(use test)

;; ex 1.9

(define inc add1)
(define dec sub1)

(define (my-+ a b)
  (if (= a 0) b (inc (my-+ (dec a) b))))

;; (+ 2 3)
;; (if (= 2 0) 3 (inc (+ (dec 2) 3)))
;; (if #f 3 (inc (+ (dec 2) 3)))
;; (inc (+ (dec 2) 3))
;; (inc (+ 1 3))
;; (inc (if (= 1 0) 3 (inc (+ (dec 1) 3))))
;; (inc (if #f 3 (inc (+ (dec 1) 3))))
;; (inc (inc (+ (dec 1) 3)))
;; (inc (inc (+ 0 3)))
;; (inc (inc (if (= 0 0) 3 (inc (+ (dec 0) 3)))))
;; (inc (inc (if #t 3 (inc (+ (dec 0) 3)))))
;; (inc (inc 3))
;; (inc 4)
;; 5

;; => series of recursive processes

(test 5 (my-+ 2 3))

(define (my-+ a b)
  (if (= a 0) b (my-+ (dec a) (inc b))))

;; (+ 2 3)
;; (if (= 2 0) 3 (+ (dec 2) (inc 3)))
;; (if #f 3 (+ (dec 2) (inc 3)))
;; (+ (dec 2) (inc 3))
;; (+ 1 4)
;; (if (= 1 0) 4 (+ (dec 1) (inc 4)))
;; (if #f 4 (+ (dec 1) (inc 4)))
;; (+ (dec 1) (inc 4))
;; (+ 0 5)
;; (if (= 0 0) 5 (+ (dec 0) (inc 5)))
;; (if #t 5 (+ (dec 0) (inc 5)))
;; 5

;; => series of iterative processes

(test 5 (my-+ 2 3))

;; ex 1.10

(define (A x y)
  (cond ((= y 0) 0)
        ((= x 0) (* 2 y))
        ((= y 1) 2)
        (else (A (- x 1) (A x (- y 1))))))

;; (A 1 10)
;; (A 0 (A 1 9))
;; (A 0 (A 0 (A 1 8)))
;; (A 0 (A 0 (A 0 (A 1 7))))
;; (A 0 (A 0 (A 0 (A 0 (A 1 6)))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2)))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 32)))))
;; (A 0 (A 0 (A 0 (A 0 64))))
;; (A 0 (A 0 (A 0 128)))
;; (A 0 (A 0 256))
;; (A 0 512)
;; 1024

(test 1024 (A 1 10))

;; (A 2 4)
;; (A 1 (A 2 3))
;; (A 1 (A 1 (A 2 2)))
;; (A 1 (A 1 (A 1 (A 2 1))))
;; (A 1 (A 1 (A 1 2)))
;; (A 1 (A 1 (A 0 (A 1 1))))
;; (A 1 (A 1 (A 0 2)))
;; (A 1 (A 1 4))
;; (A 1 (A 0 (A 1 3)))
;; (A 1 (A 0 (A 0 (A 1 2))))
;; (A 1 (A 0 (A 0 (A 0 (A 1 1)))))
;; (A 1 (A 0 (A 0 (A 0 2))))
;; (A 1 (A 0 (A 0 4)))
;; (A 1 (A 0 8))
;; (A 1 16)
;; (A 0 (A 1 15))
;; (A 0 (A 0 (A 1 14)))
;; (A 0 (A 0 (A 0 (A 1 13))))
;; (A 0 (A 0 (A 0 (A 0 (A 1 12)))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10)))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8)))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6)))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2)))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1))))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32)))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128)))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512)))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 2048)))))
;; (A 0 (A 0 (A 0 (A 0 4096))))
;; (A 0 (A 0 (A 0 8192)))
;; (A 0 (A 0 16384))
;; (A 0 32768)
;; 65536

(test 65536 (A 2 4))

;; (A 3 3)
;; (A 2 (A 3 2))
;; (A 2 (A 2 (A 3 1)))
;; (A 2 (A 2 2))
;; (A 2 (A 1 (A 2 1)))
;; (A 2 (A 1 2))
;; (A 2 (A 0 (A 1 1)))
;; (A 2 (A 0 2))
;; (A 2 4)
;; (A 1 (A 2 3))
;; (A 1 (A 1 (A 2 2)))
;; (A 1 (A 1 (A 1 (A 2 1))))
;; (A 1 (A 1 (A 1 2)))
;; (A 1 (A 1 (A 0 (A 1 1))))
;; (A 1 (A 1 (A 0 2)))
;; (A 1 (A 1 4))
;; (A 1 (A 0 (A 1 3)))
;; (A 1 (A 0 (A 0 (A 1 2))))
;; (A 1 (A 0 (A 0 (A 0 (A 1 1)))))
;; (A 1 (A 0 (A 0 (A 0 2))))
;; (A 1 (A 0 (A 0 4)))
;; (A 1 (A 0 8))
;; (A 1 16)
;; (A 0 (A 1 15))
;; (A 0 (A 0 (A 1 14)))
;; (A 0 (A 0 (A 0 (A 1 13))))
;; (A 0 (A 0 (A 0 (A 0 (A 1 12)))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10)))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8)))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6)))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2)))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1))))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32)))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64))))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128)))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256))))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512)))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024))))))
;; (A 0 (A 0 (A 0 (A 0 (A 0 2048)))))
;; (A 0 (A 0 (A 0 (A 0 4096))))
;; (A 0 (A 0 (A 0 8192)))
;; (A 0 (A 0 16384))
;; (A 0 32768)
;; 65536

(test 65536 (A 3 3))

(define (f n) (A 0 n)) ;=> 2n
(test 2 (f 1))
(test 4 (f 2))
(test 6 (f 3))
(test 8 (f 4))

(define (g n) (A 1 n)) ;=> 2^n
(test 2 (g 1))
(test 4 (g 2))
(test 8 (g 3))
(test 16 (g 4))

(define (h n) (A 2 n)) ;=> 2^2^...^2 (exponentiate two repeatedly by two n times)
(test 2 (h 1))
(test 4 (h 2))
(test 16 (h 3))
(test 65536 (h 4))