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Istreqv

Case-insensitive string equivalence test.

Signature
(istreqv x y) → bool

(istreqv x y) determines if x and y are case-insensitively equivalent strings. That is, x and y must have the same length and their elements must be ichareqv to one another.

Logically this is identical to

(icharlisteqv (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 icharlisteqv of the coerces as our normal form.

Definitions and Theorems

Function: istreqv-aux

(defun istreqv-aux (x y n l)
  (declare (type string x)
           (type string y)
           (type (integer 0 *) n)
           (type (integer 0 *) l)
           (xargs :guard (and (natp n)
                              (natp l)
                              (equal (length x) l)
                              (equal (length y) l)
                              (<= n l))))
  (mbe :logic
       (if (zp (- (nfix l) (nfix n)))
           t
         (and (ichareqv (char x n) (char y n))
              (istreqv-aux x y (+ (nfix n) 1) l)))
       :exec
       (if (eql n l)
           t
         (and (ichareqv (the character (char x n))
                        (the character (char y n)))
              (istreqv-aux x y (the (integer 0 *) (+ 1 n))
                           l)))))

Theorem: istreqv-aux-removal

(defthm istreqv-aux-removal
  (implies (and (stringp x)
                (stringp y)
                (natp n)
                (<= n l)
                (= l (length x))
                (= l (length y)))
           (equal (istreqv-aux x y n l)
                  (icharlisteqv (nthcdr n (explode x))
                                (nthcdr n (explode y))))))

Function: istreqv$inline

(defun istreqv$inline (x y)
  (declare (type string x)
           (type string y))
  (let ((acl2::__function__ 'istreqv))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (icharlisteqv (explode x) (explode y))
         :exec
         (b* (((the (integer 0 *) xl) (length x))
              ((the (integer 0 *) yl) (length y)))
           (and (eql xl yl)
                (istreqv-aux x y 0 xl))))))

Theorem: istreqv-is-an-equivalence

(defthm istreqv-is-an-equivalence
  (and (booleanp (istreqv x y))
       (istreqv x x)
       (implies (istreqv x y) (istreqv y x))
       (implies (and (istreqv x y) (istreqv y z))
                (istreqv x z)))
  :rule-classes (:equivalence))

Theorem: streqv-refines-istreqv

(defthm streqv-refines-istreqv
  (implies (streqv x y) (istreqv x y))
  :rule-classes (:refinement))

Theorem: istreqv-implies-ichareqv-char-1

(defthm istreqv-implies-ichareqv-char-1
  (implies (istreqv x x-equiv)
           (ichareqv (char x n) (char x-equiv n)))
  :rule-classes (:congruence))

Theorem: istreqv-implies-icharlisteqv-explode-1

(defthm istreqv-implies-icharlisteqv-explode-1
  (implies (istreqv x x-equiv)
           (icharlisteqv (explode x)
                         (explode x-equiv)))
  :rule-classes (:congruence))

Theorem: istreqv-implies-istreqv-string-append-1

(defthm istreqv-implies-istreqv-string-append-1
  (implies (istreqv x x-equiv)
           (istreqv (string-append x y)
                    (string-append x-equiv y)))
  :rule-classes (:congruence))

Theorem: istreqv-implies-istreqv-string-append-2

(defthm istreqv-implies-istreqv-string-append-2
  (implies (istreqv y y-equiv)
           (istreqv (string-append x y)
                    (string-append x y-equiv)))
  :rule-classes (:congruence))