--- a/src/HOL/Data_Structures/Trie_Fun.thy Sat May 11 19:08:26 2019 +0200
+++ b/src/HOL/Data_Structures/Trie_Fun.thy Sat May 11 22:19:28 2019 +0200
@@ -10,6 +10,9 @@
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)"
@@ -17,7 +20,7 @@
fun insert :: "('a::linorder) 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> Nd False (\<lambda>_. None) | Some t \<Rightarrow> t))))"
+ 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
"delete [] (Nd b m) = Nd False m" |
@@ -73,7 +76,7 @@
qed
interpretation S: Set
-where empty = "Nd False (\<lambda>_. None)" and isin = isin and insert = insert and delete = delete
+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)