diff --git a/src/HOL/Data_Structures/Trie_Fun.thy b/src/HOL/Data_Structures/Trie_Fun.thy
--- a/src/HOL/Data_Structures/Trie_Fun.thy
+++ b/src/HOL/Data_Structures/Trie_Fun.thy
@@ -1,90 +1,90 @@
 section \<open>Tries via Functions\<close>
 
 theory Trie_Fun
 imports
   Set_Specs
 begin
 
 text \<open>A trie where each node maps a key to sub-tries via a function.
 Nice abstract model. Not efficient because of the function space.\<close>
 
 datatype 'a trie = Nd bool "'a \<Rightarrow> 'a trie option"
 
 definition empty :: "'a trie" where
 [simp]: "empty = Nd False (\<lambda>_. None)"
 
 fun isin :: "'a trie \<Rightarrow> 'a list \<Rightarrow> bool" where
 "isin (Nd b m) [] = b" |
 "isin (Nd b m) (k # xs) = (case m k of None \<Rightarrow> False | Some t \<Rightarrow> isin t xs)"
 
-fun insert :: "('a::linorder) list \<Rightarrow> 'a trie \<Rightarrow> 'a trie" where
+fun insert :: "'a list \<Rightarrow> 'a trie \<Rightarrow> 'a trie" where
 "insert [] (Nd b m) = Nd True m" |
 "insert (x#xs) (Nd b m) =
    Nd b (m(x := Some(insert xs (case m x of None \<Rightarrow> empty | Some t \<Rightarrow> t))))"
 
-fun delete :: "('a::linorder) list \<Rightarrow> 'a trie \<Rightarrow> 'a trie" where
+fun delete :: "'a list \<Rightarrow> 'a trie \<Rightarrow> 'a trie" where
 "delete [] (Nd b m) = Nd False m" |
 "delete (x#xs) (Nd b m) = Nd b
    (case m x of
       None \<Rightarrow> m |
       Some t \<Rightarrow> m(x := Some(delete xs t)))"
 
 text \<open>The actual definition of \<open>set\<close> is a bit cryptic but canonical, to enable
 primrec to prove termination:\<close>
 
 primrec set :: "'a trie \<Rightarrow> 'a list set" where
 "set (Nd b m) = (if b then {[]} else {}) \<union>
     (\<Union>a. case (map_option set o m) a of None \<Rightarrow> {} | Some t \<Rightarrow> (#) a ` t)"
 
 text \<open>This is the more human-readable version:\<close>
 
 lemma set_Nd:
   "set (Nd b m) =
      (if b then {[]} else {}) \<union>
      (\<Union>a. case m a of None \<Rightarrow> {} | Some t \<Rightarrow> (#) a ` set t)"
 by (auto simp: split: option.splits)
 
 lemma isin_set: "isin t xs = (xs \<in> set t)"
 apply(induction t xs rule: isin.induct)
 apply (auto split: option.split)
 done
 
 lemma set_insert: "set (insert xs t) = set t \<union> {xs}"
 proof(induction xs t rule: insert.induct)
   case 1 thus ?case by simp
 next
   case 2
   thus ?case
     apply(simp)
     apply(subst set_eq_iff)
     apply(auto split!: if_splits option.splits)
      apply fastforce
     by (metis imageI option.sel)
 qed
 
 lemma set_delete: "set (delete xs t) = set t - {xs}"
 proof(induction xs t rule: delete.induct)
   case 1 thus ?case by (force split: option.splits)
 next
   case 2
   show ?case
     apply (auto simp add: image_iff 2 split!: if_splits option.splits)
      apply (metis DiffI empty_iff insert_iff option.inject)
      apply (metis DiffI empty_iff insert_iff option.inject)
     done
 qed
 
 interpretation S: Set
 where empty = empty and isin = isin and insert = insert and delete = delete
 and set = set and invar = "\<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: set_insert)
 next
   case 4 thus ?case by(simp add: set_delete)
 qed (rule TrueI)+
 
 end