Church numerals in Racket

church.rkt

#lang racket

;; numbers are represented as:
;; (lambda (f) (lambda (x) (f ... (f x))))
;; where there are n calls to f.
(define zero (lambda (f) (lambda (x) x)))
(define one  (lambda (f) (lambda (x) (f x))))
(define two  (lambda (f) (lambda (x) (f (f x)))))

;; do add1 n times, starting from 0
;; (add1 (add1 ... (add1 0) ...))
(define (church->nat n)
  ((n add1) 0))

(define succ
  ;; the *argument*
  (lambda (n)
    ;; the thing we're *returning* should do f "n+1 times"
    ;; ((n f) x) "applies f n times" and returns a result
    ;; 
    (lambda (f) (lambda (x) (f ((n f) x))))))

(define plus
  (lambda (n)
    (lambda (k) 
      (lambda (f) (lambda (x) ((k f) ((n f) x)))))))

(define mult
  (lambda (n)
    (lambda (k)
      (lambda (f) (lambda (x) (((n k) f) x))))))

(displayln (format "we computed ~a" (church->nat ((plus two) two))))

There is a minor mistake in this piece of code. The function defined as mult is actually k to the power n. The actual code:

#lang racket

;; numbers are represented as:
;; (lambda (f) (lambda (x) (f ... (f x))))
;; where there are n calls to f.
(define zero (lambda (f) (lambda (x) x)))
(define one  (lambda (f) (lambda (x) (f x))))
(define two  (lambda (f) (lambda (x) (f (f x)))))

;; do add1 n times, starting from 0
;; (add1 (add1 ... (add1 0) ...))
(define (church->nat n)
  ((n add1) 0))

(define succ
  ;; the *argument*
  (lambda (n)
    ;; the thing we're *returning* should do f "n+1 times"
    ;; ((n f) x) "applies f n times" and returns a result
    ;; 
    (lambda (f) (lambda (x) (f ((n f) x))))))

(define plus
  (lambda (n)
    (lambda (k) 
      (lambda (f) (lambda (x) ((k f) ((n f) x)))))))

(define mult
  (lambda (n)
    (lambda (k)
      (lambda (f) (lambda (x) ((n (k f)) x))))))
      
(define power
  (lambda (n)
    (lambda (k)
      (lambda (f) (lambda (x) (((n k) f) x))))))

(displayln (format "we computed ~a" (church->nat ((plus two) two))))
(displayln (format "we computed ~a" (church->nat ((mult two) one))))