diff --git a/src/HOL/Data_Structures/AVL_Map.thy b/src/HOL/Data_Structures/AVL_Map.thy
--- a/src/HOL/Data_Structures/AVL_Map.thy
+++ b/src/HOL/Data_Structures/AVL_Map.thy
@@ -1,201 +1,186 @@
 (* Author: Tobias Nipkow *)
 
 section "AVL Tree Implementation of Maps"
 
 theory AVL_Map
 imports
   AVL_Set
   Lookup2
 begin
 
 fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
 "update x y Leaf = Node Leaf ((x,y), 1) Leaf" |
 "update x y (Node l ((a,b), h) r) = (case cmp x a of
    EQ \<Rightarrow> Node l ((x,y), h) r |
    LT \<Rightarrow> balL (update x y l) (a,b) r |
    GT \<Rightarrow> balR l (a,b) (update x y r))"
 
 fun delete :: "'a::linorder \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
 "delete _ Leaf = Leaf" |
 "delete x (Node l ((a,b), h) r) = (case cmp x a of
    EQ \<Rightarrow> if l = Leaf then r
          else let (l', ab') = split_max l in balR l' ab' r |
    LT \<Rightarrow> balR (delete x l) (a,b) r |
    GT \<Rightarrow> balL l (a,b) (delete x r))"
 
 
 subsection \<open>Functional Correctness\<close>
 
 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_balL inorder_balR)
 
 
 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_balL inorder_balR
       inorder_split_maxD split: prod.splits)
 
 
 subsection \<open>AVL invariants\<close>
 
 
 subsubsection \<open>Insertion maintains AVL balance\<close>
 
 theorem avl_update:
   assumes "avl t"
   shows "avl(update x y t)"
         "(height (update x y t) = height t \<or> height (update x y t) = height t + 1)"
 using assms
 proof (induction x y t rule: update.induct)
   case eq2: (2 x y l a b h r)
   case 1
   show ?case
   proof(cases "x = a")
     case True with eq2 1 show ?thesis by simp
   next
     case False
     with eq2 1 show ?thesis 
     proof(cases "x<a")
-      case True with eq2 1 show ?thesis by (auto simp add:avl_balL)
+      case True with eq2 1 show ?thesis by (auto intro!: avl_balL)
     next
-      case False with eq2 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balR)
+      case False with eq2 1 \<open>x\<noteq>a\<close> show ?thesis by (auto intro!: avl_balR)
     qed
   qed
   case 2
   show ?case
   proof(cases "x = a")
     case True with eq2 1 show ?thesis by simp
   next
     case False
     show ?thesis 
     proof(cases "x<a")
       case True
       show ?thesis
       proof(cases "height (update x y l) = height r + 2")
         case False with eq2 2 \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
       next
         case True 
         hence "(height (balL (update x y l) (a,b) r) = height r + 2) \<or>
           (height (balL (update x y l) (a,b) r) = height r + 3)" (is "?A \<or> ?B")
-          using eq2 2 \<open>x<a\<close> by (intro height_balL) simp_all
+          using eq2 2 \<open>x<a\<close> height_balL[OF _ _ True] by simp
         thus ?thesis
         proof
           assume ?A with 2 \<open>x < a\<close> show ?thesis by (auto)
         next
           assume ?B with True 1 eq2(2) \<open>x < a\<close> show ?thesis by (simp) arith
         qed
       qed
     next
       case False
       show ?thesis
       proof(cases "height (update x y r) = height l + 2")
         case False with eq2 2 \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
       next
         case True 
         hence "(height (balR l (a,b) (update x y r)) = height l + 2) \<or>
           (height (balR l (a,b) (update x y r)) = height l + 3)"  (is "?A \<or> ?B")
-          using eq2 2 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> by (intro height_balR) simp_all
+          using eq2 2 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> height_balR[OF _ _ True] by simp
         thus ?thesis 
         proof
           assume ?A with 2 \<open>\<not>x < a\<close> show ?thesis by (auto)
         next
           assume ?B with True 1 eq2(4) \<open>\<not>x < a\<close> show ?thesis by (simp) arith
         qed
       qed
     qed
   qed
 qed simp_all
 
