + 1)"
unfolding n_defs by (simp add: algebra_simps)
finally show ?thesis by (simp add: algebra_simps)
qed
subsubsection \<open>\<open>T_get_min\<close>\<close>
time_fun root
lemma T_root[simp]: "T_root t = 0"
by(cases t)(simp_all)
time_fun min
time_fun get_min
lemma T_get_min_estimate: "ts\<noteq>[] \<Longrightarrow> T_get_min ts = length ts"
by (induction ts rule: T_get_min.induct) auto
lemma T_get_min_bound:
assumes "invar ts"
assumes "ts\<noteq>[]"
shows "T_get_min ts \<le> log 2 (size (mset_trees ts) + 1)"
proof -
have 1: "T_get_min ts = length ts" using assms T_get_min_estimate by auto
also note size_mset_trees[OF \<open>invar ts\<close>]
finally show ?thesis .
qed
subsubsection \<open>\<open>T_del_min\<close>\<close>
time_fun get_min_rest
(*fun T_get_min_rest :: "'a::linorder trees \<Rightarrow> nat" where
"T_get_min_rest [t] = 1"
| "T_get_min_rest (t#ts) = 1 + T_get_min_rest ts"*)
lemma T_get_min_rest_estimate: "ts\<noteq>[] \<Longrightarrow> T_get_min_rest ts = length ts"
by (induction ts rule: T_get_min_rest.induct) auto
lemma T_get_min_rest_bound:
assumes "invar ts"
assumes "ts\<noteq>[]"
shows "T_get_min_rest ts \<le> log 2 (size (mset_trees ts) + 1)"
proof -
have 1: "T_get_min_rest ts = length ts" using assms T_get_min_rest_estimate by auto
also note size_mset_trees[OF \<open>invar ts\<close>]
finally show ?thesis .
qed
text\<open>Note that although the definition of function \<^const>\<open>rev\<close> has quadratic complexity,
it can and is implemented (via suitable code lemmas) as a linear time function.
Thus the following definition is justified:\<close>
definition "T_rev xs = length xs + 1"
time_fun del_min
lemma T_del_min_bound:
fixes ts
defines "n \<equiv> size (mset_trees ts)"
assumes "invar ts" and "ts\<noteq>[]"
shows "T_del_min ts \<le> 6 * log 2 (n+1) + 2"
proof -
obtain r x ts\<^sub>1 ts\<^sub>2 where GM: "get_min_rest ts = (Node r x ts\<^sub>1, ts\<^sub>2)"
by (metis surj_pair tree.exhaust_sel)
have I1: "invar (rev ts\<^sub>1)" and I2: "invar ts\<^sub>2"
using invar_get_min_rest[OF GM \<open>ts\<noteq>[]\<close> \<open>invar ts\<close>] invar_children
by auto
define n\<^sub>1 where "n\<^sub>1 = size (mset_trees ts\<^sub>1)"
define n\<^sub>2 where "n\<^sub>2 = size (mset_trees ts\<^sub>2)"
have "n\<^sub>1 \<le> n" "n\<^sub>1 + n\<^sub>2 \<le> n" unfolding n_def n\<^sub>1_def n\<^sub>2_def
using mset_get_min_rest[OF GM \<open>ts\<noteq>[]\<close>]
by (auto simp: mset_trees_def)
have "T_del_min ts = real (T_get_min_rest ts) + real (T_rev ts\<^sub>1) + real (T_merge (rev ts\<^sub>1) ts\<^sub>2)"
unfolding T_del_min.simps GM
by simp
also have "T_get_min_rest ts \<le> log 2 (n+1)"
using T_get_min_rest_bound[OF \<open>invar ts\<close> \<open>ts\<noteq>[]\<close>] unfolding n_def by simp
also have "T_rev ts\<^sub>1 \<le> 1 + log 2 (n\<^sub>1 + 1)"
unfolding T_rev_def n\<^sub>1_def using size_mset_trees[OF I1] by simp
also have "T_merge (rev ts\<^sub>1) ts\<^sub>2 \<le> 4*log 2 (n\<^sub>1 + n\<^sub>2 + 1) + 1"
unfolding n\<^sub>1_def n\<^sub>2_def using T_merge_bound[OF I1 I2] by (simp add: algebra_simps)
finally have "T_del_min ts \<le> log 2 (n+1) + log 2 (n\<^sub>1 + 1) + 4*log 2 (real (n\<^sub>1 + n\<^sub>2) + 1) + 2"
by (simp add: algebra_simps)
also note \<open>n\<^sub>1 + n\<^sub>2 \<le> n\<close>
also note \<open>n\<^sub>1 \<le> n\<close>
finally show ?thesis by (simp add: algebra_simps)
qed
end