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Istr<

Case-insensitive string less-than test.

(icharlist< x y) determines whether the string x preceeds y in alphabetical order without regards to case. The characters are compared with ichar< and shorter strings are considered smaller than longer strings.

Logically, this is identical to:

(icharlist< (explode x) (explode y))

But we use a more efficient implementation which avoids coercing the strings into lists.

NOTE: for reasoning, we leave this function enabled and prefer to work with icharlist< of the explodes as our normal form.

Definitions and Theorems

Function: istr<-aux

(defun istr<-aux (x y n xl yl)
 (declare (type string x)
          (type string y)
          (type integer n)
          (type integer xl)
          (type integer yl)
          (xargs :guard (and (stringp x)
                             (stringp y)
                             (natp n)
                             (<= n (length x))
                             (<= n (length y))
                             (equal xl (length x))
                             (equal yl (length y)))))
 (mbe
  :logic (cond ((zp (- (nfix yl) (nfix n))) nil)
               ((zp (- (nfix xl) (nfix n))) t)
               ((ichar< (char x n) (char y n)) t)
               ((ichar< (char y n) (char x n)) nil)
               (t (istr<-aux x y (+ (nfix n) 1) xl yl)))
  :exec
  (cond
   ((= (the integer n) (the integer yl))
    nil)
   ((= (the integer n) (the integer xl)) t)
   (t
    (let* ((xc (the (unsigned-byte 8)
                    (char-code (the character
                                    (char (the string x)
                                          (the integer n))))))
           (yc (the (unsigned-byte 8)
                    (char-code (the character
                                    (char (the string y)
                                          (the integer n))))))
           (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))))
      (cond ((< (the (unsigned-byte 8) xc-fix)
                (the (unsigned-byte 8) yc-fix))
             t)
            ((< (the (unsigned-byte 8) yc-fix)
                (the (unsigned-byte 8) xc-fix))
             nil)
            (t (istr<-aux (the string x)
                          (the string y)
                          (the integer (+ (the integer n) 1))
                          (the integer xl)
                          (the integer yl)))))))))

Function: istr<$inline

(defun istr<$inline (x y)
  (declare (type string x)
           (type string y))
  (mbe :logic (icharlist< (explode x) (explode y))
       :exec (istr<-aux (the string x)
                        (the string y)
                        (the integer 0)
                        (the integer (length (the string x)))
                        (the integer (length (the string y))))))

Theorem: istr<-aux-correct

(defthm istr<-aux-correct
  (implies (and (stringp x)
                (stringp y)
                (natp n)
                (<= n (length x))
                (<= n (length y))
                (equal xl (length x))
                (equal yl (length y)))
           (equal (istr<-aux x y n xl yl)
                  (icharlist< (nthcdr n (coerce x 'list))
                              (nthcdr n (coerce y 'list))))))

Theorem: istreqv-implies-equal-istr<-1

(defthm istreqv-implies-equal-istr<-1
  (implies (istreqv x x-equiv)
           (equal (istr< x y) (istr< x-equiv y)))
  :rule-classes (:congruence))

Theorem: istreqv-implies-equal-istr<-2

(defthm istreqv-implies-equal-istr<-2
  (implies (istreqv y y-equiv)
           (equal (istr< x y) (istr< x y-equiv)))
  :rule-classes (:congruence))