diff --git a/src/HOL/Data_Structures/Binomial_Heap.thy b/src/HOL/Data_Structures/Binomial_Heap.thy
--- a/src/HOL/Data_Structures/Binomial_Heap.thy
+++ b/src/HOL/Data_Structures/Binomial_Heap.thy
@@ -1,720 +1,700 @@
 (* Author: Peter Lammich
            Tobias Nipkow (tuning)
 *)
 
 section \<open>Binomial Heap\<close>
 
 theory Binomial_Heap
 imports
   "HOL-Library.Pattern_Aliases"
   Complex_Main
   Priority_Queue_Specs
 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 heap = "'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_heap :: "'a::linorder heap \<Rightarrow> 'a multiset" where
   "mset_heap ts = (\<Sum>t\<in>#mset ts. mset_tree t)"
 
 lemma mset_tree_simp_alt[simp]:
   "mset_tree (Node r a ts) = {#a#} + mset_heap ts"
   unfolding mset_heap_def by auto
 declare mset_tree.simps[simp del]
 
 lemma mset_tree_nonempty[simp]: "mset_tree t \<noteq> {#}"
 by (cases t) auto
 
 lemma mset_heap_Nil[simp]:
   "mset_heap [] = {#}"
 by (auto simp: mset_heap_def)
 
 lemma mset_heap_Cons[simp]: "mset_heap (t#ts) = mset_tree t + mset_heap ts"
 by (auto simp: mset_heap_def)
 
 lemma mset_heap_empty_iff[simp]: "mset_heap ts = {#} \<longleftrightarrow> ts=[]"
 by (auto simp: mset_heap_def)
 
 lemma root_in_mset[simp]: "root t \<in># mset_tree t"
 by (cases t) auto
 
 lemma mset_heap_rev_eq[simp]: "mset_heap (rev ts) = mset_heap ts"
 by (auto simp: mset_heap_def)
 
 subsubsection \<open>Invariants\<close>
 
 text \<open>Binomial tree\<close>
 fun invar_btree :: "'a::linorder tree \<Rightarrow> bool" where
 "invar_btree (Node r x ts) \<longleftrightarrow>
    (\<forall>t\<in>set ts. invar_btree t) \<and> map rank ts = rev [0..<r]"
 
 text \<open>Ordering (heap) invariant\<close>
 fun invar_otree :: "'a::linorder tree \<Rightarrow> bool" where
 "invar_otree (Node _ x ts) \<longleftrightarrow> (\<forall>t\<in>set ts. invar_otree t \<and> x \<le> root t)"
 
 definition "invar_tree t \<longleftrightarrow> invar_btree t \<and> invar_otree t"
 
 text \<open>Binomial Heap invariant\<close>
 definition "invar ts \<longleftrightarrow> (\<forall>t\<in>set ts. invar_tree t) \<and> (sorted_wrt (<) (map rank ts))"
 
 
 text \<open>The children of a node are a valid heap\<close>
 lemma invar_children:
   "invar_tree (Node r v ts) \<Longrightarrow> invar (rev ts)"
   by (auto simp: invar_tree_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 "invar_tree t\<^sub>1"
   assumes "invar_tree t\<^sub>2"
   assumes "rank t\<^sub>1 = rank t\<^sub>2"
   shows "invar_tree (link t\<^sub>1 t\<^sub>2)"
 using assms unfolding invar_tree_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 heap \<Rightarrow> 'a heap" 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 invar_tree0[simp]: "invar_tree (Node 0 x [])"
 unfolding invar_tree_def by auto
 
 lemma invar_Cons[simp]:
   "invar (t#ts)
   \<longleftrightarrow> invar_tree t \<and> invar ts \<and> (\<forall>t'\<in>set ts. rank t < rank t')"
 by (auto simp: invar_def)
 
 lemma invar_ins_tree:
   assumes "invar_tree 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_heap_ins_tree[simp]:
   "mset_heap (ins_tree t ts) = mset_tree t + mset_heap 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 heap \<Rightarrow> 'a heap" 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_heap_insert[simp]: "mset_heap (insert x t) = {#x#} + mset_heap t"
 by(auto simp: insert_def)
 
