diff --git a/src/HOL/Data_Structures/RBT_Map.thy b/src/HOL/Data_Structures/RBT_Map.thy
--- a/src/HOL/Data_Structures/RBT_Map.thy
+++ b/src/HOL/Data_Structures/RBT_Map.thy
@@ -1,125 +1,129 @@
 (* Author: Tobias Nipkow *)
 
 section \<open>Red-Black Tree Implementation of Maps\<close>
 
 theory RBT_Map
 imports
   RBT_Set
   Lookup2
 begin
 
 fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
 "upd x y Leaf = R Leaf (x,y) Leaf" |
 "upd x y (B l (a,b) r) = (case cmp x a of
   LT \<Rightarrow> baliL (upd x y l) (a,b) r |
   GT \<Rightarrow> baliR l (a,b) (upd x y r) |
   EQ \<Rightarrow> B l (x,y) r)" |
 "upd x y (R l (a,b) r) = (case cmp x a of
   LT \<Rightarrow> R (upd x y l) (a,b) r |
   GT \<Rightarrow> R l (a,b) (upd x y r) |
   EQ \<Rightarrow> R l (x,y) r)"
 
 definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
 "update x y t = paint Black (upd x y t)"
 
 fun del :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt" where
 "del x Leaf = Leaf" |
-"del x (Node l ((a,b), c) r) = (case cmp x a of
+"del x (Node l (ab, _) r) = (case cmp x (fst ab) of
      LT \<Rightarrow> if l \<noteq> Leaf \<and> color l = Black
-           then baldL (del x l) (a,b) r else R (del x l) (a,b) r |
+           then baldL (del x l) ab r else R (del x l) ab r |
      GT \<Rightarrow> if r \<noteq> Leaf\<and> color r = Black
-           then baldR l (a,b) (del x r) else R l (a,b) (del x r) |
+           then baldR l ab (del x r) else R l ab (del x r) |
   EQ \<Rightarrow> join l r)"
 
 definition delete :: "'a::linorder \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
 "delete x t = paint Black (del x t)"
 
 
 subsection "Functional Correctness Proofs"
 
 lemma inorder_upd:
   "sorted1(inorder t) \<Longrightarrow> inorder(upd x y t) = upd_list x y (inorder t)"
 by(induction x y t rule: upd.induct)
   (auto simp: upd_list_simps inorder_baliL inorder_baliR)
 
 lemma inorder_update:
   "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
 by(simp add: update_def inorder_upd inorder_paint)
 
+(* This lemma became necessary below when \<open>del\<close> was converted from pattern-matching to \<open>fst\<close> *)
+lemma del_list_id: "\<forall>ab\<in>set ps. y < fst ab \<Longrightarrow> x \<le> y \<Longrightarrow> del_list x ps = ps"
+by(rule del_list_idem) auto
+
 lemma inorder_del:
  "sorted1(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_join inorder_baldL inorder_baldR)
+  (auto simp: del_list_simps del_list_id inorder_join inorder_baldL inorder_baldR)
 
 lemma inorder_delete:
   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
 by(simp add: delete_def inorder_del inorder_paint)
 
 
 subsection \<open>Structural invariants\<close>
 
 subsubsection \<open>Update\<close>
 
 lemma invc_upd: assumes "invc t"
   shows "color t = Black \<Longrightarrow> invc (upd x y t)" "invc2 (upd x y t)"
 using assms
 by (induct x y t rule: upd.induct) (auto simp: invc_baliL invc_baliR invc2I)
 
 lemma invh_upd: assumes "invh t"
   shows "invh (upd x y t)" "bheight (upd x y t) = bheight t"
 using assms
 by(induct x y t rule: upd.induct)
   (auto simp: invh_baliL invh_baliR bheight_baliL bheight_baliR)
 
 theorem rbt_update: "rbt t \<Longrightarrow> rbt (update x y t)"
 by (simp add: invc_upd(2) invh_upd(1) color_paint_Black invh_paint rbt_def update_def)
 
 
 subsubsection \<open>Deletion\<close>
 
 lemma del_invc_invh: "invh t \<Longrightarrow> invc t \<Longrightarrow> invh (del x t) \<and>
    (color t = Red \<and> bheight (del x t) = bheight t \<and> invc (del x t) \<or>
     color t = Black \<and> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
 proof (induct x t rule: del.induct)
-case (2 x _ y _ c)
-  have "x = y \<or> x < y \<or> x > y" by auto
+case (2 x _ ab c)
+  have "x = fst ab \<or> x < fst ab \<or> x > fst ab" by auto
   thus ?case proof (elim disjE)
-    assume "x = y"
+    assume "x = fst ab"
     with 2 show ?thesis
     by (cases c) (simp_all add: invh_join invc_join)
   next
-    assume "x < y"
+    assume "x < fst ab"
     with 2 show ?thesis
       by(cases c)
         (auto simp: invh_baldL_invc invc_baldL invc2_baldL dest: neq_LeafD)
   next
-    assume "y < x"
+    assume "fst ab < x"
     with 2 show ?thesis
       by(cases c)
         (auto simp: invh_baldR_invc invc_baldR invc2_baldR dest: neq_LeafD)
   qed
 qed auto
 
 theorem rbt_delete: "rbt t \<Longrightarrow> rbt (delete k t)"
 by (metis delete_def rbt_def color_paint_Black del_invc_invh invc2I invh_paint)
 
 interpretation M: Map_by_Ordered
 where empty = empty and lookup = lookup and update = update 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: 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 thus ?case by (simp add: rbt_def empty_def) 
 next
   case 6 thus ?case by (simp add: rbt_update) 
 next
   case 7 thus ?case by (simp add: rbt_delete) 
 qed
 
 end