src/HOL/Data_Structures/Leftist_Heap.thy
 changeset 68413 b56ed5010e69 parent 68021 b91a043c0dcb child 68492 c7e0a7bcacaf
```     1.1 --- a/src/HOL/Data_Structures/Leftist_Heap.thy	Mon Jun 11 08:15:43 2018 +0200
1.2 +++ b/src/HOL/Data_Structures/Leftist_Heap.thy	Mon Jun 11 16:29:27 2018 +0200
1.3 @@ -12,7 +12,7 @@
1.4
1.5  fun mset_tree :: "('a,'b) tree \<Rightarrow> 'a multiset" where
1.6  "mset_tree Leaf = {#}" |
1.7 -"mset_tree (Node _ l a r) = {#a#} + mset_tree l + mset_tree r"
1.8 +"mset_tree (Node l a _ r) = {#a#} + mset_tree l + mset_tree r"
1.9
1.10  type_synonym 'a lheap = "('a,nat)tree"
1.11
1.12 @@ -22,27 +22,27 @@
1.13
1.14  fun rk :: "'a lheap \<Rightarrow> nat" where
1.15  "rk Leaf = 0" |
1.16 -"rk (Node n _ _ _) = n"
1.17 +"rk (Node _ _ n _) = n"
1.18
1.19  text\<open>The invariants:\<close>
1.20
1.21  fun (in linorder) heap :: "('a,'b) tree \<Rightarrow> bool" where
1.22  "heap Leaf = True" |
1.23 -"heap (Node _ l m r) =
1.24 +"heap (Node l m _ r) =
1.25    (heap l \<and> heap r \<and> (\<forall>x \<in> set_mset(mset_tree l + mset_tree r). m \<le> x))"
1.26
1.27  fun ltree :: "'a lheap \<Rightarrow> bool" where
1.28  "ltree Leaf = True" |
1.29 -"ltree (Node n l a r) =
1.30 +"ltree (Node l a n r) =
1.31   (n = rank r + 1 \<and> rank l \<ge> rank r \<and> ltree l & ltree r)"
1.32
1.33  definition node :: "'a lheap \<Rightarrow> 'a \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
1.34  "node l a r =
1.35   (let rl = rk l; rr = rk r
1.36 -  in if rl \<ge> rr then Node (rr+1) l a r else Node (rl+1) r a l)"
1.37 +  in if rl \<ge> rr then Node l a (rr+1) r else Node r a (rl+1) l)"
1.38
1.39  fun get_min :: "'a lheap \<Rightarrow> 'a" where
1.40 -"get_min(Node n l a r) = a"
1.41 +"get_min(Node l a n r) = a"
1.42
1.43  text \<open>For function \<open>merge\<close>:\<close>
1.44  unbundle pattern_aliases
1.45 @@ -51,25 +51,25 @@
1.46  fun merge :: "'a::ord lheap \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
1.47  "merge Leaf t2 = t2" |
1.48  "merge t1 Leaf = t1" |
1.49 -"merge (Node n1 l1 a1 r1 =: t1) (Node n2 l2 a2 r2 =: t2) =
1.50 +"merge (Node l1 a1 n1 r1 =: t1) (Node l2 a2 n2 r2 =: t2) =
1.51     (if a1 \<le> a2 then node l1 a1 (merge r1 t2)
1.52      else node l2 a2 (merge r2 t1))"
1.53
1.54  lemma merge_code: "merge t1 t2 = (case (t1,t2) of
1.55    (Leaf, _) \<Rightarrow> t2 |
1.56    (_, Leaf) \<Rightarrow> t1 |
1.57 -  (Node n1 l1 a1 r1, Node n2 l2 a2 r2) \<Rightarrow>
1.58 +  (Node l1 a1 n1 r1, Node l2 a2 n2 r2) \<Rightarrow>
1.59      if a1 \<le> a2 then node l1 a1 (merge r1 t2) else node l2 a2 (merge r2 t1))"
1.60  by(induction t1 t2 rule: merge.induct) (simp_all split: tree.split)
1.61
1.62  hide_const (open) insert
1.63
1.64  definition insert :: "'a::ord \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
1.65 -"insert x t = merge (Node 1 Leaf x Leaf) t"
1.66 +"insert x t = merge (Node Leaf x 1 Leaf) t"
1.67
1.68  fun del_min :: "'a::ord lheap \<Rightarrow> 'a lheap" where