 
 subsubsection \<open>Deletion maintains AVL balance\<close>
 
 theorem avl_delete:
   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 rule: tree2_induct)
-  case (Node l n h r)
-  obtain a b where [simp]: "n = (a,b)" by fastforce
+  case (Node l ab h r)
+  obtain a b where [simp]: "ab = (a,b)" by fastforce
   case 1
   show ?case
   proof(cases "x = a")
     case True with Node 1 show ?thesis
-      using avl_split_max[of l] by (auto simp: avl_balR split: prod.split)
+      using avl_split_max[of l] by (auto intro!: avl_balR split: prod.split)
   next
     case False
     show ?thesis 
     proof(cases "x<a")
-      case True with Node 1 show ?thesis by (auto simp add:avl_balR)
+      case True with Node 1 show ?thesis by (auto intro!: avl_balR)
     next
-      case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balL)
+      case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto intro!: avl_balL)
     qed
   qed
   case 2
   show ?case
   proof(cases "x = a")
     case True then show ?thesis using 1 avl_split_max[of l]
       by(auto simp: balR_def max_absorb2 split!: if_splits prod.split tree.split)
   next
     case False
     show ?thesis 
     proof(cases "x<a")
       case True
       show ?thesis
       proof(cases "height r = height (delete x l) + 2")
         case False with Node 1 \<open>x < a\<close> show ?thesis by(auto simp: balR_def)
       next
         case True 
-        hence "(height (balR (delete x l) n r) = height (delete x l) + 2) \<or>
-          height (balR (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
-          using Node 2 by (intro height_balR) auto
-        thus ?thesis 
-        proof
-          assume ?A with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
-        next
-          assume ?B with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
-        qed
+        thus ?thesis using height_balR[OF _ _ True, of ab] 2 Node(1,2) \<open>x < a\<close> by simp linarith
       qed
     next
       case False
       show ?thesis
       proof(cases "height l = height (delete x r) + 2")
         case False with Node 1 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> show ?thesis by(auto simp: balL_def)
       next
-        case True 
-        hence "(height (balL l n (delete x r)) = height (delete x r) + 2) \<or>
-          height (balL l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
-          using Node 2 by (intro height_balL) auto
-        thus ?thesis 
-        proof
-          assume ?A with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
-        next
-          assume ?B with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
-        qed
+        case True
+        thus ?thesis
+          using height_balL[OF _ _ True, of ab] 2 Node(3,4) \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> by auto
       qed
     qed
   qed
 qed simp_all
 
 
 interpretation M: Map_by_Ordered
 where empty = empty and lookup = lookup and update = update and delete = delete
 and inorder = inorder and inv = avl
 proof (standard, goal_cases)
   case 1 show ?case by (simp add: empty_def)
 next
   case 2 thus ?case by(simp add: lookup_map_of)
 next
   case 3 thus ?case by(simp add: inorder_update)
 next
   case 4 thus ?case by(simp add: inorder_delete)
 next
   case 5 show ?case by (simp add: empty_def)
 next
   case 6 thus ?case by(simp add: avl_update(1))
 next
   case 7 thus ?case by(simp add: avl_delete(1))
 qed
 
 end
diff --git a/src/HOL/Data_Structures/AVL_Set.thy b/src/HOL/Data_Structures/AVL_Set.thy
--- a/src/HOL/Data_Structures/AVL_Set.thy
+++ b/src/HOL/Data_Structures/AVL_Set.thy
@@ -1,352 +1,358 @@
-(*
-Author: Tobias Nipkow
-*)
+(* Author: Tobias Nipkow *)
 
-subsection \<open>Invariant expressed via 3-way cases\<close>
+subsection \<open>Invariant\<close>
 
 theory AVL_Set
 imports
   AVL_Set_Code
   "HOL-Number_Theory.Fib"
 begin
 
 fun avl :: "'a avl_tree \<Rightarrow> bool" where
 "avl Leaf = True" |
 "avl (Node l (a,n) r) =
- ((height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1) \<and> 
+ (abs(int(height l) - int(height r)) \<le> 1 \<and>
   n = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
 
-
 subsubsection \<open>Insertion maintains AVL balance\<close>
 
 declare Let_def [simp]
 
 lemma ht_height[simp]: "avl t \<Longrightarrow> ht t = height t"
 by (cases t rule: tree2_cases) simp_all
 
