### Y-Combinator defined and used in Clojure

Below is Y-Combinator (a fixed-point combinator) encoded in Clojure
and examples of its use. Note
that this is the call-by-value version of Y-Combinator since the one
given in the book converges only in the call-by-name lambda-calculus,
and results in divergence in call-by-value.

```
(ns YCombinator)
;; the Y-Combinator
(def Y (fn [f] ((fn [x] (f (fn [y] ((x x) y))))
(fn [x] (f (fn [y] ((x x) y)))))))
;; the function that is used to define recursive factorial.
;; Note that f is a parameter and the function itself is anonymous.
(def fact-funct (fn [f] (fn [n] (if (<= n 0) 1 (* n (f (- n 1)))))))
;; using fact-funct to define the factorial function as the fixed
;; point.
(def fact (Y fact-funct))
;; checking that fact works as expected
(println (fact 5))
;; the function for recursion on collections
(def sum-funct (fn [f] (fn [coll] (if (empty? coll) 0
(+ (first coll) (f (rest coll)))))))
;; defining the sum-coll function using Y-Combinator
(def sum-coll (Y sum-funct))
(println (sum-coll [2 3 4 5]))
```

UMM CSci 4651

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