merged
authornipkow
Tue Jun 12 07:18:18 2018 +0200 (11 months ago)
changeset 68423c1db7503dbaa
parent 68421 e082a36dc35d
parent 68422 0a3a36fa1d63
child 68427 f75d765a281f
child 68431 b294e095f64c
merged
     1.1 --- a/src/HOL/Data_Structures/AVL_Map.thy	Mon Jun 11 22:43:52 2018 +0100
     1.2 +++ b/src/HOL/Data_Structures/AVL_Map.thy	Tue Jun 12 07:18:18 2018 +0200
     1.3 @@ -23,7 +23,7 @@
     1.4     GT \<Rightarrow> balL l (a,b) (delete x r))"
     1.5  
     1.6  
     1.7 -subsection \<open>Functional Correctness Proofs\<close>
     1.8 +subsection \<open>Functional Correctness\<close>
     1.9  
    1.10  theorem inorder_update:
    1.11    "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
    1.12 @@ -36,9 +36,153 @@
    1.13    (auto simp: del_list_simps inorder_balL inorder_balR
    1.14       inorder_del_root inorder_split_maxD split: prod.splits)
    1.15  
    1.16 +
    1.17 +subsection \<open>AVL invariants\<close>
    1.18 +
    1.19 +
    1.20 +subsubsection \<open>Insertion maintains AVL balance\<close>
    1.21 +
    1.22 +theorem avl_update:
    1.23 +  assumes "avl t"
    1.24 +  shows "avl(update x y t)"
    1.25 +        "(height (update x y t) = height t \<or> height (update x y t) = height t + 1)"
    1.26 +using assms
    1.27 +proof (induction x y t rule: update.induct)
    1.28 +  case eq2: (2 x y l a b h r)
    1.29 +  case 1
    1.30 +  show ?case
    1.31 +  proof(cases "x = a")
    1.32 +    case True with eq2 1 show ?thesis by simp
    1.33 +  next
    1.34 +    case False
    1.35 +    with eq2 1 show ?thesis 
    1.36 +    proof(cases "x<a")
    1.37 +      case True with eq2 1 show ?thesis by (auto simp add:avl_balL)
    1.38 +    next
    1.39 +      case False with eq2 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balR)
    1.40 +    qed
    1.41 +  qed
    1.42 +  case 2
    1.43 +  show ?case
    1.44 +  proof(cases "x = a")
    1.45 +    case True with eq2 1 show ?thesis by simp
    1.46 +  next
    1.47 +    case False
    1.48 +    show ?thesis 
    1.49 +    proof(cases "x<a")
    1.50 +      case True
    1.51 +      show ?thesis
    1.52 +      proof(cases "height (update x y l) = height r + 2")
    1.53 +        case False with eq2 2 \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
    1.54 +      next
    1.55 +        case True 
    1.56 +        hence "(height (balL (update x y l) (a,b) r) = height r + 2) \<or>
    1.57 +          (height (balL (update x y l) (a,b) r) = height r + 3)" (is "?A \<or> ?B")
    1.58 +          using eq2 2 \<open>x<a\<close> by (intro height_balL) simp_all
    1.59 +        thus ?thesis
    1.60 +        proof
    1.61 +          assume ?A with 2 \<open>x < a\<close> show ?thesis by (auto)
    1.62 +        next
    1.63 +          assume ?B with True 1 eq2(2) \<open>x < a\<close> show ?thesis by (simp) arith
    1.64 +        qed
    1.65 +      qed
    1.66 +    next
    1.67 +      case False
    1.68 +      show ?thesis
    1.69 +      proof(cases "height (update x y r) = height l + 2")
    1.70 +        case False with eq2 2 \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
    1.71 +      next
    1.72 +        case True 
    1.73 +        hence "(height (balR l (a,b) (update x y r)) = height l + 2) \<or>
    1.74 +          (height (balR l (a,b) (update x y r)) = height l + 3)"  (is "?A \<or> ?B")
    1.75 +          using eq2 2 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> by (intro height_balR) simp_all
    1.76 +        thus ?thesis 
    1.77 +        proof
    1.78 +          assume ?A with 2 \<open>\<not>x < a\<close> show ?thesis by (auto)
    1.79 +        next
