-- Autumn School Proof and Computation -- 3-8 October 2016 -- Fischbachau, Bayern, DE -- -- Andreas Abel, Agda Tutorial -- -- Part 1: How to Keep Your Neighbours in Order (Conor McBride) module TreeSortOrdSolution 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 0 _ = le _ compare (suc x) 0 = 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) (# y) = R x y ext R ⊥ _ = True ext R _ _ = False module _ {A : Set} (R : Rel A) (compare : Total R) where -- Binary search trees data BST (l u : Ext A) : Set where leaf : ext R 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 : ext R l (# p)) (p≤u : ext R (# 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 ... | le p≤q = node q (insert p l≤p p≤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 nil = leaf _ tree (cons x xs) = insert x _ _ (tree xs) -- Ordered lists data OList (l u : Ext A) : Set where onil : ext R l u → OList l u ocons : (p : A) (l≤p : ext R l (# p)) (ps : OList (# p) u) → OList l u -- Flattening a BST _++_ : ∀{l m u} (xs : OList l m) (ys : ∀{k} (k≤m : ext R k m) → OList k u) → OList l u ocons x l≤x xs ++ ys = ocons x l≤x (xs ++ ys) onil l≤m ++ ys = ys l≤m infixr 1 _++_ flatten : ∀{l u} (t : BST l u) → OList l u flatten (leaf l≤u) = onil l≤u flatten (node p lt rt) = flatten lt ++ λ prf → ocons p prf (flatten rt) -- Sorting is flatten ∘ tree sort : (xs : List A) → OList ⊥ ⊤ sort xs = flatten (tree xs)