--- a/src/HOL/Data_Structures/Tree_Set.thy Wed Nov 04 15:07:23 2015 +0100
+++ b/src/HOL/Data_Structures/Tree_Set.thy Thu Nov 05 08:27:14 2015 +0100
@@ -5,31 +5,34 @@
theory Tree_Set
imports
"~~/src/HOL/Library/Tree"
+ Cmp
Set_by_Ordered
begin
-fun isin :: "'a::linorder tree \<Rightarrow> 'a \<Rightarrow> bool" where
+fun isin :: "'a::cmp tree \<Rightarrow> 'a \<Rightarrow> bool" where
"isin Leaf x = False" |
-"isin (Node l a r) x = (x < a \<and> isin l x \<or> x=a \<or> isin r x)"
+"isin (Node l a r) x =
+ (case cmp x a of LT \<Rightarrow> isin l x | EQ \<Rightarrow> True | GT \<Rightarrow> isin r x)"
hide_const (open) insert
-fun insert :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+fun insert :: "'a::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"insert x Leaf = Node Leaf x Leaf" |
-"insert x (Node l a r) =
- (if x < a then Node (insert x l) a r else
- if x = a then Node l a r
- else Node l a (insert x r))"
+"insert x (Node l a r) = (case cmp x a of
+ LT \<Rightarrow> Node (insert x l) a r |
+ EQ \<Rightarrow> Node l a r |
+ GT \<Rightarrow> Node l a (insert x r))"
fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
"del_min (Node Leaf a r) = (a, r)" |
"del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))"
-fun delete :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+fun delete :: "'a::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"delete x Leaf = Leaf" |
-"delete x (Node l a r) = (if x < a then Node (delete x l) a r else
- if x > a then Node l a (delete x r) else
- if r = Leaf then l else let (a',r') = del_min r in Node l a' r')"
+"delete x (Node l a r) = (case cmp x a of
+ LT \<Rightarrow> Node (delete x l) a r |
+ GT \<Rightarrow> Node l a (delete x r) |
+ EQ \<Rightarrow> if r = Leaf then l else let (a',r') = del_min r in Node l a' r')"
subsection "Functional Correctness Proofs"
@@ -56,7 +59,6 @@
"sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
by(induction t) (auto simp: del_list_simps del_minD split: prod.splits)
-
interpretation Set_by_Ordered
where empty = Leaf and isin = isin and insert = insert and delete = delete
and inorder = inorder and wf = "\<lambda>_. True"