    1.80 +          assume ?B with True 1 eq2(4) \<open>\<not>x < a\<close> show ?thesis by (simp) arith
    1.81 +        qed
    1.82 +      qed
    1.83 +    qed
    1.84 +  qed
    1.85 +qed simp_all
    1.86 +
    1.87 +
    1.88 +subsubsection \<open>Deletion maintains AVL balance\<close>
    1.89 +
    1.90 +theorem avl_delete:
    1.91 +  assumes "avl t" 
    1.92 +  shows "avl(delete x t)" and "height t = (height (delete x t)) \<or> height t = height (delete x t) + 1"
    1.93 +using assms
    1.94 +proof (induct t)
    1.95 +  case (Node l n h r)
    1.96 +  obtain a b where [simp]: "n = (a,b)" by fastforce
    1.97 +  case 1
    1.98 +  show ?case
    1.99 +  proof(cases "x = a")
   1.100 +    case True with Node 1 show ?thesis by (auto simp:avl_del_root)
   1.101 +  next
   1.102 +    case False
   1.103 +    show ?thesis 
   1.104 +    proof(cases "x<a")
   1.105 +      case True with Node 1 show ?thesis by (auto simp add:avl_balR)
   1.106 +    next
   1.107 +      case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balL)
   1.108 +    qed
   1.109 +  qed
   1.110 +  case 2
   1.111 +  show ?case
   1.112 +  proof(cases "x = a")
   1.113 +    case True
   1.114 +    with 1 have "height (Node l n h r) = height(del_root (Node l n h r))
   1.115 +      \<or> height (Node l n h r) = height(del_root (Node l n h r)) + 1"
   1.116 +      by (subst height_del_root,simp_all)
   1.117 +    with True show ?thesis by simp
   1.118 +  next
   1.119 +    case False
   1.120 +    show ?thesis 
   1.121 +    proof(cases "x<a")
   1.122 +      case True
   1.123 +      show ?thesis
   1.124 +      proof(cases "height r = height (delete x l) + 2")
   1.125 +        case False with Node 1 \<open>x < a\<close> show ?thesis by(auto simp: balR_def)
   1.126 +      next
   1.127 +        case True 
   1.128 +        hence "(height (balR (delete x l) n r) = height (delete x l) + 2) \<or>
   1.129 +          height (balR (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
   1.130 +          using Node 2 by (intro height_balR) auto
   1.131 +        thus ?thesis 
   1.132 +        proof
   1.133 +          assume ?A with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
   1.134 +        next
   1.135 +          assume ?B with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
   1.136 +        qed
   1.137 +      qed
   1.138 +    next
   1.139 +      case False
   1.140 +      show ?thesis
   1.141 +      proof(cases "height l = height (delete x r) + 2")
   1.142 +        case False with Node 1 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> show ?thesis by(auto simp: balL_def)
   1.143 +      next
   1.144 +        case True 
   1.145 +        hence "(height (balL l n (delete x r)) = height (delete x r) + 2) \<or>
   1.146 +          height (balL l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
   1.147 +          using Node 2 by (intro height_balL) auto
   1.148 +        thus ?thesis 
   1.149 +        proof
   1.150 +          assume ?A with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
   1.151 +        next
   1.152 +          assume ?B with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
   1.153 +        qed
   1.154 +      qed
   1.155 +    qed
   1.156 +  qed
   1.157 +qed simp_all
   1.158 +
   1.159 +
   1.160  interpretation Map_by_Ordered
   1.161  where empty = Leaf and lookup = lookup and update = update and delete = delete
   1.162 -and inorder = inorder and inv = "\<lambda>_. True"
   1.163 +and inorder = inorder and inv = avl
   1.164  proof (standard, goal_cases)
   1.165    case 1 show ?case by simp
   1.166  next
   1.167 @@ -47,6 +191,12 @@
   1.168    case 3 thus ?case by(simp add: inorder_update)
