--- a/src/HOL/Data_Structures/Leftist_Heap.thy Mon Jun 11 08:15:43 2018 +0200
+++ b/src/HOL/Data_Structures/Leftist_Heap.thy Mon Jun 11 16:29:27 2018 +0200
@@ -12,7 +12,7 @@
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"
+"mset_tree (Node l a _ r) = {#a#} + mset_tree l + mset_tree r"
type_synonym 'a lheap = "('a,nat)tree"
@@ -22,27 +22,27 @@
fun rk :: "'a lheap \<Rightarrow> nat" where
"rk Leaf = 0" |
-"rk (Node n _ _ _) = n"
+"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 (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 n l a r) =
+"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 (rr+1) l a r else Node (rl+1) r a l)"
+ 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 n l a r) = a"
+"get_min(Node l a n r) = a"
text \<open>For function \<open>merge\<close>:\<close>
unbundle pattern_aliases
@@ -51,25 +51,25 @@
fun merge :: "'a::ord lheap \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
"merge Leaf t2 = t2" |
"merge t1 Leaf = t1" |
-"merge (Node n1 l1 a1 r1 =: t1) (Node n2 l2 a2 r2 =: t2) =
+"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 r2 t1))"
lemma merge_code: "merge t1 t2 = (case (t1,t2) of
(Leaf, _) \<Rightarrow> t2 |
(_, Leaf) \<Rightarrow> t1 |
- (Node n1 l1 a1 r1, Node n2 l2 a2 r2) \<Rightarrow>
+ (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 r2 t1))"
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 1 Leaf x Leaf) t"
+"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 n l x r) = merge l r"
+"del_min (Node l x n r) = merge l r"
subsection "Lemmas"
@@ -104,7 +104,7 @@
lemma ltree_merge: "\<lbrakk> ltree l; ltree r \<rbrakk> \<Longrightarrow> ltree (merge l r)"
proof(induction l r rule: merge.induct)
- case (3 n1 l1 a1 r1 n2 l2 a2 r2)
+ case (3 l1 a1 n1 r1 l2 a2 n2 r2)
show ?case (is "ltree(merge ?t1 ?t2)")
proof cases
assume "a1 \<le> a2"
@@ -173,14 +173,14 @@
proof(induction t)
case Leaf show ?case by simp
next
- case (Node n l a r)
+ 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>n, l, a, r\<rangle> = 2 ^ rank r + 2 ^ rank r"
+ 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>n, l, a, r\<rangle>" by simp
+ also have "\<dots> = size1 \<langle>l, a, n, r\<rangle>" by simp
finally show ?case .
qed
@@ -189,16 +189,16 @@
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 n1 l1 a1 r1 =: t1) (Node n2 l2 a2 r2 =: t2) =
+"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 r2 t1)"
definition t_insert :: "'a::ord \<Rightarrow> 'a lheap \<Rightarrow> nat" where
-"t_insert x t = t_merge (Node 1 Leaf x Leaf) t"
+"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 n l a r) = t_merge l r"
+"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)
@@ -213,7 +213,7 @@
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 1 Leaf x Leaf" t]
+using t_merge_log[of "Node Leaf x 1 Leaf" t]
by(simp add: t_insert_def split: tree.split)
(* FIXME mv ? *)
@@ -234,7 +234,7 @@
proof(cases t)
case Leaf thus ?thesis using assms by simp
next
- case [simp]: (Node _ t1 _ t2)
+ 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)