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• Equivalences

Charlisteqv

Case-sensitive character-list equivalence test.

Signature

(charlisteqv x y) → equivp

Arguments

x — Guard (character-listp x).

y — Guard (character-listp y).

(charlisteqv x y) determines if x and y are equivalent when interpreted as character lists. That is, x and y must have the same length and their elements must be chareqv to one another.

See also icharlisteqv for a case-insensitive alternative.

Definitions and Theorems

Function: charlisteqv$inline

(defun charlisteqv$inline (x y)
       (declare (xargs :guard (and (character-listp x)
                                   (character-listp y))))
       (let ((acl2::__function__ 'charlisteqv))
            (declare (ignorable acl2::__function__))
            (mbe :logic (equal (make-character-list x)
                               (make-character-list y))
                 :exec (equal x y))))

Theorem: charlisteqv-is-an-equivalence

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

Theorem: list-equiv-refines-charlisteqv

(defthm list-equiv-refines-charlisteqv
        (implies (list-equiv x y)
                 (charlisteqv x y))
        :rule-classes (:refinement))

Theorem: charlisteqv-implies-chareqv-car-1

(defthm charlisteqv-implies-chareqv-car-1
        (implies (charlisteqv x x-equiv)
                 (chareqv (car x) (car x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-cdr-1

(defthm charlisteqv-implies-charlisteqv-cdr-1
        (implies (charlisteqv x x-equiv)
                 (charlisteqv (cdr x) (cdr x-equiv)))
        :rule-classes (:congruence))

Theorem: chareqv-implies-charlisteqv-cons-1

(defthm chareqv-implies-charlisteqv-cons-1
        (implies (chareqv a a-equiv)
                 (charlisteqv (cons a x)
                              (cons a-equiv x)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-cons-2

(defthm charlisteqv-implies-charlisteqv-cons-2
        (implies (charlisteqv x x-equiv)
                 (charlisteqv (cons a x)
                              (cons a x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-equal-len-1

(defthm charlisteqv-implies-equal-len-1
        (implies (charlisteqv x x-equiv)
                 (equal (len x) (len x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-list-fix-1

(defthm charlisteqv-implies-charlisteqv-list-fix-1
        (implies (charlisteqv x x-equiv)
                 (charlisteqv (list-fix x)
                              (list-fix x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-chareqv-nth-2

(defthm charlisteqv-implies-chareqv-nth-2
        (implies (charlisteqv x x-equiv)
                 (chareqv (nth n x) (nth n x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-take-2

(defthm charlisteqv-implies-charlisteqv-take-2
        (implies (charlisteqv x x-equiv)
                 (charlisteqv (take n x)
                              (take n x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-nthcdr-2

(defthm charlisteqv-implies-charlisteqv-nthcdr-2
        (implies (charlisteqv x x-equiv)
                 (charlisteqv (nthcdr n x)
                              (nthcdr n x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-append-1

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

Theorem: charlisteqv-implies-charlisteqv-append-2

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

Theorem: charlisteqv-implies-charlisteqv-rev-1

(defthm charlisteqv-implies-charlisteqv-rev-1
        (implies (charlisteqv x x-equiv)
                 (charlisteqv (rev x) (rev x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-revappend-2

(defthm charlisteqv-implies-charlisteqv-revappend-2
        (implies (charlisteqv y y-equiv)
                 (charlisteqv (revappend x y)
                              (revappend x y-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-charlisteqv-revappend-1

(defthm charlisteqv-implies-charlisteqv-revappend-1
        (implies (charlisteqv x x-equiv)
                 (charlisteqv (revappend x y)
                              (revappend x-equiv y)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-equal-make-character-list-1

(defthm charlisteqv-implies-equal-make-character-list-1
        (implies (charlisteqv x x-equiv)
                 (equal (make-character-list x)
                        (make-character-list x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-implies-equal-implode-1

(defthm charlisteqv-implies-equal-implode-1
        (implies (charlisteqv x x-equiv)
                 (equal (implode x) (implode x-equiv)))
        :rule-classes (:congruence))

Theorem: charlisteqv-when-not-consp-left

(defthm charlisteqv-when-not-consp-left
        (implies (not (consp x))
                 (equal (charlisteqv x y) (atom y))))

Theorem: charlisteqv-when-not-consp-right

(defthm charlisteqv-when-not-consp-right
        (implies (not (consp y))
                 (equal (charlisteqv x y) (atom x))))

Theorem: charlisteqv-of-cons-right

(defthm charlisteqv-of-cons-right
        (equal (charlisteqv x (cons a y))
               (and (consp x)
                    (chareqv (car x) (double-rewrite a))
                    (charlisteqv (cdr x)
                                 (double-rewrite y)))))

Theorem: charlisteqv-of-cons-left

(defthm charlisteqv-of-cons-left
        (equal (charlisteqv (cons a x) y)
               (and (consp y)
                    (chareqv (double-rewrite a) (car y))
                    (charlisteqv (double-rewrite x)
                                 (cdr y)))))

Theorem: charlisteqv-when-not-same-lens

(defthm charlisteqv-when-not-same-lens
        (implies (not (equal (len x) (len y)))
                 (not (charlisteqv x y))))

Theorem: make-character-list-is-identity-under-charlisteqv

(defthm make-character-list-is-identity-under-charlisteqv
        (charlisteqv (make-character-list x) x))

Theorem: charlisteqv*

(defthm
     charlisteqv*
     (equal (charlisteqv x y)
            (if (consp x)
                (and (consp y)
                     (chareqv (car x) (car y))
                     (charlisteqv (cdr x) (cdr y)))
                (atom y)))
     :rule-classes
     ((:definition :controller-alist ((charlisteqv$inline t nil)))))