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• Nat-to-hex-chars

Basic-nat-to-hex-chars

Logically simple definition that is similar to nat-to-hex-chars.

Signature

(basic-nat-to-hex-chars n) → chars

Arguments

n — Guard (natp n).

Returns

chars — Type (hex-digit-char-list*p chars).

This almost computes (nat-to-hex-chars n), but when n is zero it returns nil instead of (#\0). You would normally never call this function directly, but it is convenient for reasoning about nat-to-hex-chars.

Definitions and Theorems

Function: basic-nat-to-hex-chars

(defun basic-nat-to-hex-chars (n)
  (declare (xargs :guard (natp n)))
  (let ((acl2::__function__ 'basic-nat-to-hex-chars))
    (declare (ignorable acl2::__function__))
    (if (zp n)
        nil
      (cons (hex-digit-to-char (logand n 15))
            (basic-nat-to-hex-chars (ash n -4))))))

Theorem: hex-digit-char-list*p-of-basic-nat-to-hex-chars

(defthm hex-digit-char-list*p-of-basic-nat-to-hex-chars
  (b* ((chars (basic-nat-to-hex-chars n)))
    (hex-digit-char-list*p chars))
  :rule-classes :rewrite)

Theorem: basic-nat-to-hex-chars-when-zp

(defthm basic-nat-to-hex-chars-when-zp
  (implies (zp n)
           (equal (basic-nat-to-hex-chars n) nil)))

Theorem: true-listp-of-basic-nat-to-hex-chars

(defthm true-listp-of-basic-nat-to-hex-chars
  (true-listp (basic-nat-to-hex-chars n))
  :rule-classes :type-prescription)

Theorem: character-listp-of-basic-nat-to-hex-chars

(defthm character-listp-of-basic-nat-to-hex-chars
  (character-listp (basic-nat-to-hex-chars n)))

Theorem: basic-nat-to-hex-chars-under-iff

(defthm basic-nat-to-hex-chars-under-iff
  (iff (basic-nat-to-hex-chars n)
       (not (zp n))))

Theorem: consp-of-basic-nat-to-hex-chars

(defthm consp-of-basic-nat-to-hex-chars
  (equal (consp (basic-nat-to-hex-chars n))
         (if (basic-nat-to-hex-chars n) t nil)))

Theorem: basic-nat-to-hex-chars-one-to-one

(defthm basic-nat-to-hex-chars-one-to-one
  (equal (equal (basic-nat-to-hex-chars n)
                (basic-nat-to-hex-chars m))
         (equal (nfix n) (nfix m))))