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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)))))