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(ns church-factorial)
; def#
; ----
(defmacro def#
"A light wrapper around `def`, that keeps track of the
_source code_ for each definition
This let's us _unwrap_ all the definitions later : >"
[name v]
`(do
(def ~name ~v)
(alter-meta! (var ~name) assoc :source {:name '~name :v '~v})
(var ~name)))
; pair
; ----
(def# church-pair (fn [a b]
(fn [selector]
(selector a b))))
(def# take-first-arg (fn [a b] a))
(def# church-first (fn [pair]
(pair take-first-arg)))
(def# take-second-arg (fn [a b] b))
(def# church-second (fn [pair]
(pair take-second-arg)))
(comment
(def# ex-pair (church-pair 0 1))
(church-first ex-pair)
(church-second ex-pair))
; factorial-clj
; -------------
(defn factorial-clj [n]
(if (zero? n)
1
(* n (factorial-clj (dec n)))))
(comment (factorial-clj 5))
; church-numerals: 0-2
; ------------
(def# zero (fn [f v] v))
(def# one (fn [f v]
(f (zero f v))))
(def# two (fn [f v]
(f (one f v))))
; church-numeral->int
; ---------------
(defn church-numeral->int [church-numeral]
(church-numeral inc 0))
(comment
(map church-numeral->int [zero one two]))
; church-inc
; --------------
(def# church-inc
(fn [church-numeral]
(fn [f v]
(f (church-numeral f v)))))
(comment (church-numeral->int (church-inc (church-inc one))))
; int->church-numeral
; -----------------------------
(defn int->church-numeral [clojure-int]
(if (zero? clojure-int)
zero
(church-inc (int->church-numeral (dec clojure-int)))))
(comment
(church-numeral->int (int->church-numeral 5)))
; shift-and-inc
; -------------
(def# shift-and-inc (fn [pair]
(church-pair
(church-second pair)
(church-inc (church-second pair)))))
(comment
(let [p (shift-and-inc (church-pair one two))]
(map church-numeral->int [(church-first p) (church-second p)])))
; church-dec
; -----------
(def# church-dec
(fn [church-numeral]
(church-first
(church-numeral shift-and-inc
(church-pair zero zero)))))
(comment
(church-numeral->int (church-dec (int->church-numeral 10))))
; church-*
; --------------
(def# church-*
(fn [num-a num-b]
(fn [f v]
(num-a (partial num-b f) v))))
(comment
(church-numeral->int
(church-* (int->church-numeral 5) (int->church-numeral 5))))
; church-booleans
; ---------------
(def# church-true (fn [when-true when-false]
when-true))
(def# church-false (fn [when-true when-false]
when-false))
; church-bool->bool
; -----------------------
(defn church-bool->bool [church-bool]
(church-bool true false))
(comment
(church-bool->bool church-true)
(church-bool->bool church-false))
; church-if
; ---------
(def# church-if (fn [church-bool when-true when-false]
(church-bool when-true when-false)))
(comment
(church-numeral->int
(church-if church-true one two))
(church-numeral->int
(church-if church-false one two)))
; church-zero?
; ------------
(def# church-zero?
(fn [church-numeral]
(church-numeral (fn [v] church-false) church-true)))
(comment
(church-bool->bool (church-zero? zero))
(church-bool->bool (church-zero? one)))
; factorial-v0
; ------------
(def# factorial-v0
(fn [church-numeral-n]
((church-if
(church-zero? church-numeral-n)
(fn [] one)
(fn []
(church-*
church-numeral-n
(factorial-v0 (church-dec church-numeral-n))))))))
(comment
(church-numeral->int (factorial-v0 (int->church-numeral 5))))
; y-combinator
; ------------
(def# make-recursable
(fn [injectable-f]
((fn [recursion-handler] (recursion-handler recursion-handler))
(fn [recursion-handler]
(injectable-f (fn [next-arg]
((recursion-handler recursion-handler) next-arg)))))))
; factorial-yc
; ------------
(def# injectable-factorial
(fn [factorial-cb]
(fn [church-numeral-n]
((church-if
(church-zero? church-numeral-n)
(fn [] one)
(fn []
(church-*
church-numeral-n
(factorial-cb (church-dec church-numeral-n)))))))))
(def# factorial-yc (make-recursable injectable-factorial))
(comment
(church-numeral->int (factorial-yc (int->church-numeral 5))))
; expand
; ------
(defn expand
"This takes a form like
(church-numeral->int (factorial-yc (int->church-numeral 5)))
And expands all the function definitions, to give
us the intuition for how our 'lambda calculus' way would look!"
[form]
(cond
(symbol? form)
(if-let [source (some-> (str *ns* "/" form)
symbol
find-var
meta
:source)]
(expand (:v source))
form)
(seq? form)
(map expand form)
:else form))
(comment (expand '(factorial-yc (church-inc (church-inc (church-inc two))))))