src/HOL/Data_Structures/AVL_Set.thy

+(*
+Author:     Tobias Nipkow
+Derived from AFP entry AVL.
+*)
+
+section "AVL Tree Implementation of Sets"
+
+theory AVL_Set
+imports Isin2
+begin
+
+type_synonym 'a avl_tree = "('a,nat) tree"
+
+text {* Invariant: *}
+
+fun avl :: "'a avl_tree \<Rightarrow> bool" where
+"avl Leaf = True" |
+"avl (Node h l a r) =
+ ((height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1) \<and> 
+  h = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
+
+fun ht :: "'a avl_tree \<Rightarrow> nat" where
+"ht Leaf = 0" |
+"ht (Node h l a r) = h"
+
+definition node :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+"node l a r = Node (max (ht l) (ht r) + 1) l a r"
+
+definition node_bal_l :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+"node_bal_l l a r = (
+  if ht l = ht r + 2 then (case l of 
+    Node _ bl b br \<Rightarrow> (if ht bl < ht br
+    then case br of
+      Node _ cl c cr \<Rightarrow> node (node bl b cl) c (node cr a r)
+    else node bl b (node br a r)))
+  else node l a r)"
+
+definition node_bal_r :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+"node_bal_r l a r = (
+  if ht r = ht l + 2 then (case r of
+    Node _ bl b br \<Rightarrow> (if ht bl > ht br
+    then case bl of
+      Node _ cl c cr \<Rightarrow> node (node l a cl) c (node cr b br)
+    else node (node l a bl) b br))
+  else node l a r)"
+
+fun insert :: "'a::order \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+"insert x Leaf = Node 1 Leaf x Leaf" |
+"insert x (Node h l a r) = 
+   (if x=a then Node h l a r
+    else if x<a
+      then node_bal_l (insert x l) a r
+      else node_bal_r l a (insert x r))"
+
+fun delete_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
+"delete_max (Node _ l a Leaf) = (l,a)" |
+"delete_max (Node _ l a r) = (
+  let (r',a') = delete_max r in
+  (node_bal_l l a r', a'))"
+
+lemmas delete_max_induct = delete_max.induct[case_names Leaf Node]
+
+fun delete_root :: "'a avl_tree \<Rightarrow> 'a avl_tree" where
+"delete_root (Node h Leaf a r) = r" |
+"delete_root (Node h l a Leaf) = l" |
+"delete_root (Node h l a r) =
+  (let (l', a') = delete_max l in node_bal_r l' a' r)"
+
+lemmas delete_root_cases = delete_root.cases[case_names Leaf_t Node_Leaf Node_Node]
+
+fun delete :: "'a::order \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+"delete _ Leaf = Leaf" |
+"delete x (Node h l a r) = (
+   if x = a then delete_root (Node h l a r)
+   else if x < a then node_bal_r (delete x l) a r
+   else node_bal_l l a (delete x r))"
+
+
+subsection {* Functional Correctness Proofs *}
+
+text{* Very different from the AFP/AVL proofs *}
+
+
+subsubsection "Proofs for insert"
+
+lemma inorder_node_bal_l:
+  "inorder (node_bal_l l a r) = inorder l @ a # inorder r"
+by (auto simp: node_def node_bal_l_def split:tree.splits)
+
+lemma inorder_node_bal_r:
+  "inorder (node_bal_r l a r) = inorder l @ a # inorder r"
+by (auto simp: node_def node_bal_r_def split:tree.splits)
+
+theorem inorder_insert:
+  "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
+by (induct t) 
+   (auto simp: ins_list_simps inorder_node_bal_l inorder_node_bal_r)
+
+
+subsubsection "Proofs for delete"
+
+lemma inorder_delete_maxD:
+  "\<lbrakk> delete_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
+   inorder t' @ [a] = inorder t"
+by(induction t arbitrary: t' rule: delete_max.induct)
+  (auto simp: inorder_node_bal_l split: prod.splits tree.split)
+
+lemma inorder_delete_root:
+  "inorder (delete_root (Node h l a r)) = inorder l @ inorder r"
+by(induction "Node h l a r" arbitrary: l a r h rule: delete_root.induct)
+  (auto simp: inorder_node_bal_r inorder_delete_maxD split: prod.splits)
+
+theorem inorder_delete:
+  "sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
+by(induction t)
+  (auto simp: del_list_simps inorder_node_bal_l inorder_node_bal_r
+    inorder_delete_root inorder_delete_maxD split: prod.splits)
+
+
+subsubsection "Overall functional correctness"
+
+interpretation Set_by_Ordered
+where empty = Leaf and isin = isin and insert = insert and delete = delete
+and inorder = inorder and wf = "\<lambda>_. True"
+proof (standard, goal_cases)
+  case 1 show ?case by simp
+next
+  case 2 thus ?case by(simp add: isin_set)
+next
+  case 3 thus ?case by(simp add: inorder_insert)
+next
+  case 4 thus ?case by(simp add: inorder_delete)
+next
+  case 5 thus ?case ..
