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Downcase-char

Convert a character to lower-case.

Signature
(downcase-char x) → char
Returns: char — Type (characterp char).

(downcase-char x) converts upper-case characters into their lower-case equivalents, and returns other characters unchanged.

ACL2 has a built-in alternative to this function, common-lisp::char-downcase, which however can also convert characters that are not standard characters. (Historical note: Before May 2024, the guard for acl2::char-downcase required standard-char-p yet also converted only standard characters; this provided motivation for the development of downcase-char.)

Definitions and Theorems

Function: downcase-char$inline

(defun downcase-char$inline (x)
  (declare (type character x))
  (let ((acl2::__function__ 'downcase-char))
    (declare (ignorable acl2::__function__))
    (b* (((the (unsigned-byte 8) code)
          (char-code x)))
      (if (and (<= (big-a) code)
               (<= code (big-z)))
          (code-char (the (unsigned-byte 8)
                          (+ code (case-delta))))
        (mbe :logic (char-fix x) :exec x)))))

Theorem: characterp-of-downcase-char

(defthm characterp-of-downcase-char
  (b* ((char (downcase-char$inline x)))
    (characterp char))
  :rule-classes :type-prescription)

Theorem: chareqv-implies-equal-downcase-char-1

(defthm chareqv-implies-equal-downcase-char-1
  (implies (chareqv x x-equiv)
           (equal (downcase-char x)
                  (downcase-char x-equiv)))
  :rule-classes (:congruence))

Theorem: downcase-char-does-nothing-unless-up-alpha-p

(defthm downcase-char-does-nothing-unless-up-alpha-p
  (implies (not (up-alpha-p x))
           (equal (downcase-char x) (char-fix x))))

Theorem: downcase-char-of-downcase-char

(defthm downcase-char-of-downcase-char
  (equal (downcase-char (downcase-char x))
         (downcase-char x)))

Theorem: downcase-char-of-upcase-char

(defthm downcase-char-of-upcase-char
  (equal (downcase-char (upcase-char x))
         (downcase-char x)))

Theorem: upcase-char-of-downcase-char

(defthm upcase-char-of-downcase-char
  (equal (upcase-char (downcase-char x))
         (upcase-char x)))

Theorem: char-downcase-is-downcase-char

(defthm char-downcase-is-downcase-char
  (implies (standard-char-p x)
           (equal (common-lisp::char-downcase x)
                  (downcase-char (double-rewrite x)))))