section \<open>Unbalanced Tree Implementation of Map\<close>
theory Tree_Map imports
Tree_Set
Map_Specs begin
fun lookup :: "('a::linorder*'b) tree \ 'a \ 'b option" where "lookup Leaf x = None" | "lookup (Node l (a,b) r) x =
(case cmp x a of LT \<Rightarrow> lookup l x | GT \<Rightarrow> lookup r x | EQ \<Rightarrow> Some b)"
fun update :: "'a::linorder \ 'b \ ('a*'b) tree \ ('a*'b) tree" where "update x y Leaf = Node Leaf (x,y) Leaf" | "update x y (Node l (a,b) r) = (case cmp x a of
LT \<Rightarrow> Node (update x y l) (a,b) r |
EQ \<Rightarrow> Node l (x,y) r |
GT \<Rightarrow> Node l (a,b) (update x y r))"
fun delete :: "'a::linorder \ ('a*'b) tree \ ('a*'b) tree" where "delete x Leaf = Leaf" | "delete x (Node l (a,b) r) = (case cmp x a of
LT \<Rightarrow> Node (delete x l) (a,b) r |
GT \<Rightarrow> Node l (a,b) (delete x r) |
EQ \<Rightarrow> if r = Leaf then l else let (ab',r') = split_min r in Node l ab' r')"
subsection "Functional Correctness Proofs"
lemma lookup_map_of: "sorted1(inorder t) \ lookup t x = map_of (inorder t) x" by (induction t) (auto simp: map_of_simps split: option.split)
lemma inorder_update: "sorted1(inorder t) \ inorder(update a b t) = upd_list a b (inorder t)" by(induction t) (auto simp: upd_list_simps)
interpretation M: Map_by_Ordered where empty = empty and lookup = lookup and update = update 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: lookup_map_of) next case 3 thus ?caseby(simp add: inorder_update) next case 4 thus ?caseby(simp add: inorder_delete) qed auto
end
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