 text \<open>First, a fast but relatively manual proof with many lemmas:\<close>
 
 lemma height_balL:
   "\<lbrakk> avl l; avl r; height l = height r + 2 \<rbrakk> \<Longrightarrow>
-   height (balL l a r) = height r + 2 \<or>
-   height (balL l a r) = height r + 3"
-by (cases l) (auto simp:node_def balL_def split:tree.split)
-       
+   height (balL l a r) \<in> {height r + 2, height r + 3}"
+apply (cases l)
+ apply (auto simp:node_def balL_def split:tree.split)
+ by arith+
+
 lemma height_balR:
   "\<lbrakk> avl l; avl r; height r = height l + 2 \<rbrakk> \<Longrightarrow>
-   height (balR l a r) = height l + 2 \<or>
-   height (balR l a r) = height l + 3"
-by (cases r) (auto simp add:node_def balR_def split:tree.split)
+   height (balR l a r) : {height l + 2, height l + 3}"
+apply (cases r)
+ apply(auto simp add:node_def balR_def split:tree.split)
+ by arith+
 
 lemma height_node[simp]: "height(node l a r) = max (height l) (height r) + 1"
 by (simp add: node_def)
 
 lemma height_balL2:
   "\<lbrakk> avl l; avl r; height l \<noteq> height r + 2 \<rbrakk> \<Longrightarrow>
    height (balL l a r) = 1 + max (height l) (height r)"
 by (cases l, cases r) (simp_all add: balL_def)
 
 lemma height_balR2:
   "\<lbrakk> avl l;  avl r;  height r \<noteq> height l + 2 \<rbrakk> \<Longrightarrow>
    height (balR l a r) = 1 + max (height l) (height r)"
 by (cases l, cases r) (simp_all add: balR_def)
 
 lemma avl_balL: 
   "\<lbrakk> avl l; avl r; height r - 1 \<le> height l \<and> height l \<le> height r + 2 \<rbrakk> \<Longrightarrow> avl(balL l a r)"
 by(cases l, cases r)
   (auto simp: balL_def node_def split!: if_split tree.split)
 
 lemma avl_balR: 
   "\<lbrakk> avl l; avl r; height l - 1 \<le> height r \<and> height r \<le> height l + 2 \<rbrakk> \<Longrightarrow> avl(balR l a r)"
 by(cases r, cases l)
   (auto simp: balR_def node_def split!: if_split tree.split)
 
 text\<open>Insertion maintains the AVL property. Requires simultaneous proof.\<close>
 
 theorem avl_insert:
-  assumes "avl t"
-  shows "avl(insert x t)"
-        "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
-using assms
+  "avl t \<Longrightarrow> avl(insert x t)"
+  "avl t \<Longrightarrow> height (insert x t) \<in> {height t, height t + 1}"
 proof (induction t rule: tree2_induct)
   case (Node l a _ r)
   case 1
   show ?case
   proof(cases "x = a")
-    case True with Node 1 show ?thesis by simp
+    case True with 1 show ?thesis by simp
   next
     case False
     show ?thesis 
     proof(cases "x<a")
-      case True with Node 1 show ?thesis by (auto simp add:avl_balL)
+      case True with 1 Node(1,2) show ?thesis by (auto intro!:avl_balL)
     next
-      case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balR)
+      case False with 1 Node(3,4) \<open>x\<noteq>a\<close> show ?thesis by (auto intro!:avl_balR)
     qed
   qed
   case 2
   show ?case
   proof(cases "x = a")
-    case True with Node 1 show ?thesis by simp
+    case True with 2 show ?thesis by simp
   next
     case False
     show ?thesis 
     proof(cases "x<a")
       case True
       show ?thesis
       proof(cases "height (insert x l) = height r + 2")
-        case False with Node 2 \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
+        case False with 2 Node(1,2) \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
       next
         case True 
         hence "(height (balL (insert x l) a r) = height r + 2) \<or>
           (height (balL (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
-          using Node 2 by (intro height_balL) simp_all
+          using 2 Node(1,2) height_balL[OF _ _ True] by simp
         thus ?thesis
         proof
           assume ?A with 2 \<open>x < a\<close> show ?thesis by (auto)
         next
-          assume ?B with True 1 Node(2) \<open>x < a\<close> show ?thesis by (simp) arith
+          assume ?B with 2 Node(2) True \<open>x < a\<close> show ?thesis by (simp) arith
         qed
       qed
     next
       case False
-      show ?thesis 
+      show ?thesis
       proof(cases "height (insert x r) = height l + 2")
-        case False with Node 2 \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
+        case False with 2 Node(3,4) \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
       next
         case True 
         hence "(height (balR l a (insert x r)) = height l + 2) \<or>
           (height (balR l a (insert x r)) = height l + 3)"  (is "?A \<or> ?B")
-          using Node 2 by (intro height_balR) simp_all
-        thus ?thesis 
+          using 2 Node(3) height_balR[OF _ _ True] by simp
+        thus ?thesis
         proof
           assume ?A with 2 \<open>\<not>x < a\<close> show ?thesis by (auto)
         next
-          assume ?B with True 1 Node(4) \<open>\<not>x < a\<close> show ?thesis by (simp) arith
+          assume ?B with 2 Node(4) True \<open>\<not>x < a\<close> show ?thesis by (simp) arith
         qed
       qed
     qed
   qed
 qed simp_all
 
