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• Cases

Downcase-first-charlist

Convert the first character of a character list to lower case.

Signature

(downcase-first-charlist x) → downcased

Arguments

x — Guard (character-listp x).

Returns

downcased — Type (character-listp downcased).

Definitions and Theorems

Function: downcase-first-charlist

(defun
     downcase-first-charlist (x)
     (declare (xargs :guard (character-listp x)))
     (let ((acl2::__function__ 'downcase-first-charlist))
          (declare (ignorable acl2::__function__))
          (mbe :logic (if (atom x)
                          nil
                          (cons (downcase-char (car x))
                                (make-character-list (cdr x))))
               :exec (cond ((atom x) nil)
                           ((up-alpha-p (car x))
                            (cons (downcase-char (car x)) (cdr x)))
                           (t x)))))

Theorem: character-listp-of-downcase-first-charlist

(defthm character-listp-of-downcase-first-charlist
        (b* ((downcased (downcase-first-charlist x)))
            (character-listp downcased))
        :rule-classes :rewrite)

Theorem: charlisteqv-implies-equal-downcase-first-charlist-1

(defthm charlisteqv-implies-equal-downcase-first-charlist-1
        (implies (charlisteqv x x-equiv)
                 (equal (downcase-first-charlist x)
                        (downcase-first-charlist x-equiv)))
        :rule-classes (:congruence))

Theorem: icharlisteqv-implies-icharlisteqv-downcase-first-charlist-1

(defthm icharlisteqv-implies-icharlisteqv-downcase-first-charlist-1
        (implies (icharlisteqv x x-equiv)
                 (icharlisteqv (downcase-first-charlist x)
                               (downcase-first-charlist x-equiv)))
        :rule-classes (:congruence))

Theorem: downcase-first-charlist-when-atom

(defthm downcase-first-charlist-when-atom
        (implies (atom x)
                 (equal (downcase-first-charlist x)
                        nil)))

Theorem: len-of-downcase-first-charlist

(defthm len-of-downcase-first-charlist
        (equal (len (downcase-first-charlist x))
               (len x)))

Theorem: consp-of-downcase-first-charlist

(defthm consp-of-downcase-first-charlist
        (equal (consp (downcase-first-charlist x))
               (consp x)))

Theorem: downcase-first-charlist-under-iff

(defthm downcase-first-charlist-under-iff
        (iff (downcase-first-charlist x)
             (consp x)))