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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))