 text \<open>Now an automatic proof without lemmas:\<close>
 
 theorem avl_insert_auto: "avl t \<Longrightarrow>
   avl(insert x t) \<and> height (insert x t) \<in> {height t, height t + 1}"
 apply (induction t rule: tree2_induct)
  (* if you want to save a few secs: apply (auto split!: if_split) *)
  apply (auto simp: balL_def balR_def node_def max_absorb2 split!: if_split tree.split)
 done
 
 
 subsubsection \<open>Deletion maintains AVL balance\<close>
 
 lemma avl_split_max:
   "\<lbrakk> avl t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
   avl (fst (split_max t)) \<and>
   height t \<in> {height(fst (split_max t)), height(fst (split_max t)) + 1}"
 by(induct t rule: split_max_induct)
   (auto simp: balL_def node_def max_absorb2 split!: prod.split if_split tree.split)
 
 text\<open>Deletion maintains the AVL property:\<close>
 
 theorem avl_delete:
-  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
+  "avl t \<Longrightarrow> avl(delete x t)"
+  "avl t \<Longrightarrow> height t \<in> {height (delete x t), height (delete x t) + 1}"
 proof (induct t rule: tree2_induct)
   case (Node l a n r)
   case 1
-  thus ?case
-    using Node avl_split_max[of l] by (auto simp: avl_balL avl_balR split: prod.split)
+  show ?case
+  proof(cases "x = a")
+    case True thus ?thesis
+      using 1 avl_split_max[of l] by (auto intro!: avl_balR split: prod.split)
+  next
+    case False thus ?thesis
+      using Node 1 by (auto intro!: avl_balL avl_balR)
+  qed
   case 2
   show ?case
   proof(cases "x = a")
-    case True then show ?thesis using 1 avl_split_max[of l]
+    case True thus ?thesis using 2 avl_split_max[of l]
       by(auto simp: balR_def max_absorb2 split!: if_splits prod.split tree.split)
   next
     case False
-    show ?thesis 
+    show ?thesis
     proof(cases "x<a")
       case True
       show ?thesis
       proof(cases "height r = height (delete x l) + 2")
-        case False with Node 1 \<open>x < a\<close> show ?thesis by(auto simp: balR_def)
+        case False
+        thus ?thesis using 2 Node(1,2) \<open>x < a\<close> by(auto simp: balR_def)
       next
-        case True 
-        hence "(height (balR (delete x l) a r) = height (delete x l) + 2) \<or>
-          height (balR (delete x l) a r) = height (delete x l) + 3" (is "?A \<or> ?B")
-          using Node 2 by (intro height_balR) auto
-        thus ?thesis 
-        proof
-          assume ?A with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
-        next
-          assume ?B with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
-        qed
+        case True
+        thus ?thesis using height_balR[OF _ _ True, of a] 2 Node(1,2) \<open>x < a\<close> by simp linarith
       qed
     next
       case False
       show ?thesis
       proof(cases "height l = height (delete x r) + 2")
-        case False with Node 1 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> show ?thesis by(auto simp: balL_def)
+        case False
+        thus ?thesis using 2 Node(3,4) \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> by(auto simp: balL_def)
       next
-        case True 
-        hence "(height (balL l a (delete x r)) = height (delete x r) + 2) \<or>
-          height (balL l a (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
-          using Node 2 by (intro height_balL) auto
-        thus ?thesis 
-        proof
-          assume ?A with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
-        next
-          assume ?B with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
-        qed
+        case True
+        thus ?thesis
+          using height_balL[OF _ _ True, of a] 2 Node(3,4) \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> by simp linarith
       qed
     qed
   qed
 qed simp_all
 
 text \<open>A more automatic proof.
 Complete automation as for insertion seems hard due to resource requirements.\<close>
 
