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Charlist-codelist-conversions

Nats=>chars

Convert a true list of natural numbers below 256 to the corresponding true list of characters.

Signature
(nats=>chars nats) → chars
Arguments: nats — Guard (unsigned-byte-listp 8 nats).
Returns: chars — Type (character-listp chars).

This operation has a natural-recursive definition for logic reasoning and a tail-recursive executional for execution.

Definitions and Theorems

Function: nats=>chars-exec

(defun nats=>chars-exec (nats rev-chars)
  (declare (xargs :guard (and (unsigned-byte-listp 8 nats)
                              (character-listp rev-chars))))
  (let ((__function__ 'nats=>chars-exec))
    (declare (ignorable __function__))
    (cond ((endp nats) (rev rev-chars))
          (t (nats=>chars-exec (cdr nats)
                               (cons (code-char (car nats))
                                     rev-chars))))))

Function: nats=>chars

(defun nats=>chars (nats)
  (declare (xargs :guard (unsigned-byte-listp 8 nats)))
  (let ((__function__ 'nats=>chars))
    (declare (ignorable __function__))
    (mbe :logic (cond ((endp nats) nil)
                      (t (cons (code-char (car nats))
                               (nats=>chars (cdr nats)))))
         :exec (nats=>chars-exec nats nil))))

Theorem: character-listp-of-nats=>chars

(defthm character-listp-of-nats=>chars
  (b* ((chars (nats=>chars nats)))
    (character-listp chars))
  :rule-classes :rewrite)

Theorem: len-of-nats=>chars

(defthm len-of-nats=>chars
  (equal (len (nats=>chars nats))
         (len nats)))

Theorem: nats=>chars-of-cons

(defthm nats=>chars-of-cons
  (equal (nats=>chars (cons nat nats))
         (cons (code-char nat)
               (nats=>chars nats))))

Theorem: nats=>chars-of-append

(defthm nats=>chars-of-append
  (equal (nats=>chars (append nats1 nats2))
         (append (nats=>chars nats1)
                 (nats=>chars nats2))))

Theorem: nats=>chars-of-repeat

(defthm nats=>chars-of-repeat
  (equal (nats=>chars (repeat n char))
         (repeat n (code-char char))))

Theorem: nth-of-nats=>chars

(defthm nth-of-nats=>chars
  (equal (nth i (nats=>chars nats))
         (if (< (nfix i) (len nats))
             (code-char (nth i nats))
           nil)))

Theorem: nats=>chars-of-revappend

(defthm nats=>chars-of-revappend
  (equal (nats=>chars (revappend x y))
         (revappend (nats=>chars x)
                    (nats=>chars y))))

Theorem: nats=>chars-of-nthcdr

(defthm nats=>chars-of-nthcdr
  (equal (nats=>chars (nthcdr n nats))
         (nthcdr n (nats=>chars nats))))

Theorem: nats=>chars-of-take

(defthm nats=>chars-of-take
  (implies (<= (nfix n) (len nats))
           (equal (nats=>chars (take n nats))
                  (take n (nats=>chars nats)))))

Subtopics

Nats<=>chars-inverses-theorems: nats=>chars and chars=>nats are mutual inverses.