# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1443005535 -3600
# Node ID 5c9a9504f7131cf60c673f9e6859731c1ec57b5b
# Parent  b033aeaab1b8a8372f40dc7e68f37e49e3efcebc# Parent  c46faf9762f7c1a7ff0deeee2e9e3554a0a42a5c
Merge

diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/AList_Upd_Del.thy
--- a/src/HOL/Data_Structures/AList_Upd_Del.thy	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/Data_Structures/AList_Upd_Del.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -17,7 +17,7 @@
 "map_of [] = (\<lambda>a. None)" |
 "map_of ((x,y)#ps) = (\<lambda>a. if x=a then Some y else map_of ps a)"
 
-text \<open>Updating into an association list:\<close>
+text \<open>Updating an association list:\<close>
 
 fun upd_list :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) list \<Rightarrow> ('a*'b) list" where
 "upd_list a b [] = [(a,b)]" |
@@ -58,6 +58,7 @@
 by (meson le_less_trans map_of_None2 not_less sorted_snoc_iff)
 
 lemmas map_of_sorteds = map_of_sorted_Cons map_of_sorted_snoc
+lemmas map_of_simps = sorted_lems map_of_append map_of_sorteds
 
 
 subsection \<open>Lemmas for @{const upd_list}\<close>
@@ -136,4 +137,6 @@
 lemmas del_list_sorted =
   del_list_sorted1 del_list_sorted2 del_list_sorted3 del_list_sorted4 del_list_sorted5
 
+lemmas del_list_simps = sorted_lems del_list_sorted
+
 end
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/AVL_Map.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/AVL_Map.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -0,0 +1,54 @@
+(* Author: Tobias Nipkow *)
+
+section "AVL Tree Implementation of Maps"
+
+theory AVL_Map
+imports
+  AVL_Set
+  Lookup2
+begin
+
+fun update :: "'a::order \<Rightarrow> 'b \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
+"update x y Leaf = Node 1 Leaf (x,y) Leaf" |
+"update x y (Node h l (a,b) r) = 
+   (if x = a then Node h l (x,y) r else
+    if x < a then node_bal_l (update x y l) (a,b) r
+    else node_bal_r l (a,b) (update x y r))"
+
+fun delete :: "'a::order \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
+"delete _ Leaf = Leaf" |
+"delete x (Node h l (a,b) r) = (
+   if x = a then delete_root (Node h l (a,b) r) else
+   if x < a then node_bal_r (delete x l) (a,b) r
+   else node_bal_l l (a,b) (delete x r))"
+
+
+subsection {* Functional Correctness Proofs *}
+
+theorem inorder_update:
+  "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
+by (induct t) 
+   (auto simp: upd_list_simps inorder_node_bal_l inorder_node_bal_r)
+
+
+theorem inorder_delete:
+  "sorted1(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)
+
+
+interpretation Map_by_Ordered
+where empty = Leaf and lookup = lookup and update = update 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: lookup_eq)
+next
+  case 3 thus ?case by(simp add: inorder_update)
+next
+  case 4 thus ?case by(simp add: inorder_delete)
+qed (rule TrueI)+
+
+end
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/AVL_Set.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/AVL_Set.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -0,0 +1,457 @@
+(*
+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
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/List_Ins_Del.thy
--- a/src/HOL/Data_Structures/List_Ins_Del.thy	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/Data_Structures/List_Ins_Del.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -74,7 +74,7 @@
   ins_list x (xs @ y # ys) = ins_list x xs @ (y#ys)"
 by(induction xs) (auto simp: sorted_lems)
 
-lemmas ins_simps = sorted_lems ins_list_sorted1 ins_list_sorted2
+lemmas ins_list_simps = sorted_lems ins_list_sorted1 ins_list_sorted2
 
 
 subsection \<open>Delete one occurrence of an element from a list:\<close>
@@ -123,7 +123,7 @@
    del_list a (xs @ x # ys @ y # zs @ z # us) @ u # vs" 
 by (induction xs) (auto simp: sorted_Cons_iff del_list_sorted4)
 