   1.169  next
   1.170    case 4 thus ?case by(simp add: inorder_delete)
   1.171 -qed auto
   1.172 +next
   1.173 +  case 5 show ?case by simp
   1.174 +next
   1.175 +  case 6 thus ?case by(simp add: avl_update(1))
   1.176 +next
   1.177 +  case 7 thus ?case by(simp add: avl_delete(1))
   1.178 +qed
   1.179  
   1.180  end
     2.1 --- a/src/HOL/Data_Structures/AVL_Set.thy	Mon Jun 11 22:43:52 2018 +0100
     2.2 +++ b/src/HOL/Data_Structures/AVL_Set.thy	Tue Jun 12 07:18:18 2018 +0200
     2.3 @@ -121,22 +121,6 @@
     2.4      inorder_del_root inorder_split_maxD split: prod.splits)
     2.5  
     2.6  
     2.7 -subsubsection "Overall functional correctness"
     2.8 -
     2.9 -interpretation Set_by_Ordered
    2.10 -where empty = Leaf and isin = isin and insert = insert and delete = delete
    2.11 -and inorder = inorder and inv = "\<lambda>_. True"
    2.12 -proof (standard, goal_cases)
    2.13 -  case 1 show ?case by simp
    2.14 -next
    2.15 -  case 2 thus ?case by(simp add: isin_set_inorder)
    2.16 -next
    2.17 -  case 3 thus ?case by(simp add: inorder_insert)
    2.18 -next
    2.19 -  case 4 thus ?case by(simp add: inorder_delete)
    2.20 -qed (rule TrueI)+
    2.21 -
    2.22 -
    2.23  subsection \<open>AVL invariants\<close>
    2.24  
    2.25  text\<open>Essentially the AFP/AVL proofs\<close>
    2.26 @@ -224,7 +208,7 @@
    2.27  
    2.28  text\<open>Insertion maintains the AVL property:\<close>
    2.29  
    2.30 -theorem avl_insert_aux:
    2.31 +theorem avl_insert:
    2.32    assumes "avl t"
    2.33    shows "avl(insert x t)"
    2.34          "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
    2.35 @@ -232,32 +216,28 @@
    2.36  proof (induction t)
    2.37    case (Node l a h r)
    2.38    case 1
    2.39 -  with Node show ?case
    2.40 +  show ?case
    2.41    proof(cases "x = a")
    2.42 -    case True
    2.43 -    with Node 1 show ?thesis by simp
    2.44 +    case True with Node 1 show ?thesis by simp
    2.45    next
    2.46      case False
    2.47 -    with Node 1 show ?thesis 
    2.48 +    show ?thesis 
    2.49      proof(cases "x<a")
    2.50 -      case True
    2.51 -      with Node 1 show ?thesis by (auto simp add:avl_balL)
    2.52 +      case True with Node 1 show ?thesis by (auto simp add:avl_balL)
    2.53      next
    2.54 -      case False
    2.55 -      with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balR)
    2.56 +      case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balR)
    2.57      qed
    2.58    qed
    2.59    case 2
    2.60 -  from 2 Node show ?case
    2.61 +  show ?case
    2.62    proof(cases "x = a")
    2.63 -    case True
    2.64 -    with Node 1 show ?thesis by simp
    2.65 +    case True with Node 1 show ?thesis by simp
    2.66    next
    2.67      case False
    2.68 -    with Node 1 show ?thesis 
    2.69 -     proof(cases "x<a")
    2.70 +    show ?thesis 
    2.71 +    proof(cases "x<a")
    2.72        case True
    2.73 -      with Node 2 show ?thesis
    2.74 +      show ?thesis
    2.75        proof(cases "height (insert x l) = height r + 2")
    2.76          case False with Node 2 \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
    2.77        next
    2.78 @@ -267,19 +247,16 @@
    2.79            using Node 2 by (intro height_balL) simp_all
    2.80          thus ?thesis
    2.81          proof
    2.82 -          assume ?A
    2.83 -          with 2 \<open>x < a\<close> show ?thesis by (auto)
    2.84 +          assume ?A with 2 \<open>x < a\<close> show ?thesis by (auto)
    2.85          next
    2.86 -          assume ?B
