diff --git a/src/HOL/Data_Structures/RBT_Set2.thy b/src/HOL/Data_Structures/RBT_Set2.thy
--- a/src/HOL/Data_Structures/RBT_Set2.thy
+++ b/src/HOL/Data_Structures/RBT_Set2.thy
@@ -1,161 +1,144 @@
 (* Author: Tobias Nipkow *)
 
 section \<open>Alternative Deletion in Red-Black Trees\<close>
 
 theory RBT_Set2
 imports RBT_Set
 begin
 
 text \<open>This is a conceptually simpler version of deletion. Instead of the tricky \<open>combine\<close>
 function this version follows the standard approach of replacing the deleted element
 (in function \<open>del\<close>) by the minimal element in its right subtree.\<close>
 
 fun split_min :: "'a rbt \<Rightarrow> 'a \<times> 'a rbt" where
 "split_min (Node l (a, _) r) =
   (if l = Leaf then (a,r)
    else let (x,l') = split_min l
         in (x, if color l = Black then baldL l' a r else R l' a r))"
 
 fun del :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
 "del x Leaf = Leaf" |
 "del x (Node l (a, _) r) =
   (case cmp x a of
      LT \<Rightarrow> let l' = del x l in if l \<noteq> Leaf \<and> color l = Black
            then baldL l' a r else R l' a r |
      GT \<Rightarrow> let r' = del x r in if r \<noteq> Leaf \<and> color r = Black
            then baldR l a r' else R l a r' |
      EQ \<Rightarrow> if r = Leaf then l else let (a',r') = split_min r in
            if color r = Black then baldR l a' r' else R l a' r')"
 
 text \<open>The first two \<open>let\<close>s speed up the automatic proof of \<open>inv_del\<close> below.\<close>
 
 definition delete :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
 "delete x t = paint Black (del x t)"
 
 
 subsection "Functional Correctness Proofs"
 
 lemma split_minD:
   "split_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> x # inorder t' = inorder t"
 by(induction t arbitrary: t' rule: split_min.induct)
   (auto simp: inorder_baldL sorted_lems split: prod.splits if_splits)
 
 lemma inorder_del:
  "sorted(inorder t) \<Longrightarrow>  inorder(del x t) = del_list x (inorder t)"
 by(induction x t rule: del.induct)
   (auto simp: del_list_simps inorder_baldL inorder_baldR split_minD Let_def split: prod.splits)
 
 lemma inorder_delete:
   "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
 by (auto simp: delete_def inorder_del inorder_paint)
 
 
 subsection \<open>Structural invariants\<close>
 
 lemma neq_Red[simp]: "(c \<noteq> Red) = (c = Black)"
 by (cases c) auto
 
 
 subsubsection \<open>Deletion\<close>
 
 lemma inv_split_min: "\<lbrakk> split_min t = (x,t'); t \<noteq> Leaf; invh t; invc t \<rbrakk> \<Longrightarrow>
    invh t' \<and>
    (color t = Red \<longrightarrow> bheight t' = bheight t \<and> invc t') \<and>
    (color t = Black \<longrightarrow> bheight t' = bheight t - 1 \<and> invc2 t')"
 apply(induction t arbitrary: x t' rule: split_min.induct)
 apply(auto simp: inv_baldR inv_baldL invc2I dest!: neq_LeafD
            split: if_splits prod.splits)
 done
 
 text \<open>An automatic proof. It is quite brittle, e.g. inlining the \<open>let\<close>s in @{const del} breaks it.\<close>
 lemma inv_del: "\<lbrakk> invh t; invc t \<rbrakk> \<Longrightarrow>
    invh (del x t) \<and>
    (color t = Red \<longrightarrow> bheight (del x t) = bheight t \<and> invc (del x t)) \<and>
    (color t = Black \<longrightarrow> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
 apply(induction x t rule: del.induct)
 apply(auto simp: inv_baldR inv_baldL invc2I Let_def dest!: inv_split_min dest: neq_LeafD
            split!: prod.splits if_splits)
 done
 