+qed
+
+
+subsection {* AVL invariants *}
+
+text{* Essentially the AFP/AVL proofs *}
+
+
+subsubsection {* Insertion maintains AVL balance *}
+
+declare Let_def [simp]
+
+lemma [simp]: "avl t \<Longrightarrow> ht t = height t"
+by (induct t) simp_all
+
+lemma height_node_bal_l:
+  "\<lbrakk> height l = height r + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
+   height (node_bal_l l a r) = height r + 2 \<or>
+   height (node_bal_l l a r) = height r + 3"
+by (cases l) (auto simp:node_def node_bal_l_def split:tree.split)
+       
+lemma height_node_bal_r:
+  "\<lbrakk> height r = height l + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
+   height (node_bal_r l a r) = height l + 2 \<or>
+   height (node_bal_r l a r) = height l + 3"
+by (cases r) (auto simp add:node_def node_bal_r_def split:tree.split)
+
+lemma [simp]: "height(node l a r) = max (height l) (height r) + 1"
+by (simp add: node_def)
+
+lemma avl_node:
+  "\<lbrakk> avl l; avl r;
+     height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1
+   \<rbrakk> \<Longrightarrow> avl(node l a r)"
+by (auto simp add:max_def node_def)
+
+lemma height_node_bal_l2:
+  "\<lbrakk> avl l; avl r; height l \<noteq> height r + 2 \<rbrakk> \<Longrightarrow>
+   height (node_bal_l l a r) = (1 + max (height l) (height r))"
+by (cases l, cases r) (simp_all add: node_bal_l_def)
+
+lemma height_node_bal_r2:
+  "\<lbrakk> avl l;  avl r;  height r \<noteq> height l + 2 \<rbrakk> \<Longrightarrow>
+   height (node_bal_r l a r) = (1 + max (height l) (height r))"
+by (cases l, cases r) (simp_all add: node_bal_r_def)
+
+lemma avl_node_bal_l: 
+  assumes "avl l" "avl r" and "height l = height r \<or> height l = height r + 1
+    \<or> height r = height l + 1 \<or> height l = height r + 2" 
+  shows "avl(node_bal_l l a r)"
+proof(cases l)
+  case Leaf
+  with assms show ?thesis by (simp add: node_def node_bal_l_def)
+next
+  case (Node ln ll lr lh)
+  with assms show ?thesis
+  proof(cases "height l = height r + 2")
+    case True
+    from True Node assms show ?thesis
+      by (auto simp: node_bal_l_def intro!: avl_node split: tree.split) arith+
+  next
+    case False
+    with assms show ?thesis by (simp add: avl_node node_bal_l_def)
+  qed
+qed
+
+lemma avl_node_bal_r: 
+  assumes "avl l" and "avl r" and "height l = height r \<or> height l = height r + 1
+    \<or> height r = height l + 1 \<or> height r = height l + 2" 
+  shows "avl(node_bal_r l a r)"
+proof(cases r)
+  case Leaf
+  with assms show ?thesis by (simp add: node_def node_bal_r_def)
+next
+  case (Node rn rl rr rh)
+  with assms show ?thesis
+  proof(cases "height r = height l + 2")
+    case True
+      from True Node assms show ?thesis
+        by (auto simp: node_bal_r_def intro!: avl_node split: tree.split) arith+
+  next
+    case False
+    with assms show ?thesis by (simp add: node_bal_r_def avl_node)
+  qed
+qed
+
+(* It appears that these two properties need to be proved simultaneously: *)
+
+text{* Insertion maintains the AVL property: *}
+
+theorem avl_insert_aux:
+  assumes "avl t"
+  shows "avl(insert x t)"
+        "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
+using assms
+proof (induction t)
+  case (Node h l a r)
+  case 1
+  with Node show ?case
+  proof(cases "x = a")
+    case True
+    with Node 1 show ?thesis by simp
+  next
+    case False
+    with Node 1 show ?thesis 
+    proof(cases "x<a")
+      case True
+      with Node 1 show ?thesis by (auto simp add:avl_node_bal_l)
+    next
+      case False
+      with Node 1 `x\<noteq>a` show ?thesis by (auto simp add:avl_node_bal_r)
+    qed
+  qed
+  case 2
+  from 2 Node show ?case
+  proof(cases "x = a")
+    case True
+    with Node 1 show ?thesis by simp
