#lang s-exp "../cur.rkt"
(require "sugar.rkt" "bool.rkt")
;; TODO: override (all-defined-out) to enable exporting all these
;; properly.
(provide Nat z s add1 sub1 plus )
(module+ test
  (require rackunit))

(data Nat : Type
  (z : Nat)
  (s : (-> Nat Nat)))

(define (add1 (n : Nat)) (s n))
(module+ test
  (check-equal? (add1 (s z)) (s (s z))))

(define (sub1 (n : Nat))
  (case* Nat n (lambda (x : Nat) Nat)
    [z z]
    [(s (x : Nat)) IH: ((ih-n : Nat)) x]))
(module+ test
  (check-equal? (sub1 (s z)) z))

(define (plus (n1 : Nat) (n2 : Nat))
  (case* Nat n1 (lambda (x : Nat) Nat)
    [z n2]
    [(s (x : Nat)) IH: ((ih-n1 : Nat))
     (s ih-n1)]))
(module+ test
  (check-equal? (plus z z) z)
  (check-equal? (plus (s (s z)) (s (s z))) (s (s (s (s z))))))

;; Credit to this function goes to Max
(define (nat-equal? (n1 : Nat))
  (elim Nat n1 (lambda (x : Nat) (-> Nat Bool))
    (lambda (n2 : Nat)
      (elim Nat n2 (lambda (x : Nat) Bool)
        true
        (lambda* (x : Nat) (ih-n2 : Bool) false)))
    (lambda* (x : Nat) (ih : (-> Nat Bool))
      (lambda (n2 : Nat)
        (elim Nat n2 (lambda (x : Nat) Bool)
          false
          (lambda* (x : Nat) (ih-bla : Bool)
            (ih x)))))))
(module+ test
  (check-equal? (nat-equal? z z) true)
  (check-equal? (nat-equal? z (s z)) false)
  (check-equal? (nat-equal? (s z) (s z)) true))