module TreeSortOrdSolution where
open import Prelude
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
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
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)
tree : (xs : List A) → BST ⊥ ⊤
tree nil = leaf _
tree (cons x xs) = insert x _ _ (tree xs)
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
_++_ : ∀{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)
sort : (xs : List A) → OList ⊥ ⊤
sort xs = flatten (tree xs)