 subsubsection \<open>\<open>merge\<close>\<close>
 
 context
 includes pattern_aliases
 begin
 
 fun merge :: "'a::linorder heap \<Rightarrow> 'a heap \<Rightarrow> 'a heap" 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]: 
     "invar_tree t\<^sub>1" "invar_tree 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_heap_merge[simp]:
   "mset_heap (merge ts\<^sub>1 ts\<^sub>2) = mset_heap ts\<^sub>1 + mset_heap 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 heap \<Rightarrow> 'a" where
   "get_min [t] = root t"
 | "get_min (t#ts) = min (root t) (get_min ts)"
 
 lemma invar_tree_root_min:
   assumes "invar_tree t"
   assumes "x \<in># mset_tree t"
   shows "root t \<le> x"
 using assms unfolding invar_tree_def
 by (induction t arbitrary: x rule: mset_tree.induct) (fastforce simp: mset_heap_def)
 
 lemma get_min_mset:
   assumes "ts\<noteq>[]"
   assumes "invar ts"
   assumes "x \<in># mset_heap ts"
   shows "get_min ts \<le> x"
   using assms
 apply (induction ts arbitrary: x rule: get_min.induct)
 apply (auto
       simp: invar_tree_root_min min_def intro: order_trans;
       meson linear order_trans invar_tree_root_min
       )+
 done
 
 lemma get_min_member:
   "ts\<noteq>[] \<Longrightarrow> get_min ts \<in># mset_heap ts"
 by (induction ts rule: get_min.induct) (auto simp: min_def)
 
 lemma get_min:
   assumes "mset_heap ts \<noteq> {#}"
   assumes "invar ts"
   shows "get_min ts = Min_mset (mset_heap 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 heap \<Rightarrow> 'a tree \<times> 'a heap" 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 "invar_tree t'" and "invar ts'"
 proof -
   have "invar_tree 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 "invar_tree t'" and "invar ts'" by auto
 qed
 
 subsubsection \<open>\<open>del_min\<close>\<close>
 
 definition del_min :: "'a::linorder heap \<Rightarrow> 'a::linorder heap" 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_heap_del_min:
   assumes "ts \<noteq> []"
   shows "mset_heap ts = mset_heap (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_heap_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 binheap: 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_heap
 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_heap_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 "invar_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_heap 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 "invar_tree t"
   shows "size (mset_tree t) = 2^rank t"
 using assms unfolding invar_tree_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_heap:
   assumes "invar ts"
   shows "2^length ts \<le> size (mset_heap ts) + 1"
 proof -
   from \<open>invar ts\<close> have
     ASC: "sorted_wrt (<) (map rank ts)" and
     TINV: "\<forall>t\<in>set ts. invar_tree 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> \<le> (\<Sum>t\<leftarrow>ts. 2^rank t) + 1"
     using sorted_wrt_less_sum_mono_lowerbound[OF _ ASC, of "(^) (2::nat)"]
     using power_increasing[where a="2::nat"]
     by (auto simp: o_def)
   also have "\<dots> = (\<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_heap ts) + 1"
     unfolding mset_heap_def by (induction ts) auto
   finally show ?thesis .
 qed
 
 subsubsection \<open>Timing Functions\<close>
 
 text \<open>
   We define timing functions for each operation, and provide
   estimations of their complexity.
 \<close>
 definition T_link :: "'a::linorder tree \<Rightarrow> 'a tree \<Rightarrow> nat" where
 [simp]: "T_link _ _ = 1"
 
 fun T_ins_tree :: "'a::linorder tree \<Rightarrow> 'a heap \<Rightarrow> nat" where
   "T_ins_tree t [] = 1"
 | "T_ins_tree t\<^sub>1 (t\<^sub>2 # rest) = (
     (if rank t\<^sub>1 < rank t\<^sub>2 then 1
      else T_link t\<^sub>1 t\<^sub>2 + T_ins_tree (link t\<^sub>1 t\<^sub>2) rest)
   )"
 
 definition T_insert :: "'a::linorder \<Rightarrow> 'a heap \<Rightarrow> nat" where
 "T_insert x ts = T_ins_tree (Node 0 x []) ts"
 