1.69  "del_min Leaf = Leaf" |
1.70 -"del_min (Node n l x r) = merge l r"
1.71 +"del_min (Node l x n r) = merge l r"
1.72
1.73
1.74  subsection "Lemmas"
1.75 @@ -104,7 +104,7 @@
1.76
1.77  lemma ltree_merge: "\<lbrakk> ltree l; ltree r \<rbrakk> \<Longrightarrow> ltree (merge l r)"
1.78  proof(induction l r rule: merge.induct)
1.79 -  case (3 n1 l1 a1 r1 n2 l2 a2 r2)
1.80 +  case (3 l1 a1 n1 r1 l2 a2 n2 r2)
1.81    show ?case (is "ltree(merge ?t1 ?t2)")
1.82    proof cases
1.83      assume "a1 \<le> a2"
1.84 @@ -173,14 +173,14 @@
1.85  proof(induction t)
1.86    case Leaf show ?case by simp
1.87  next
1.88 -  case (Node n l a r)
1.89 +  case (Node l a n r)
1.90    hence "rank r \<le> rank l" by simp
1.91    hence *: "(2::nat) ^ rank r \<le> 2 ^ rank l" by simp
1.92 -  have "(2::nat) ^ rank \<langle>n, l, a, r\<rangle> = 2 ^ rank r + 2 ^ rank r"
1.93 +  have "(2::nat) ^ rank \<langle>l, a, n, r\<rangle> = 2 ^ rank r + 2 ^ rank r"
1.95    also have "\<dots> \<le> size1 l + size1 r"
1.96      using Node * by (simp del: power_increasing_iff)
1.97 -  also have "\<dots> = size1 \<langle>n, l, a, r\<rangle>" by simp
1.98 +  also have "\<dots> = size1 \<langle>l, a, n, r\<rangle>" by simp
1.99    finally show ?case .
1.100  qed
1.101
1.102 @@ -189,16 +189,16 @@
1.103  fun t_merge :: "'a::ord lheap \<Rightarrow> 'a lheap \<Rightarrow> nat" where
1.104  "t_merge Leaf t2 = 1" |
1.105  "t_merge t2 Leaf = 1" |
1.106 -"t_merge (Node n1 l1 a1 r1 =: t1) (Node n2 l2 a2 r2 =: t2) =
1.107 +"t_merge (Node l1 a1 n1 r1 =: t1) (Node l2 a2 n2 r2 =: t2) =
1.108    (if a1 \<le> a2 then 1 + t_merge r1 t2
1.109     else 1 + t_merge r2 t1)"
1.110
1.111  definition t_insert :: "'a::ord \<Rightarrow> 'a lheap \<Rightarrow> nat" where
1.112 -"t_insert x t = t_merge (Node 1 Leaf x Leaf) t"
1.113 +"t_insert x t = t_merge (Node Leaf x 1 Leaf) t"
1.114
1.115  fun t_del_min :: "'a::ord lheap \<Rightarrow> nat" where
1.116  "t_del_min Leaf = 1" |
1.117 -"t_del_min (Node n l a r) = t_merge l r"
1.118 +"t_del_min (Node l a n r) = t_merge l r"
1.119
1.120  lemma t_merge_rank: "t_merge l r \<le> rank l + rank r + 1"
1.121  proof(induction l r rule: merge.induct)
1.122 @@ -213,7 +213,7 @@
1.123  by linarith
1.124
1.125  corollary t_insert_log: "ltree t \<Longrightarrow> t_insert x t \<le> log 2 (size1 t) + 2"
1.126 -using t_merge_log[of "Node 1 Leaf x Leaf" t]
1.127 +using t_merge_log[of "Node Leaf x 1 Leaf" t]
1.128  by(simp add: t_insert_def split: tree.split)
1.129
1.130  (* FIXME mv ? *)
1.131 @@ -234,7 +234,7 @@
1.132  proof(cases t)
1.133    case Leaf thus ?thesis using assms by simp
1.134  next
1.135 -  case [simp]: (Node _ t1 _ t2)
1.136 +  case [simp]: (Node t1 _ _ t2)
1.137    have "t_del_min t = t_merge t1 t2" by simp
1.138    also have "\<dots> \<le> log 2 (size1 t1) + log 2 (size1 t2) + 1"
1.139      using \<open>ltree t\<close> by (auto simp: t_merge_log simp del: t_merge.simps)
```