 text\<open>A structured proof where one can see what is used in each case.\<close>
 lemma inv_del2: "\<lbrakk> invh t; invc t \<rbrakk> \<Longrightarrow>
    invh (del x t) \<and>
    (color t = Red \<longrightarrow> bheight (del x t) = bheight t \<and> invc (del x t)) \<and>
    (color t = Black \<longrightarrow> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
 proof(induction x t rule: del.induct)
   case (1 x)
   then show ?case by simp
 next
   case (2 x l a c r)
   note if_split[split del]
   show ?case
   proof cases
     assume "x < a"
     show ?thesis
-    proof cases
-      assume "l \<noteq> Leaf \<and> color l = Black"
-      then show ?thesis using \<open>x < a\<close> "2.IH"(1) "2.prems"
-        by(auto simp: inv_baldL dest: neq_LeafD)
+    proof cases (* For readability; could be automated more: *)
+      assume *: "l \<noteq> Leaf \<and> color l = Black"
+      hence "bheight l > 0" using neq_LeafD[of l] by auto
+      thus ?thesis using \<open>x < a\<close> "2.IH"(1) "2.prems" inv_baldL[of "del x l"] * by(auto)
     next
       assume "\<not>(l \<noteq> Leaf \<and> color l = Black)"
-      then show ?thesis using \<open>x < a\<close> "2.IH"(1) "2.prems"
-        by(auto)
+      thus ?thesis using \<open>x < a\<close> "2.prems" "2.IH"(1) by(auto)
     qed
-  next
+  next (* more automation: *)
     assume "\<not> x < a"
     show ?thesis
     proof cases
       assume "x > a"
-      show ?thesis
-      proof cases
-        assume "r \<noteq> Leaf \<and> color r = Black"
-        then show ?thesis using \<open>a < x\<close> "2.IH"(2) "2.prems"
-          by(auto simp: inv_baldR dest: neq_LeafD)
-      next
-        assume "\<not>(r \<noteq> Leaf \<and> color r = Black)"
-        then show ?thesis using \<open>a < x\<close> "2.IH"(2) "2.prems"
-          by(auto)
-      qed
+      show ?thesis using \<open>a < x\<close> "2.IH"(2) "2.prems" neq_LeafD[of r] inv_baldR[of _ "del x r"]
+          by(auto simp: Let_def split: if_split)
+    
     next
       assume "\<not> x > a"
-      show ?thesis
-      proof cases
-        assume "r = Leaf"
-        then show ?thesis using "2.prems" \<open>\<not> x < a\<close> \<open>\<not> x > a\<close>
-          by(auto simp: invc2I)
-      next
-        assume "\<not> r = Leaf"
-        then show ?thesis using "2.prems" \<open>\<not> x < a\<close> \<open>\<not> x > a\<close>
-          by(auto simp: inv_baldR dest!: inv_split_min dest: neq_LeafD split: prod.split if_split)
-      qed
-    next
+      show ?thesis using "2.prems" \<open>\<not> x < a\<close> \<open>\<not> x > a\<close>
+          by(auto simp: inv_baldR invc2I dest!: inv_split_min dest: neq_LeafD split: prod.split if_split)
     qed
   qed
 qed
 
 theorem rbt_delete: "rbt t \<Longrightarrow> rbt (delete x t)"
 by (metis delete_def rbt_def color_paint_Black inv_del invh_paint)
 
 text \<open>Overall correctness:\<close>
 
 interpretation S: Set_by_Ordered
 where empty = empty and isin = isin and insert = insert and delete = delete
 and inorder = inorder and inv = rbt
 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: rbt_def empty_def) 
 next
   case 6 thus ?case by (simp add: rbt_insert) 
 next
   case 7 thus ?case by (simp add: rbt_delete) 
 qed
 
 
 end