 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_heap ts) + 1) + 1"
 proof -
 
   have 1: "T_insert x ts \<le> length ts + 1"
     unfolding T_insert_def by (rule T_ins_tree_simple_bound)
   also have "\<dots> \<le> log 2 (2 * (size (mset_heap ts) + 1))"
   proof -
     from size_mset_heap[of ts] assms
     have "2 ^ length ts \<le> size (mset_heap ts) + 1"
       unfolding invar_def by auto
     hence "2 ^ (length ts + 1) \<le> 2 * (size (mset_heap ts) + 1)" by auto
     thus ?thesis using le_log2_of_power by blast
   qed
   finally show ?thesis
     by (simp only: log_mult of_nat_mult) auto
 qed
 
 subsubsection \<open>\<open>T_merge\<close>\<close>
 
 context
 includes pattern_aliases
 begin
 
 fun T_merge :: "'a::linorder heap \<Rightarrow> 'a heap \<Rightarrow> nat" where
   "T_merge ts\<^sub>1 [] = 1"
 | "T_merge [] ts\<^sub>2 = 1"
 | "T_merge (t\<^sub>1#ts\<^sub>1 =: h\<^sub>1) (t\<^sub>2#ts\<^sub>2 =: h\<^sub>2) = 1 + (
     if rank t\<^sub>1 < rank t\<^sub>2 then T_merge ts\<^sub>1 h\<^sub>2
     else if rank t\<^sub>2 < rank t\<^sub>1 then T_merge h\<^sub>1 ts\<^sub>2
     else T_ins_tree (link t\<^sub>1 t\<^sub>2) (merge ts\<^sub>1 ts\<^sub>2) + T_merge ts\<^sub>1 ts\<^sub>2
   )"
 
 end
 
 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:
   "length (merge ts\<^sub>1 ts\<^sub>2) + T_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: T_merge.induct)
    (auto simp: T_ins_tree_length algebra_simps)
 
 text \<open>Finally, we get the desired logarithmic bound\<close>
-lemma T_merge_bound_aux:
+lemma T_merge_bound:
   fixes ts\<^sub>1 ts\<^sub>2
   defines "n\<^sub>1 \<equiv> size (mset_heap ts\<^sub>1)"
   defines "n\<^sub>2 \<equiv> size (mset_heap ts\<^sub>2)"
-  assumes BINVARS: "invar ts\<^sub>1" "invar 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) + 2"
 proof -
   define n where "n = n\<^sub>1 + n\<^sub>2"
 
+  note BINVARS = \<open>invar ts\<^sub>1\<close> \<open>invar ts\<^sub>2\<close>
+
   from T_merge_length[of ts\<^sub>1 ts\<^sub>2]
   have "T_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1" by auto
   hence "(2::nat)^T_merge ts\<^sub>1 ts\<^sub>2 \<le> 2^(2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1)"
     by (rule power_increasing) auto
   also have "\<dots> = 2*(2^length ts\<^sub>1)\<^sup>2*(2^length ts\<^sub>2)\<^sup>2"
     by (auto simp: algebra_simps power_add power_mult)
   also note BINVARS(1)[THEN size_mset_heap]
   also note BINVARS(2)[THEN size_mset_heap]
   finally have "2 ^ T_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * (n\<^sub>1 + 1)\<^sup>2 * (n\<^sub>2 + 1)\<^sup>2"
     by (auto simp: power2_nat_le_eq_le n\<^sub>1_def n\<^sub>2_def)
   from le_log2_of_power[OF this] have "T_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 \<dots>"
     by simp
   also have "\<dots> = log 2 2 + 2*log 2 (n\<^sub>1 + 1) + 2*log 2 (n\<^sub>2 + 1)"
     by (simp add: log_mult log_nat_power)
   also have "n\<^sub>2 \<le> n" by (auto simp: n_def)
   finally have "T_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 2 + 2*log 2 (n\<^sub>1 + 1) + 2*log 2 (n + 1)"
     by auto
   also have "n\<^sub>1 \<le> n" by (auto simp: n_def)
   finally have "T_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 2 + 4*log 2 (n + 1)"
     by auto
   also have "log 2 2 \<le> 2" by auto
   finally have "T_merge ts\<^sub>1 ts\<^sub>2 \<le> 4*log 2 (n + 1) + 2" by auto
   thus ?thesis unfolding n_def by (auto simp: algebra_simps)
 qed
 