+  next
+    case False
+    with Node 1 show ?thesis 
+     proof(cases "x<a")
+      case True
+      with Node 2 show ?thesis
+      proof(cases "height (insert x l) = height r + 2")
+        case False with Node 2 `x < a` show ?thesis by (auto simp: height_node_bal_l2)
+      next
+        case True 
+        hence "(height (node_bal_l (insert x l) a r) = height r + 2) \<or>
+          (height (node_bal_l (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
+          using Node 2 by (intro height_node_bal_l) simp_all
+        thus ?thesis
+        proof
+          assume ?A
+          with 2 `x < a` show ?thesis by (auto)
+        next
+          assume ?B
+          with True 1 Node(2) `x < a` show ?thesis by (simp) arith
+        qed
+      qed
+    next
+      case False
+      with Node 2 show ?thesis 
+      proof(cases "height (insert x r) = height l + 2")
+        case False
+        with Node 2 `\<not>x < a` show ?thesis by (auto simp: height_node_bal_r2)
+      next
+        case True 
+        hence "(height (node_bal_r l a (insert x r)) = height l + 2) \<or>
+          (height (node_bal_r l a (insert x r)) = height l + 3)"  (is "?A \<or> ?B")
+          using Node 2 by (intro height_node_bal_r) simp_all
+        thus ?thesis 
+        proof
+          assume ?A
+          with 2 `\<not>x < a` show ?thesis by (auto)
+        next
+          assume ?B
+          with True 1 Node(4) `\<not>x < a` show ?thesis by (simp) arith
+        qed
+      qed
+    qed
+  qed
+qed simp_all
+
+
+subsubsection {* Deletion maintains AVL balance *}
+
+lemma avl_delete_max:
+  assumes "avl x" and "x \<noteq> Leaf"
+  shows "avl (fst (delete_max x))" "height x = height(fst (delete_max x)) \<or>
+         height x = height(fst (delete_max x)) + 1"
+using assms
+proof (induct x rule: delete_max_induct)
+  case (Node h l a rh rl b rr)
+  case 1
+  with Node have "avl l" "avl (fst (delete_max (Node rh rl b rr)))" by auto
+  with 1 Node have "avl (node_bal_l l a (fst (delete_max (Node rh rl b rr))))"
+    by (intro avl_node_bal_l) fastforce+
+  thus ?case 
+    by (auto simp: height_node_bal_l height_node_bal_l2
+      linorder_class.max.absorb1 linorder_class.max.absorb2
+      split:prod.split)
+next
+  case (Node h l a rh rl b rr)
+  case 2
+  let ?r = "Node rh rl b rr"
+  let ?r' = "fst (delete_max ?r)"
+  from `avl x` Node 2 have "avl l" and "avl ?r" by simp_all
+  thus ?case using Node 2 height_node_bal_l[of l ?r' a] height_node_bal_l2[of l ?r' a]
+    apply (auto split:prod.splits simp del:avl.simps) by arith+
+qed auto
+
+lemma avl_delete_root:
+  assumes "avl t" and "t \<noteq> Leaf"
+  shows "avl(delete_root t)" 
+using assms
+proof (cases t rule:delete_root_cases)
+  case (Node_Node h lh ll ln lr n rh rl rn rr)
+  let ?l = "Node lh ll ln lr"
+  let ?r = "Node rh rl rn rr"
+  let ?l' = "fst (delete_max ?l)"
+  from `avl t` and Node_Node have "avl ?r" by simp
+  from `avl t` and Node_Node have "avl ?l" by simp
+  hence "avl(?l')" "height ?l = height(?l') \<or>
+         height ?l = height(?l') + 1" by (rule avl_delete_max,simp)+
+  with `avl t` Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
+            \<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
+  with `avl ?l'` `avl ?r` have "avl(node_bal_r ?l' (snd(delete_max ?l)) ?r)"
+    by (rule avl_node_bal_r)
+  with Node_Node show ?thesis by (auto split:prod.splits)
+qed simp_all
+
+lemma height_delete_root:
+  assumes "avl t" and "t \<noteq> Leaf" 
+  shows "height t = height(delete_root t) \<or> height t = height(delete_root t) + 1"
+using assms
+proof (cases t rule: delete_root_cases)
+  case (Node_Node h lh ll ln lr n rh rl rn rr)
+  let ?l = "Node lh ll ln lr"
