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Ordering

Ichar<

Case-insensitive character less-than test.

(ichar< x y) determines if x is "smaller" than y, but ignoring case. Our approach is basically to downcase upper-case letters and then compare the resulting character codes.

Something subtle about this is that, in the ASCII character ordering, the upper-case characters do not immediately preceede the lower-case ones. That is, upper-case Z is at 90, but lower-case a does not start until 97. So, in our approach, the 6 intervening characters (the square brackets, backslash, hat, underscore, and backtick) are considered "smaller" than letters, even though in regular ASCII ordering they are "larger" than the upper-case letters.

Definitions and Theorems

Function: ichar<$inline

(defun ichar<$inline (x y)
 (declare (type character x)
          (type character y))
 (mbe
    :logic
    (< (char-code (downcase-char x))
       (char-code (downcase-char y)))
    :exec
    (let* ((xc (the (unsigned-byte 8)
                    (char-code (the character x))))
           (yc (the (unsigned-byte 8)
                    (char-code (the character y))))
           (xc-fix (if (and (<= (big-a) (the (unsigned-byte 8) xc))
                            (<= (the (unsigned-byte 8) xc) (big-z)))
                       (the (unsigned-byte 8)
                            (+ (the (unsigned-byte 8) xc) 32))
                     (the (unsigned-byte 8) xc)))
           (yc-fix (if (and (<= (big-a) (the (unsigned-byte 8) yc))
                            (<= (the (unsigned-byte 8) yc) (big-z)))
                       (the (unsigned-byte 8)
                            (+ (the (unsigned-byte 8) yc) 32))
                     (the (unsigned-byte 8) yc))))
      (< (the (unsigned-byte 8) xc-fix)
         (the (unsigned-byte 8) yc-fix)))))

Theorem: ichar<-irreflexive

(defthm ichar<-irreflexive
  (not (ichar< x x)))

Theorem: ichar<-antisymmetric

(defthm ichar<-antisymmetric
  (implies (ichar< x y)
           (not (ichar< y x))))

Theorem: ichar<-transitive

(defthm ichar<-transitive
  (implies (and (ichar< x y) (ichar< y z))
           (ichar< x z)))

Theorem: ichar<-transitive-two

(defthm ichar<-transitive-two
  (implies (and (ichar< y z) (ichar< x y))
           (ichar< x z)))

Theorem: ichar<-trichotomy-weak

(defthm ichar<-trichotomy-weak
  (implies (and (not (ichar< x y))
                (not (ichar< y x)))
           (equal (ichareqv x y) t)))

Theorem: ichareqv-implies-equal-ichar<-1

(defthm ichareqv-implies-equal-ichar<-1
  (implies (ichareqv x x-equiv)
           (equal (ichar< x y) (ichar< x-equiv y)))
  :rule-classes (:congruence))

Theorem: ichareqv-implies-equal-ichar<-2

(defthm ichareqv-implies-equal-ichar<-2
  (implies (ichareqv y y-equiv)
           (equal (ichar< x y) (ichar< x y-equiv)))
  :rule-classes (:congruence))

Theorem: ichar<-trichotomy-strong

(defthm ichar<-trichotomy-strong
  (equal (ichar< x y)
         (and (not (ichareqv x y))
              (not (ichar< y x))))
  :rule-classes ((:rewrite :loop-stopper ((x y)))))