-lemma T_merge_bound:
-  fixes ts\<^sub>1 ts\<^sub>2
-  defines "n\<^sub>1 \<equiv> size (mset_heap ts\<^sub>1)"
-  defines "n\<^sub>2 \<equiv> size (mset_heap 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) + 2"
-using assms T_merge_bound_aux unfolding invar_def by blast
-
 subsubsection \<open>\<open>T_get_min\<close>\<close>
 
 fun T_get_min :: "'a::linorder heap \<Rightarrow> nat" where
   "T_get_min [t] = 1"
 | "T_get_min (t#ts) = 1 + T_get_min ts"
 
 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_heap ts) + 1)"
 proof -
   have 1: "T_get_min ts = length ts" using assms T_get_min_estimate by auto
   also have "\<dots> \<le> log 2 (size (mset_heap ts) + 1)"
   proof -
     from size_mset_heap[of ts] assms have "2 ^ length ts \<le> size (mset_heap ts) + 1"
       unfolding invar_def by auto
     thus ?thesis using le_log2_of_power by blast
   qed
   finally show ?thesis by auto
 qed
 
 subsubsection \<open>\<open>T_del_min\<close>\<close>
 
 fun T_get_min_rest :: "'a::linorder heap \<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_aux:
+lemma T_get_min_rest_bound:
   assumes "invar ts"
   assumes "ts\<noteq>[]"
   shows "T_get_min_rest ts \<le> log 2 (size (mset_heap ts) + 1)"
 proof -
   have 1: "T_get_min_rest ts = length ts" using assms T_get_min_rest_estimate by auto
   also have "\<dots> \<le> log 2 (size (mset_heap ts) + 1)"
   proof -
     from size_mset_heap[of ts] assms have "2 ^ length ts \<le> size (mset_heap ts) + 1"
       by auto
     thus ?thesis using le_log2_of_power by blast
   qed
   finally show ?thesis by auto
 qed
 
-lemma T_get_min_rest_bound:
-  assumes "invar ts"
-  assumes "ts\<noteq>[]"
-  shows "T_get_min_rest ts \<le> log 2 (size (mset_heap ts) + 1)"
-using assms T_get_min_rest_bound_aux unfolding invar_def by blast
-
 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"
 
 definition T_del_min :: "'a::linorder heap \<Rightarrow> nat" where
   "T_del_min ts = T_get_min_rest ts + (case get_min_rest ts of (Node _ x ts\<^sub>1, ts\<^sub>2)
                     \<Rightarrow> T_rev ts\<^sub>1 + T_merge (rev ts\<^sub>1) ts\<^sub>2
   )"
 
 lemma T_rev_ts1_bound_aux:
   fixes ts
   defines "n \<equiv> size (mset_heap ts)"
   assumes BINVAR: "invar (rev ts)"
   shows "T_rev ts \<le> 1 + log 2 (n+1)"
 proof -
   have "T_rev ts = length ts + 1" by (auto simp: T_rev_def)
   hence "2^T_rev ts = 2*2^length ts" by auto
   also have "\<dots> \<le> 2*n+2" using size_mset_heap[OF BINVAR] by (auto simp: n_def)
   finally have "2 ^ T_rev ts \<le> 2 * n + 2" .
   from le_log2_of_power[OF this] have "T_rev ts \<le> log 2 (2 * (n + 1))"
     by auto
   also have "\<dots> = 1 + log 2 (n+1)"
     by (simp only: of_nat_mult log_mult) auto
   finally show ?thesis by (auto simp: algebra_simps)
 qed
 
