-- 8th Summer School on Formal Techniques -- Menlo College, Atherton, California, US -- -- 21-25 May 2018 -- -- Lecture 1: Introduction to Agda -- -- File 2: How to Keep Your Neighbours in Order (Conor McBride) module TreeSortOrd where open import Prelude -- Comparing natural numbers Total : ∀{A} (R : Rel A) → Set Total R = ∀ x y → R x y ⊎ R y x pattern le z = inl z pattern ge z = inr z compare : Total _≤_ compare zero y = le _ compare (suc x) zero = ge _ compare (suc x) (suc y) = compare x y -- Extension by a least and a greatest element data Ext (A : Set) : Set where ⊤ : Ext A # : A → Ext A ⊥ : Ext A ext : ∀{A} → Rel A → Rel (Ext A) ext R ⊤ ⊤ = True ext R ⊤ (# x) = False ext R ⊤ ⊥ = False ext R (# x) ⊤ = True ext R (# x) (# y) = R x y ext R (# x) ⊥ = False ext R ⊥ y = True module _ {A : Set} (R : Rel A) (compare : Total R) (let _≤_ = ext R) where -- Binary search trees data BST (l u : Ext A) : Set where leaf : l ≤ u → BST l u node : (p : A) (lt : BST l (# p)) (rt : BST (# p) u) → BST l u insert : ∀{l u : Ext A} (p : A) (l≤p : l ≤ # p) (p≤u : # p ≤ u) (t : BST l u) → BST l u insert p l≤p p≤u (leaf l≤u) = node p (leaf l≤p) (leaf p≤u) insert p l≤p p≤u (node q lt rt) with compare p q insert p l≤p p≤u (node q lt rt) | le p≤q = node q (insert p l≤p p≤q lt) rt insert p l≤p p≤u (node q lt rt) | ge q≤p = node q lt (insert p q≤p p≤u rt) -- Building a BST tree : (xs : List A) → BST ⊥ ⊤ tree [] = leaf _ tree (x ∷ xs) = insert x _ _ (tree xs) {- -- Ordered lists data OList (l u : Ext A) : Set where onil : l ≤ u → OList l u ocons : (p : A) (l≤p : l ≤ # p) (ps : OList (# p) u) → OList l u -- Flattening a BST _++_ : ∀{l m u} (xs : OList l m) (ys : ∀{k} (k≤m : k ≤ m) → OList k u) → OList l u ocons x l≤x xs ++ ys = {!!} onil l≤m ++ ys = {!!} infixr 1 _++_ flatten : ∀{l u} (t : BST l u) → OList l u flatten (leaf l≤u) = {!!} flatten (node p lt rt) = {!!} -- Sorting is flatten ∘ tree sort : (xs : List A) → OList ⊥ ⊤ sort xs = flatten (tree xs) -- -}