--- a/src/HOL/Data_Structures/Tree_Set.thy Wed Nov 11 16:42:30 2015 +0100
+++ b/src/HOL/Data_Structures/Tree_Set.thy Wed Nov 11 18:32:26 2015 +0100
@@ -1,75 +1,75 @@
-(* Author: Tobias Nipkow *)
-
-section {* Tree Implementation of Sets *}
-
-theory Tree_Set
-imports
- "~~/src/HOL/Library/Tree"
- Cmp
- Set_by_Ordered
-begin
-
-fun isin :: "'a::cmp tree \<Rightarrow> 'a \<Rightarrow> bool" where
-"isin Leaf x = False" |
-"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::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
-"insert x Leaf = Node Leaf x Leaf" |
-"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::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
-"delete x Leaf = Leaf" |
-"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"
-
-lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps1)
-
-lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps2)
-
-
-lemma inorder_insert:
- "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
-by(induction t) (auto simp: ins_list_simps)
-
-
-lemma del_minD:
- "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted(inorder t) \<Longrightarrow>
- x # inorder t' = inorder t"
-by(induction t arbitrary: t' rule: del_min.induct)
- (auto simp: sorted_lems split: prod.splits)
-
-lemma inorder_delete:
- "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 inv = "\<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: inorder_insert)
-next
- case 4 thus ?case by(simp add: inorder_delete)
-qed (rule TrueI)+
-
-end
+(* Author: Tobias Nipkow *)
+
+section {* Tree Implementation of Sets *}
+
+theory Tree_Set
+imports
+ "~~/src/HOL/Library/Tree"
+ Cmp
+ Set_by_Ordered
+begin
+
+fun isin :: "'a::cmp tree \<Rightarrow> 'a \<Rightarrow> bool" where
+"isin Leaf x = False" |
+"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::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+"insert x Leaf = Node Leaf x Leaf" |
+"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::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+"delete x Leaf = Leaf" |
+"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"
+
+lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
+by (induction t) (auto simp: elems_simps1)
+
+lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
+by (induction t) (auto simp: elems_simps2)
+
+
+lemma inorder_insert:
+ "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
+by(induction t) (auto simp: ins_list_simps)
+
+
+lemma del_minD:
+ "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted(inorder t) \<Longrightarrow>
+ x # inorder t' = inorder t"
+by(induction t arbitrary: t' rule: del_min.induct)
+ (auto simp: sorted_lems split: prod.splits)
+
+lemma inorder_delete:
+ "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 inv = "\<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: inorder_insert)
+next
+ case 4 thus ?case by(simp add: inorder_delete)
+qed (rule TrueI)+
+
+end