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(in-package "ACL2")
(defun member-if-over-1 (x)
(cond ((endp x) nil)
((< 1 (car x)) x)
(t (member-if-over-1 (cdr x)))))
(defun sum (x)
(if (endp x)
0
(+ (car x)
(sum (cdr x)))))
(defthm pigeonhole
(implies (< (len x) (sum x))
(member-if-over-1 x)))
(defun list-count (e x)
(cond ((endp x) 0)
((equal e (car x)) (1+ (list-count e (cdr x))))
(t (list-count e (cdr x)))))
(defthm member-remove
(implies (not (equal x y))
(iff (member x (remove y z))
(member x z))))
(defthm no-duplicatesp-remove
(implies (no-duplicatesp x)
(no-duplicatesp (remove e x))))
(defun map-list-count (x s)
(if (endp s)
nil
(cons (list-count (car s) x)
(map-list-count (remove (car s) x) (cdr s)))))
(defthm len-map-list-count
(equal (len (map-list-count x s))
(len s)))
(defthm len-remove
(equal (len (remove e x))
(- (len x) (list-count e x))))
(defthm not-subsetp-remove
(implies (not (subsetp (remove e x) y))
(not (subsetp x (cons e y)))))
(defthm sum-map-list-count
(implies (and (no-duplicatesp s)
(subsetp x s))
(equal (sum (map-list-count x s))
(len x)))
:hints (("Goal" :induct (map-list-count x s))))
(defthm not-no-duplicatesp-list-count
(implies (< 1 (list-count e x))
(not (no-duplicatesp x)))
:rule-classes ((:rewrite :match-free :all)))
(defthm not-no-duplicatesp-remove
(implies (not (no-duplicatesp (remove e x)))
(not (no-duplicatesp x)))
:rule-classes ((:rewrite :match-free :all)))
(defthm member-if-over-1-map-list-count-is-duplicated
(implies (and (no-duplicatesp s)
(subsetp x s)
(member-if-over-1 (map-list-count x s)))
(not (no-duplicatesp x)))
:hints (("Goal" :induct (map-list-count x s)))
:rule-classes ((:rewrite :match-free :all)))
(defthm pigeonhole-for-list
(implies (and (no-duplicatesp s)
(subsetp x s)
(< (len s) (len x)))
(not (no-duplicatesp x)))
:rule-classes ((:rewrite :match-free :all)))
(in-theory (disable pigeonhole-for-list))
(defthm blood-types-are-duplicates
(implies (and (subsetp x '(A B O AB))
(<= 5 (len x)))
(not (no-duplicatesp x)))
:hints (("Goal" :use (:instance pigeonhole-for-list
(s '(A B O AB))
(x x)))))
(defthm birth-months-are-duplicates
(let ((birth-months '(1 2 3 4 5 6 7 8 9 10 11 12)))
(implies (and (subsetp x birth-months)
(<= 13 (len x)))
(not (no-duplicatesp x))))
:hints (("Goal" :use (:instance pigeonhole-for-list
(s '(1 2 3 4 5 6 7 8 9 10 11 12))
(x x)))))
(defun nats-below (n)
(if (zp n)
nil
(cons (- n 1)
(nats-below (- n 1)))))
(defthm len-nats-below
(implies (natp n)
(equal (len (nats-below n)) n)))
(defun all-under-p (x n)
(if (endp x)
t
(and (< (car x) n)
(all-under-p (cdr x) n))))
(defthm all-under-p-lemma
(implies (and (posp n)
(all-under-p x (+ -1 n)))
(all-under-p x n)))
(defthm all-under-p-nats-below
(all-under-p (nats-below n) n))
(defthm all-under-p-not-member
(implies (all-under-p x n)
(not (member n x))))
(defthm not-member-nats-below
(not (member n (nats-below n))))
(defthm no-duplicatesp-nats-below
(no-duplicatesp (nats-below n)))
(defthm duplicates-nats
(implies (and (integerp n)
(<= 0 n)
(subsetp x (nats-below n))
(< n m)
(<= m (len x)))
(not (no-duplicatesp x)))
:rule-classes ((:rewrite :match-free :all))
:hints (("Goal" :use (:instance pigeonhole-for-list
(s (nats-below n))
(x x)))))
(in-theory (disable duplicates-nats))
(defthm numbers-of-hairs-are-duplicates-in-gunma
(let ((max-number-of-hairs 250000)
(gunma-population 2000000))
(implies (and (subsetp x (nats-below max-number-of-hairs))
(= gunma-population (len x)))
(not (no-duplicatesp x))))
:hints (("Goal" :use (:instance duplicates-nats
(n 250000)
(m 2000000)
(x x)))))
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