session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
(* Author: Tobias Nipkow *)
section \<open>Unbalanced Tree Implementation of Set\<close>
theory Tree_Set
imports
"HOL-Library.Tree"
Cmp
Set_by_Ordered
begin
fun isin :: "'a::linorder 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::linorder \<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 l a r) =
(if l = Leaf then (a,r) else let (x,l') = del_min l in (x, Node l' a r))"
fun delete :: "'a::linorder \<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> x # inorder t' = inorder t"
by(induction t arbitrary: t' rule: del_min.induct)
(auto simp: sorted_lems split: prod.splits if_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