section \<open>Unbalanced Tree Implementation of Set\<close>
theory Tree_Set imports "HOL-Library.Tree"
Cmp
Set_Specs begin
definition empty :: "'a tree"where "empty = Leaf"
fun isin :: "'a::linorder tree \ 'a \ 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 \ 'a tree \ '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))"
text\<open>Deletion by replacing:\<close>
fun split_min :: "'a tree \ 'a * 'a tree" where "split_min (Node l a r) =
(if l = Leaf then (a,r) else let (x,l') = split_min l in (x, Node l' a r))"
fun delete :: "'a::linorder \ 'a tree \ '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') = split_min r in Node l a' r')"
text\<open>Deletion by joining:\<close>
fun join :: "('a::linorder)tree \ 'a tree \ 'a tree" where "join t Leaf = t" | "join Leaf t = t" | "join (Node t1 a t2) (Node t3 b t4) =
(case join t2 t3 of
Leaf \<Rightarrow> Node t1 a (Node Leaf b t4) |
Node u2 x u3 \<Rightarrow> Node (Node t1 a u2) x (Node u3 b t4))"
fun delete2 :: "'a::linorder \ 'a tree \ 'a tree" where "delete2 x Leaf = Leaf" | "delete2 x (Node l a r) =
(case cmp x a of
LT \<Rightarrow> Node (delete2 x l) a r |
GT \<Rightarrow> Node l a (delete2 x r) |
EQ \<Rightarrow> join l r)"
subsection "Functional Correctness Proofs"
lemma isin_set: "sorted(inorder t) \ isin t x = (x \ set (inorder t))" by (induction t) (auto simp: isin_simps)
lemma inorder_insert: "sorted(inorder t) \ inorder(insert x t) = ins_list x (inorder t)" by(induction t) (auto simp: ins_list_simps)
lemma split_minD: "split_min t = (x,t') \ t \ Leaf \ x # inorder t' = inorder t" by(induction t arbitrary: t' rule: split_min.induct)
(auto simp: sorted_lems split: prod.splits if_splits)
interpretation S: Set_by_Ordered where empty = empty and isin = isin and insert = insert and delete = delete and inorder = inorder and inv = "\_. True" proof (standard, goal_cases) case 1 show ?caseby (simp add: empty_def) next case 2 thus ?caseby(simp add: isin_set) next case 3 thus ?caseby(simp add: inorder_insert) next case 4 thus ?caseby(simp add: inorder_delete) qed (rule TrueI)+
lemma inorder_join: "inorder(join l r) = inorder l @ inorder r" by(induction l r rule: join.induct) (auto split: tree.split)
lemma inorder_delete2: "sorted(inorder t) \ inorder(delete2 x t) = del_list x (inorder t)" by(induction t) (auto simp: inorder_join del_list_simps)
interpretation S2: Set_by_Ordered where empty = empty and isin = isin and insert = insert and delete = delete2 and inorder = inorder and inv = "\_. True" proof (standard, goal_cases) case 1 show ?caseby (simp add: empty_def) next case 2 thus ?caseby(simp add: isin_set) next case 3 thus ?caseby(simp add: inorder_insert) next case 4 thus ?caseby(simp add: inorder_delete2) qed (rule TrueI)+
end
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