-lemmas del_simps = sorted_lems
+lemmas del_list_simps = sorted_lems
   del_list_sorted1
   del_list_sorted2
   del_list_sorted3
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/Lookup2.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Lookup2.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -0,0 +1,21 @@
+(* Author: Tobias Nipkow *)
+
+section \<open>Function \textit{lookup} for Tree2\<close>
+
+theory Lookup2
+imports
+  Tree2
+  Map_by_Ordered
+begin
+
+fun lookup :: "('a::linorder * 'b, 'c) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
+"lookup Leaf x = None" |
+"lookup (Node _ l (a,b) r) x =
+  (if x < a then lookup l x else
+   if x > a then lookup r x else Some b)"
+
+lemma lookup_eq:
+  "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
+by(induction t) (auto simp: map_of_simps split: option.split)
+
+end
\ No newline at end of file
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/RBT.thy
--- a/src/HOL/Data_Structures/RBT.thy	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/Data_Structures/RBT.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -1,6 +1,6 @@
 (* Author: Tobias Nipkow *)
 
-section \<open>Red-Black Trees\<close>
+section \<open>Red-Black Tree\<close>
 
 theory RBT
 imports Tree2
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/RBT_Map.thy
--- a/src/HOL/Data_Structures/RBT_Map.thy	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/Data_Structures/RBT_Map.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -5,15 +5,9 @@
 theory RBT_Map
 imports
   RBT_Set
-  Map_by_Ordered
+  Lookup2
 begin
 
-fun lookup :: "('a::linorder * 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b option" where
-"lookup Leaf x = None" |
-"lookup (Node _ l (a,b) r) x =
-  (if x < a then lookup l x else
-   if x > a then lookup r x else Some b)"
-
 fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
 "update x y Leaf = R Leaf (x,y) Leaf" |
 "update x y (B l (a,b) r) =
@@ -41,12 +35,6 @@
 
 subsection "Functional Correctness Proofs"
 
-lemma lookup_eq:
-  "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
-by(induction t)
-  (auto simp: sorted_lems map_of_append map_of_sorteds split: option.split)
-
-
 lemma inorder_update:
   "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
 by(induction x y t rule: update.induct)
@@ -60,7 +48,7 @@
  "sorted1(inorder t2) \<Longrightarrow>  inorder(deleteR x t1 a t2) =
     inorder t1 @ a # del_list x (inorder t2)"
 by(induction x t1 and x t1 a t2 and x t1 a t2 rule: delete_deleteL_deleteR.induct)
-  (auto simp: del_list_sorted sorted_lems inorder_combine inorder_balL inorder_balR)
+  (auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
 
 
 interpretation Map_by_Ordered
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/RBT_Set.thy
--- a/src/HOL/Data_Structures/RBT_Set.thy	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/Data_Structures/RBT_Set.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -35,29 +35,27 @@
 
 lemma inorder_bal:
   "inorder(bal l a r) = inorder l @ a # inorder r"
-by(induction l a r rule: bal.induct) (auto simp: sorted_lems)
+by(induction l a r rule: bal.induct) (auto)
 
 lemma inorder_insert:
   "sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
-by(induction a t rule: insert.induct) (auto simp: ins_simps inorder_bal)
+by(induction a t rule: insert.induct) (auto simp: ins_list_simps inorder_bal)
 
 lemma inorder_red: "inorder(red t) = inorder t"
-by(induction t) (auto simp: sorted_lems)
+by(induction t) (auto)
 
 lemma inorder_balL:
   "inorder(balL l a r) = inorder l @ a # inorder r"
-by(induction l a r rule: balL.induct)
-  (auto simp: sorted_lems inorder_bal inorder_red)
+by(induction l a r rule: balL.induct)(auto simp: inorder_bal inorder_red)
 