-lemma T_del_min_bound_aux:
+lemma T_del_min_bound:
   fixes ts
   defines "n \<equiv> size (mset_heap ts)"
-  assumes BINVAR: "invar ts"
+  assumes "invar ts"
   assumes "ts\<noteq>[]"
   shows "T_del_min ts \<le> 6 * log 2 (n+1) + 3"
 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)
 
-  note BINVAR' = invar_get_min_rest[OF GM \<open>ts\<noteq>[]\<close> BINVAR]
+  note BINVAR' = invar_get_min_rest[OF GM \<open>ts\<noteq>[]\<close> \<open>invar ts\<close>]
   hence BINVAR1: "invar (rev ts\<^sub>1)" by (blast intro: invar_children)
 
   define n\<^sub>1 where "n\<^sub>1 = size (mset_heap ts\<^sub>1)"
   define n\<^sub>2 where "n\<^sub>2 = size (mset_heap ts\<^sub>2)"
 
   have T_rev_ts1_bound: "T_rev ts\<^sub>1 \<le> 1 + log 2 (n+1)"
   proof -
     note T_rev_ts1_bound_aux[OF BINVAR1, simplified, folded n\<^sub>1_def]
     also have "n\<^sub>1 \<le> n"
       unfolding n\<^sub>1_def n_def
       using mset_get_min_rest[OF GM \<open>ts\<noteq>[]\<close>]
       by (auto simp: mset_heap_def)
     finally show ?thesis by (auto simp: algebra_simps)
   qed
 
   have "T_del_min ts = T_get_min_rest ts + T_rev ts\<^sub>1 + T_merge (rev ts\<^sub>1) ts\<^sub>2"
     unfolding T_del_min_def by (simp add: GM)
   also have "\<dots> \<le> log 2 (n+1) + T_rev ts\<^sub>1 + T_merge (rev ts\<^sub>1) ts\<^sub>2"
-    using T_get_min_rest_bound_aux[OF assms(2-)] by (auto simp: n_def)
+    using T_get_min_rest_bound[OF assms(2-)] by (auto simp: n_def)
   also have "\<dots> \<le> 2*log 2 (n+1) + T_merge (rev ts\<^sub>1) ts\<^sub>2 + 1"
     using T_rev_ts1_bound by auto
   also have "\<dots> \<le> 2*log 2 (n+1) + 4 * log 2 (n\<^sub>1 + n\<^sub>2 + 1) + 3"
-    using T_merge_bound_aux[OF \<open>invar (rev ts\<^sub>1)\<close> \<open>invar ts\<^sub>2\<close>]
+    using T_merge_bound[OF \<open>invar (rev ts\<^sub>1)\<close> \<open>invar ts\<^sub>2\<close>]
     by (auto simp: n\<^sub>1_def n\<^sub>2_def algebra_simps)
   also have "n\<^sub>1 + n\<^sub>2 \<le> n"
     unfolding n\<^sub>1_def n\<^sub>2_def n_def
     using mset_get_min_rest[OF GM \<open>ts\<noteq>[]\<close>]
     by (auto simp: mset_heap_def)
   finally have "T_del_min ts \<le> 6 * log 2 (n+1) + 3"
     by auto
   thus ?thesis by (simp add: algebra_simps)
 qed
 
-lemma T_del_min_bound:
-  fixes ts
-  defines "n \<equiv> size (mset_heap ts)"
-  assumes "invar ts"
-  assumes "ts\<noteq>[]"
-  shows "T_del_min ts \<le> 6 * log 2 (n+1) + 3"
-using assms T_del_min_bound_aux unfolding invar_def by blast
-
 end