to find increasing subsequence, for example, [1 0 1 2 3 0 4 5] => [0 1 2 3]

find-increasing-subsequence.lisp

(defun find-subsequence-a (lst)
  (let ((result '(1)))
    (do ((l lst (cdr l)))
        ((null (cdr l)) result)
      (let ((current (car l))
            (next (cadr l)))
        (if (> next current)
            (setq result (cons (1+ (car result)) result))
            (setq result (cons 1 result)))))))

(defun find-subsequence-1 (lst)
  (let* ((dp (find-subsequence-a lst))
         (max-length (reduce #'max dp))
         (index (1+ (position max-length (reverse dp)))))
    (subseq lst (- index max-length) index)))

(defun find-subsequence-b (lst)
  (reduce #'(lambda (acc y)
              (let ((x (caar acc)))
                (if (> y x)
                    `((,y ,@(car acc)) ,@(cdr acc))
                    `((,y) ,@acc))
                )
              )
          (cdr lst)
          :initial-value `((,(car lst)))))

(defun find-subsequence-2 (lst)
  (reverse (reduce #'(lambda (sub-a sub-b)
                       (if (> (length sub-a) (length sub-b))
                           sub-a
                           sub-b))
                   (find-subsequence-b lst))))