 theorem avl_delete_auto:
-  assumes "avl t" 
-  shows "avl(delete x t)" and "height t \<in> {height (delete x t), height (delete x t) + 1}"
-using assms
+  "avl t \<Longrightarrow> avl(delete x t)"
+  "avl t \<Longrightarrow> height t \<in> {height (delete x t), height (delete x t) + 1}"
 proof (induct t rule: tree2_induct)
   case (Node l a n r)
   case 1
-  show ?case
-  proof(cases "x = a")
-    case True thus ?thesis
-      using 1 avl_split_max[of l] by (auto simp: avl_balR split: prod.split)
-  next
-    case False thus ?thesis
-      using Node 1 by (auto simp: avl_balL avl_balR)
-  qed
+  thus ?case
+    using Node avl_split_max[of l] by (auto intro!: avl_balL avl_balR split: prod.split)
   case 2
   show ?case
   proof(cases "x = a")
     case True thus ?thesis
-      using 1 avl_split_max[of l]
+      using 2 avl_split_max[of l]
       by(auto simp: balR_def max_absorb2 split!: if_splits prod.split tree.split)
   next
     case False thus ?thesis 
-      using height_balL[of l "delete x r" a] height_balR[of "delete x l" r a] Node 1
+      using height_balL[of l "delete x r" a] height_balR[of "delete x l" r a] 2 Node
         by(auto simp: balL_def balR_def split!: if_split)
   qed
 qed simp_all
 
 
 subsection "Overall correctness"
 
 interpretation S: Set_by_Ordered
 where empty = empty and isin = isin and insert = insert and delete = delete
 and inorder = inorder and inv = avl
 proof (standard, goal_cases)
   case 1 show ?case by (simp add: empty_def)
 next
   case 2 thus ?case by(simp add: isin_set_inorder)
 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 by (simp add: empty_def)
 next
   case 6 thus ?case by (simp add: avl_insert(1))
 next
   case 7 thus ?case by (simp add: avl_delete(1))
 qed
 
 
 subsection \<open>Height-Size Relation\<close>
 
-text \<open>Based on theorems by Daniel St\"uwe, Manuel Eberl and Peter Lammich, much simplified.\<close>
-
 text \<open>Any AVL tree of height \<open>n\<close> has at least \<open>fib (n+2)\<close> leaves:\<close>
 
-lemma avl_fib_bound: "avl t \<Longrightarrow> height t = n \<Longrightarrow> fib (n+2) \<le> size1 t"
-proof (induction n arbitrary: t rule: fib.induct)
-  case 1 thus ?case by (simp)
-next
-  case 2 thus ?case by (cases t) (auto)
+theorem avl_fib_bound:
+  "avl t \<Longrightarrow> fib(height t + 2) \<le> size1 t"
+proof (induction rule: tree2_induct)
+  case (Node l a h r)
+  have 1: "height l + 1 \<le> height r + 2" and 2: "height r + 1 \<le> height l + 2"
+    using Node.prems by auto
+  have "fib (max (height l) (height r) + 3) \<le> size1 l + size1 r"
+  proof cases
+    assume "height l \<ge> height r"
+    hence "fib (max (height l) (height r) + 3) = fib (height l + 3)"
+      by(simp add: max_absorb1)
+    also have "\<dots> = fib (height l + 2) + fib (height l + 1)"
+      by (simp add: numeral_eq_Suc)
+    also have "\<dots> \<le> size1 l + fib (height l + 1)"
+      using Node by (simp)
+    also have "\<dots> \<le> size1 r + size1 l"
+      using Node fib_mono[OF 1] by auto
+    also have "\<dots> = size1 (Node l (a,h) r)"
+      by simp
+    finally show ?thesis 
+      by (simp)
+  next
+    assume "\<not> height l \<ge> height r"
+    hence "fib (max (height l) (height r) + 3) = fib (height r + 3)"
+      by(simp add: max_absorb1)
+    also have "\<dots> = fib (height r + 2) + fib (height r + 1)"
+      by (simp add: numeral_eq_Suc)
+    also have "\<dots> \<le> size1 r + fib (height r + 1)"
+      using Node by (simp)
+    also have "\<dots> \<le> size1 r + size1 l"
+      using Node fib_mono[OF 2] by auto
+    also have "\<dots> = size1 (Node l (a,h) r)"
+      by simp
+    finally show ?thesis 
+      by (simp)
+  qed
+  also have "\<dots> = size1 (Node l (a,h) r)"
+    by simp
+  finally show ?case by (simp del: fib.simps add: numeral_eq_Suc)
+qed auto
+
+lemma avl_fib_bound_auto: "avl t \<Longrightarrow> fib (height t + 2) \<le> size1 t"
+proof (induction t rule: tree2_induct)
+  case Leaf thus ?case by (simp)
 next
-  case (3 h)
-  from "3.prems" obtain l a r where
-    [simp]: "t = Node l (a,Suc(Suc h)) r" "avl l" "avl r"
-    and C: "
-      height r = Suc h \<and> height l = Suc h
-    \<or> height r = Suc h \<and> height l = h
-    \<or> height r = h \<and> height l = Suc h" (is "?C1 \<or> ?C2 \<or> ?C3")
-    by (cases t) (simp, fastforce)
-  {
-    assume ?C1
-    with "3.IH"(1)
-    have "fib (h + 3) \<le> size1 l" "fib (h + 3) \<le> size1 r"
-      by (simp_all add: eval_nat_numeral)
-    hence ?case by (auto simp: eval_nat_numeral)
-  } moreover {
-    assume ?C2
-    hence ?case using "3.IH"(1)[of r] "3.IH"(2)[of l] by auto
-  } moreover {
-    assume ?C3
-    hence ?case using "3.IH"(1)[of l] "3.IH"(2)[of r] by auto
-  } ultimately show ?case using C by blast
+  case (Node l a h r)
+  have 1: "height l + 1 \<le> height r + 2" and 2: "height r + 1 \<le> height l + 2"
+    using Node.prems by auto
+  have left: "height l \<ge> height r \<Longrightarrow> ?case" (is "?asm \<Longrightarrow> _")
+    using Node fib_mono[OF 1] by (simp add: max.absorb1)
+  have right: "height l \<le> height r \<Longrightarrow> ?case"
+    using Node fib_mono[OF 2] by (simp add: max.absorb2)
+  show ?case using left right using Node.prems by simp linarith
 qed
 
