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Repeat

(repeat n x) creates a list of xes with length n; it is a simpler alternative to make-list.

Definitions and Theorems

Function: repeat

(defun repeat (n x)
  (declare (xargs :guard (natp n)))
  (mbe :logic
       (if (zp n)
           nil
         (cons x (repeat (- n 1) x)))
       :exec (make-list n :initial-element x)))

Theorem: repeat-when-zp

(defthm repeat-when-zp
  (implies (zp n)
           (equal (repeat n a) nil)))

Theorem: (repeat 0 x)

(defthm |(repeat 0 x)|
  (equal (repeat 0 x) nil))

Theorem: repeat-under-iff

(defthm repeat-under-iff
  (iff (repeat n x) (not (zp n))))

Theorem: consp-of-repeat

(defthm consp-of-repeat
  (equal (consp (repeat n a))
         (not (zp n))))

Theorem: repeat-1

(defthm repeat-1
  (equal (repeat 1 a) (list a)))

Theorem: take-when-atom

(defthm take-when-atom
  (implies (atom x)
           (equal (take n x) (repeat n nil))))

Theorem: len-of-repeat

(defthm len-of-repeat
  (equal (len (repeat n x)) (nfix n)))

Theorem: repeat-of-nfix

(defthm repeat-of-nfix
  (equal (repeat (nfix n) x)
         (repeat n x)))

Theorem: car-of-repeat-increment

(defthm car-of-repeat-increment
  (implies (natp n)
           (equal (car (repeat (+ 1 n) x)) x)))

Theorem: cdr-of-repeat-increment

(defthm cdr-of-repeat-increment
  (implies (natp n)
           (equal (cdr (repeat (+ 1 n) x))
                  (repeat n x))))

Theorem: member-of-repeat

(defthm member-of-repeat
  (equal (member a (repeat n b))
         (if (equal a b) (repeat n b) nil)))

Theorem: take-of-repeat

(defthm take-of-repeat
  (equal (take n (repeat k a))
         (if (<= (nfix n) (nfix k))
             (repeat n a)
           (append (repeat k a)
                   (repeat (- (nfix n) (nfix k)) nil)))))

Theorem: nthcdr-of-repeat

(defthm nthcdr-of-repeat
  (equal (nthcdr n (repeat k a))
         (if (<= (nfix n) (nfix k))
             (repeat (- (nfix k) (nfix n)) a)
           nil)))

Theorem: append-of-repeat-to-cons-of-same

(defthm append-of-repeat-to-cons-of-same
  (equal (append (repeat n a) (cons a x))
         (cons a (append (repeat n a) x))))

Theorem: equal-of-append-repeat

(defthm equal-of-append-repeat
  (implies (case-split (<= n (len y)))
           (equal (equal (append (repeat n a) x) y)
                  (and (equal (repeat n a) (take n y))
                       (equal x (nthcdr n y))))))

Theorem: rev-of-repeat

(defthm rev-of-repeat
  (equal (rev (repeat n a)) (repeat n a)))

Theorem: subsetp-of-repeat

(defthm subsetp-of-repeat
  (iff (subsetp-equal (repeat n x) y)
       (or (zp n) (member-equal x y))))

Theorem: element-list-p-of-repeat

(defthm element-list-p-of-repeat
  (iff (element-list-p (repeat n x))
       (or (element-p x) (zp n)))
  :rule-classes :rewrite)

Subtopics

Take-of-too-many: Rewrite (take n x) when n is more than (len x).
Make-list-ac-removal: Rewrite rule that eliminates make-list-ac (and hence make-list) in favor of repeat.
Replicate: Alias for repeat.