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(** {1 Polymorphic binary trees with elements at nodes} *)
module Tree
type tree 'a = Empty | Node (tree 'a) 'a (tree 'a)
let predicate is_empty (t:tree 'a)
ensures { result <-> t = Empty }
= match t with Empty -> true | Node _ _ _ -> false end
end
(** {2 Number of nodes} *)
module Size
use Tree
use int.Int
let rec function size (t: tree 'a) : int = match t with
| Empty -> 0
| Node l _ r -> 1 + size l + size r
end
lemma size_nonneg: forall t: tree 'a. 0 <= size t
lemma size_empty: forall t: tree 'a. 0 = size t <-> t = Empty
end
(** {2 Occurrences in a binary tree} *)
module Occ
use Tree
use int.Int
function occ (x: 'a) (t: tree 'a) : int = match t with
| Empty -> 0
| Node l y r -> (if y = x then 1 else 0) + occ x l + occ x r
end
lemma occ_nonneg:
forall x: 'a, t: tree 'a. 0 <= occ x t
predicate mem (x: 'a) (t: tree 'a) =
0 < occ x t
end
(** {2 Height of a tree} *)
module Height
use Tree
use int.Int
use int.MinMax
let rec function height (t: tree 'a) : int = match t with
| Empty -> 0
| Node l _ r -> 1 + max (height l) (height r)
end
lemma height_nonneg:
forall t: tree 'a. 0 <= height t
end
(** {2 In-order traversal} *)
module Inorder
use Tree
use list.List
use list.Append
let rec function inorder (t: tree 'a) : list 'a = match t with
| Empty -> Nil
| Node l x r -> inorder l ++ Cons x (inorder r)
end
end
(** {2 Pre-order traversal} *)
module Preorder
use Tree
use list.List
use list.Append
let rec function preorder (t: tree 'a) : list 'a = match t with
| Empty -> Nil
| Node l x r -> Cons x (preorder l ++ preorder r)
end
end
module InorderLength
use Tree
use Size
use Inorder
use list.List
use list.Length
lemma inorder_length: forall t: tree 'a. length (inorder t) = size t
end
(** {2 Huet's zipper} *)
module Zipper
use Tree
type zipper 'a =
| Top
| Left (zipper 'a) 'a (tree 'a)
| Right (tree 'a) 'a (zipper 'a)
let rec function zip (t: tree 'a) (z: zipper 'a) : tree 'a =
match z with
| Top -> t
| Left z x r -> zip (Node t x r) z
| Right l x z -> zip (Node l x t) z
end
(* navigating in a tree using a zipper *)
type pointed 'a = (tree 'a, zipper 'a)
let function start (t: tree 'a) : pointed 'a = (t, Top)
let function up (p: pointed 'a) : pointed 'a = match p with
| _, Top -> p (* do nothing *)
| l, Left z x r | r, Right l x z -> (Node l x r, z)
end
let function top (p: pointed 'a) : tree 'a = let t, z = p in zip t z
let function down_left (p: pointed 'a) : pointed 'a = match p with
| Empty, _ -> p (* do nothing *)
| Node l x r, z -> (l, Left z x r)
end
let function down_right (p: pointed 'a) : pointed 'a = match p with
| Empty, _ -> p (* do nothing *)
| Node l x r, z -> (r, Right l x z)
end
end
(* monomorphic AVL trees, for the purpose of Coma tests *)
module AVL
use int.Int
use int.MinMax
type elt
val predicate lt elt elt
clone relations.TotalStrictOrder with
type t = elt, predicate rel = lt, axiom .
(* binary trees with height stored in nodes *)
type tree = E | N int tree elt tree
let function ht (t: tree) : int =
match t with E -> 0 | N h _ _ _ -> h end
(* smart constructor *)
let function node (l: tree) (x: elt) (r: tree) : tree =
N (1 + max (ht l) (ht r)) l x r
let rec ghost function height (t: tree) : int
ensures { result >= 0 }
= match t with
| E -> 0
| N _ l _ r -> 1 + max (height l) (height r)
end
predicate wf (t: tree) =
match t with
| E -> true
| N h l _ r -> h = height t && wf l && wf r
end
predicate mem (y: elt) (t: tree) =
match t with
| E -> false
| N _ l x r -> mem y l || y=x || mem y r
end
predicate tree_lt (t: tree) (y: elt) =
forall x. mem x t -> lt x y
predicate lt_tree (y: elt) (t: tree) =
forall x. mem x t -> lt y x
predicate bst (t: tree) =
match t with
| E -> true
| N _ l x r -> bst l && tree_lt l x && bst r && lt_tree x r
end
predicate avl (t: tree) =
match t with
| E -> true
| N _ l _ r -> avl l && avl r && -1 <= height l - height r <= 1
end
end
|