    2.87 -          with True 1 Node(2) \<open>x < a\<close> show ?thesis by (simp) arith
    2.88 +          assume ?B with True 1 Node(2) \<open>x < a\<close> show ?thesis by (simp) arith
    2.89          qed
    2.90        qed
    2.91      next
    2.92        case False
    2.93 -      with Node 2 show ?thesis 
    2.94 +      show ?thesis 
    2.95        proof(cases "height (insert x r) = height l + 2")
    2.96 -        case False
    2.97 -        with Node 2 \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
    2.98 +        case False with Node 2 \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
    2.99        next
   2.100          case True 
   2.101          hence "(height (balR l a (insert x r)) = height l + 2) \<or>
   2.102 @@ -287,11 +264,9 @@
   2.103            using Node 2 by (intro height_balR) simp_all
   2.104          thus ?thesis 
   2.105          proof
   2.106 -          assume ?A
   2.107 -          with 2 \<open>\<not>x < a\<close> show ?thesis by (auto)
   2.108 +          assume ?A with 2 \<open>\<not>x < a\<close> show ?thesis by (auto)
   2.109          next
   2.110 -          assume ?B
   2.111 -          with True 1 Node(4) \<open>\<not>x < a\<close> show ?thesis by (simp) arith
   2.112 +          assume ?B with True 1 Node(4) \<open>\<not>x < a\<close> show ?thesis by (simp) arith
   2.113          qed
   2.114        qed
   2.115      qed
   2.116 @@ -310,9 +285,7 @@
   2.117    case (Node l a h r)
   2.118    case 1
   2.119    thus ?case using Node
   2.120 -    by (auto simp: height_balL height_balL2 avl_balL
   2.121 -      linorder_class.max.absorb1 linorder_class.max.absorb2
   2.122 -      split:prod.split)
   2.123 +    by (auto simp: height_balL height_balL2 avl_balL split:prod.split)
   2.124  next
   2.125    case (Node l a h r)
   2.126    case 2
   2.127 @@ -360,16 +333,15 @@
   2.128    have "height t = height ?t' \<or> height t = height ?t' + 1" using  \<open>avl t\<close> Node_Node
   2.129    proof(cases "height ?r = height ?l' + 2")
   2.130      case False
   2.131 -    show ?thesis using l'_height t_height False by (subst  height_balR2[OF \<open>avl ?l'\<close> \<open>avl ?r\<close> False])+ arith
   2.132 +    show ?thesis using l'_height t_height False
   2.133 +      by (subst height_balR2[OF \<open>avl ?l'\<close> \<open>avl ?r\<close> False])+ arith
   2.134    next
   2.135      case True
   2.136      show ?thesis
   2.137      proof(cases rule: disjE[OF height_balR[OF True \<open>avl ?l'\<close> \<open>avl ?r\<close>, of "snd (split_max ?l)"]])
   2.138 -      case 1
   2.139 -      thus ?thesis using l'_height t_height True by arith
   2.140 +      case 1 thus ?thesis using l'_height t_height True by arith
   2.141      next
   2.142 -      case 2
   2.143 -      thus ?thesis using l'_height t_height True by arith
   2.144 +      case 2 thus ?thesis using l'_height t_height True by arith
   2.145      qed
   2.146    qed
   2.147    thus ?thesis using Node_Node by (auto split:prod.splits)
   2.148 @@ -377,30 +349,27 @@
   2.149  
   2.150  text\<open>Deletion maintains the AVL property:\<close>
   2.151  
   2.152 -theorem avl_delete_aux:
   2.153 +theorem avl_delete:
   2.154    assumes "avl t" 
   2.155    shows "avl(delete x t)" and "height t = (height (delete x t)) \<or> height t = height (delete x t) + 1"
   2.156  using assms
   2.157  proof (induct t)
   2.158    case (Node l n h r)
   2.159    case 1
   2.160 -  with Node show ?case
   2.161 +  show ?case
   2.162    proof(cases "x = n")
   2.163 -    case True
   2.164 -    with Node 1 show ?thesis by (auto simp:avl_del_root)
   2.165 +    case True with Node 1 show ?thesis by (auto simp:avl_del_root)