+  let ?r = "Node rh rl rn rr"
+  let ?l' = "fst (delete_max ?l)"
+  let ?t' = "node_bal_r ?l' (snd(delete_max ?l)) ?r"
+  from `avl t` and Node_Node have "avl ?r" by simp
+  from `avl t` and Node_Node have "avl ?l" by simp
+  hence "avl(?l')"  by (rule avl_delete_max,simp)
+  have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using `avl ?l` by (intro avl_delete_max) auto
+  have t_height: "height t = 1 + max (height ?l) (height ?r)" using `avl t` Node_Node by simp
+  have "height t = height ?t' \<or> height t = height ?t' + 1" using  `avl t` Node_Node
+  proof(cases "height ?r = height ?l' + 2")
+    case False
+    show ?thesis using l'_height t_height False by (subst  height_node_bal_r2[OF `avl ?l'` `avl ?r` False])+ arith
+  next
+    case True
+    show ?thesis
+    proof(cases rule: disjE[OF height_node_bal_r[OF True `avl ?l'` `avl ?r`, of "snd (delete_max ?l)"]])
+      case 1
+      thus ?thesis using l'_height t_height True by arith
+    next
+      case 2
+      thus ?thesis using l'_height t_height True by arith
+    qed
+  qed
+  thus ?thesis using Node_Node by (auto split:prod.splits)
+qed simp_all
+
+text{* Deletion maintains the AVL property: *}
+
+theorem avl_delete_aux:
+  assumes "avl t" 
+  shows "avl(delete x t)" and "height t = (height (delete x t)) \<or> height t = height (delete x t) + 1"
+using assms
+proof (induct t)
+  case (Node h l n r)
+  case 1
+  with Node show ?case
+  proof(cases "x = n")
+    case True
+    with Node 1 show ?thesis by (auto simp:avl_delete_root)
+  next
+    case False
+    with Node 1 show ?thesis 
+    proof(cases "x<n")
+      case True
+      with Node 1 show ?thesis by (auto simp add:avl_node_bal_r)
+    next
+      case False
+      with Node 1 `x\<noteq>n` show ?thesis by (auto simp add:avl_node_bal_l)
+    qed
+  qed
+  case 2
+  with Node show ?case
+  proof(cases "x = n")
+    case True
+    with 1 have "height (Node h l n r) = height(delete_root (Node h l n r))
+      \<or> height (Node h l n r) = height(delete_root (Node h l n r)) + 1"
+      by (subst height_delete_root,simp_all)
+    with True show ?thesis by simp
+  next
+    case False
+    with Node 1 show ?thesis 
+     proof(cases "x<n")
+      case True
+      show ?thesis
+      proof(cases "height r = height (delete x l) + 2")
+        case False with Node 1 `x < n` show ?thesis by(auto simp: node_bal_r_def)
+      next
+        case True 
+        hence "(height (node_bal_r (delete x l) n r) = height (delete x l) + 2) \<or>
+          height (node_bal_r (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
+          using Node 2 by (intro height_node_bal_r) auto
+        thus ?thesis 
+        proof
+          assume ?A
+          with `x < n` Node 2 show ?thesis by(auto simp: node_bal_r_def)
+        next
+          assume ?B
+          with `x < n` Node 2 show ?thesis by(auto simp: node_bal_r_def)
+        qed
+      qed
+    next
+      case False
+      show ?thesis
+      proof(cases "height l = height (delete x r) + 2")
+        case False with Node 1 `\<not>x < n` `x \<noteq> n` show ?thesis by(auto simp: node_bal_l_def)
+      next
+        case True 
+        hence "(height (node_bal_l l n (delete x r)) = height (delete x r) + 2) \<or>
+          height (node_bal_l l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
+          using Node 2 by (intro height_node_bal_l) auto
+        thus ?thesis 
+        proof
+          assume ?A
+          with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: node_bal_l_def)
+        next
+          assume ?B
+          with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: node_bal_l_def)
+        qed
+      qed
+    qed
+  qed
+qed simp_all
+
+end