 text \<open>An exponential lower bound for \<^const>\<open>fib\<close>:\<close>
 
 lemma fib_lowerbound:
   defines "\<phi> \<equiv> (1 + sqrt 5) / 2"
   shows "real (fib(n+2)) \<ge> \<phi> ^ n"
 proof (induction n rule: fib.induct)
   case 1
   then show ?case by simp
 next
   case 2
   then show ?case by (simp add: \<phi>_def real_le_lsqrt)
 next
   case (3 n) term ?case
   have "\<phi> ^ Suc (Suc n) = \<phi> ^ 2 * \<phi> ^ n"
     by (simp add: field_simps power2_eq_square)
   also have "\<dots> = (\<phi> + 1) * \<phi> ^ n"
     by (simp_all add: \<phi>_def power2_eq_square field_simps)
   also have "\<dots> = \<phi> ^ Suc n + \<phi> ^ n"
       by (simp add: field_simps)
   also have "\<dots> \<le> real (fib (Suc n + 2)) + real (fib (n + 2))"
       by (intro add_mono "3.IH")
   finally show ?case by simp
 qed
 
 text \<open>The size of an AVL tree is (at least) exponential in its height:\<close>
 
 lemma avl_size_lowerbound:
   defines "\<phi> \<equiv> (1 + sqrt 5) / 2"
   assumes "avl t"
   shows   "\<phi> ^ (height t) \<le> size1 t"
 proof -
   have "\<phi> ^ height t \<le> fib (height t + 2)"
     unfolding \<phi>_def by(rule fib_lowerbound)
   also have "\<dots> \<le> size1 t"
-    using avl_fib_bound[of t "height t"] assms by simp
+    using avl_fib_bound[of t] assms by simp
   finally show ?thesis .
 qed
 
 text \<open>The height of an AVL tree is most \<^term>\<open>(1/log 2 \<phi>)\<close> \<open>\<approx> 1.44\<close> times worse
 than \<^term>\<open>log 2 (size1 t)\<close>:\<close>
 
 lemma  avl_height_upperbound:
   defines "\<phi> \<equiv> (1 + sqrt 5) / 2"
   assumes "avl t"
   shows   "height t \<le> (1/log 2 \<phi>) * log 2 (size1 t)"
 proof -
   have "\<phi> > 0" "\<phi> > 1" by(auto simp: \<phi>_def pos_add_strict)
   hence "height t = log \<phi> (\<phi> ^ height t)" by(simp add: log_nat_power)
   also have "\<dots> \<le> log \<phi> (size1 t)"
     using avl_size_lowerbound[OF assms(2), folded \<phi>_def] \<open>1 < \<phi>\<close>
     by (simp add: le_log_of_power) 
   also have "\<dots> = (1/log 2 \<phi>) * log 2 (size1 t)"
     by(simp add: log_base_change[of 2 \<phi>])
   finally show ?thesis .
 qed
 
 end