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(defun cons-all (e x)
(declare (xargs :guard (true-listp x)))
(if (endp x)
nil
(cons (cons e (car x))
(cons-all e (cdr x)))))
(defun combinations (x n)
(declare (xargs :guard (and (true-listp x)
(natp n))))
(cond
((zp n) (list nil))
((endp x) nil)
(t
(append (cons-all (car x)
(combinations (cdr x) (- n 1)))
(combinations (cdr x) n)))))
(defun comb (n k)
(cond
((zp k) 1)
((zp n) 0)
(t (+ (comb (- n 1) k)
(comb (- n 1) (- k 1))))))
(defthm len-combinations
(equal (len (combinations x n))
(comb (len x) n)))
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