 lemma inorder_balR:
   "inorder(balR l a r) = inorder l @ a # inorder r"
-by(induction l a r rule: balR.induct)
-  (auto simp: sorted_lems inorder_bal inorder_red)
+by(induction l a r rule: balR.induct) (auto simp: inorder_bal inorder_red)
 
 lemma inorder_combine:
   "inorder(combine l r) = inorder l @ inorder r"
 by(induction l r rule: combine.induct)
-  (auto simp: sorted_lems inorder_balL inorder_balR split: tree.split color.split)
+  (auto simp: inorder_balL inorder_balR split: tree.split color.split)
 
 lemma inorder_delete:
  "sorted(inorder t) \<Longrightarrow>  inorder(delete x t) = del_list x (inorder t)" and
@@ -66,7 +64,7 @@
  "sorted(inorder r) \<Longrightarrow>  inorder(deleteR x l a r) =
     inorder l @ a # del_list x (inorder r)"
 by(induction x t and x l a r and x l a r rule: delete_deleteL_deleteR.induct)
-  (auto simp: del_simps inorder_combine inorder_balL inorder_balR)
+  (auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
 
 interpretation Set_by_Ordered
 where empty = Leaf and isin = isin and insert = insert and delete = delete
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/Tree_Map.thy
--- a/src/HOL/Data_Structures/Tree_Map.thy	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/Data_Structures/Tree_Map.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -4,7 +4,7 @@
 
 theory Tree_Map
 imports
-  "~~/src/HOL/Library/Tree"
+  Tree_Set
   Map_by_Ordered
 begin
 
@@ -20,10 +20,6 @@
     else if a=x then Node l (a,b) r
     else Node l (x,y) (update a b r))"
 
-fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
-"del_min (Node Leaf a r) = (a, r)" |
-"del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))"
-
 fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
 "delete k Leaf = Leaf" |
 "delete k (Node l (a,b) r) = (if k<a then Node (delete k l) (a,b) r else
@@ -35,9 +31,7 @@
 
 lemma lookup_eq:
   "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
-apply (induction t)
-apply (auto simp: sorted_lems map_of_append map_of_sorteds split: option.split)
-done
+by (induction t) (auto simp: map_of_simps split: option.split)
 
 
 lemma inorder_update:
@@ -49,12 +43,11 @@
   "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
    x # inorder t' = inorder t"
 by(induction t arbitrary: t' rule: del_min.induct)
-  (auto simp: sorted_lems split: prod.splits)
+  (auto simp: del_list_simps split: prod.splits)
 
 lemma inorder_delete:
   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
-by(induction t)
-  (auto simp: del_list_sorted sorted_lems dest!: del_minD split: prod.splits)
+by(induction t) (auto simp: del_list_simps del_minD split: prod.splits)
 
 
 interpretation Map_by_Ordered
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/Data_Structures/Tree_Set.thy
--- a/src/HOL/Data_Structures/Tree_Set.thy	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/Data_Structures/Tree_Set.thy	Wed Sep 23 11:52:15 2015 +0100
@@ -43,7 +43,7 @@
 
 lemma inorder_insert:
   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
-by(induction t) (auto simp: ins_simps)
+by(induction t) (auto simp: ins_list_simps)
 
 
 lemma del_minD:
@@ -54,7 +54,7 @@
 
 lemma inorder_delete:
   "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
-by(induction t) (auto simp: del_simps del_minD split: prod.splits)
+by(induction t) (auto simp: del_list_simps del_minD split: prod.splits)
 
 
 interpretation Set_by_Ordered
diff -r b033aeaab1b8 -r 5c9a9504f713 src/HOL/ROOT
--- a/src/HOL/ROOT	Wed Sep 23 11:36:07 2015 +0100
+++ b/src/HOL/ROOT	Wed Sep 23 11:52:15 2015 +0100
@@ -176,6 +176,7 @@
   theories
     Tree_Set
     Tree_Map
+    AVL_Map
     RBT_Map
   document_files "root.tex" "root.bib"