   2.166    next
   2.167      case False
   2.168 -    with Node 1 show ?thesis 
   2.169 +    show ?thesis 
   2.170      proof(cases "x<n")
   2.171 -      case True
   2.172 -      with Node 1 show ?thesis by (auto simp add:avl_balR)
   2.173 +      case True with Node 1 show ?thesis by (auto simp add:avl_balR)
   2.174      next
   2.175 -      case False
   2.176 -      with Node 1 \<open>x\<noteq>n\<close> show ?thesis by (auto simp add:avl_balL)
   2.177 +      case False with Node 1 \<open>x\<noteq>n\<close> show ?thesis by (auto simp add:avl_balL)
   2.178      qed
   2.179    qed
   2.180    case 2
   2.181 -  with Node show ?case
   2.182 +  show ?case
   2.183    proof(cases "x = n")
   2.184      case True
   2.185      with 1 have "height (Node l n h r) = height(del_root (Node l n h r))
   2.186 @@ -409,8 +378,8 @@
   2.187      with True show ?thesis by simp
   2.188    next
   2.189      case False
   2.190 -    with Node 1 show ?thesis 
   2.191 -     proof(cases "x<n")
   2.192 +    show ?thesis 
   2.193 +    proof(cases "x<n")
   2.194        case True
   2.195        show ?thesis
   2.196        proof(cases "height r = height (delete x l) + 2")
   2.197 @@ -422,11 +391,9 @@
   2.198            using Node 2 by (intro height_balR) auto
   2.199          thus ?thesis 
   2.200          proof
   2.201 -          assume ?A
   2.202 -          with \<open>x < n\<close> Node 2 show ?thesis by(auto simp: balR_def)
   2.203 +          assume ?A with \<open>x < n\<close> Node 2 show ?thesis by(auto simp: balR_def)
   2.204          next
   2.205 -          assume ?B
   2.206 -          with \<open>x < n\<close> Node 2 show ?thesis by(auto simp: balR_def)
   2.207 +          assume ?B with \<open>x < n\<close> Node 2 show ?thesis by(auto simp: balR_def)
   2.208          qed
   2.209        qed
   2.210      next
   2.211 @@ -441,11 +408,9 @@
   2.212            using Node 2 by (intro height_balL) auto
   2.213          thus ?thesis 
   2.214          proof
   2.215 -          assume ?A
   2.216 -          with \<open>\<not>x < n\<close> \<open>x \<noteq> n\<close> Node 2 show ?thesis by(auto simp: balL_def)
   2.217 +          assume ?A with \<open>\<not>x < n\<close> \<open>x \<noteq> n\<close> Node 2 show ?thesis by(auto simp: balL_def)
   2.218          next
   2.219 -          assume ?B
   2.220 -          with \<open>\<not>x < n\<close> \<open>x \<noteq> n\<close> Node 2 show ?thesis by(auto simp: balL_def)
   2.221 +          assume ?B with \<open>\<not>x < n\<close> \<open>x \<noteq> n\<close> Node 2 show ?thesis by(auto simp: balL_def)
   2.222          qed
   2.223        qed
   2.224      qed
   2.225 @@ -453,6 +418,28 @@
   2.226  qed simp_all
   2.227  
   2.228  
   2.229 +subsection "Overall correctness"
   2.230 +
   2.231 +interpretation Set_by_Ordered
   2.232 +where empty = Leaf and isin = isin and insert = insert and delete = delete
   2.233 +and inorder = inorder and inv = avl
   2.234 +proof (standard, goal_cases)
   2.235 +  case 1 show ?case by simp
   2.236 +next
   2.237 +  case 2 thus ?case by(simp add: isin_set_inorder)
   2.238 +next
   2.239 +  case 3 thus ?case by(simp add: inorder_insert)
   2.240 +next
   2.241 +  case 4 thus ?case by(simp add: inorder_delete)
   2.242 +next
   2.243 +  case 5 thus ?case by simp
   2.244 +next
   2.245 +  case 6 thus ?case by (simp add: avl_insert(1))
   2.246 +next
   2.247 +  case 7 thus ?case by (simp add: avl_delete(1))
   2.248 +qed
   2.249 +
   2.250 +
   2.251  subsection \<open>Height-Size Relation\<close>
   2.252  
   2.253  text \<open>Based on theorems by Daniel St\"uwe, Manuel Eberl and Peter Lammich.\<close>