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• Vl-defines-p

Vl-defines-p-basics

Basic theorems about vl-defines-p, generated by deflist.

Definitions and Theorems

Theorem: vl-defines-p-of-cons

(defthm vl-defines-p-of-cons
  (equal (vl-defines-p (cons acl2::a acl2::x))
         (and (vl-define-p acl2::a)
              (vl-defines-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-cdr-when-vl-defines-p

(defthm vl-defines-p-of-cdr-when-vl-defines-p
  (implies (vl-defines-p (double-rewrite acl2::x))
           (vl-defines-p (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-when-not-consp

(defthm vl-defines-p-when-not-consp
  (implies (not (consp acl2::x))
           (vl-defines-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: vl-define-p-of-car-when-vl-defines-p

(defthm vl-define-p-of-car-when-vl-defines-p
  (implies (vl-defines-p acl2::x)
           (iff (vl-define-p (car acl2::x))
                (or (consp acl2::x) (vl-define-p nil))))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-append

(defthm vl-defines-p-of-append
  (equal (vl-defines-p (append acl2::a acl2::b))
         (and (vl-defines-p acl2::a)
              (vl-defines-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-list-fix

(defthm vl-defines-p-of-list-fix
  (equal (vl-defines-p (list-fix acl2::x))
         (vl-defines-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-sfix

(defthm vl-defines-p-of-sfix
  (iff (vl-defines-p (sfix acl2::x))
       (or (vl-defines-p acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-insert

(defthm vl-defines-p-of-insert
  (iff (vl-defines-p (insert acl2::a acl2::x))
       (and (vl-defines-p (sfix acl2::x))
            (vl-define-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-delete

(defthm vl-defines-p-of-delete
  (implies (vl-defines-p acl2::x)
           (vl-defines-p (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-mergesort

(defthm vl-defines-p-of-mergesort
  (iff (vl-defines-p (mergesort acl2::x))
       (vl-defines-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-union

(defthm vl-defines-p-of-union
  (iff (vl-defines-p (union acl2::x acl2::y))
       (and (vl-defines-p (sfix acl2::x))
            (vl-defines-p (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-intersect-1

(defthm vl-defines-p-of-intersect-1
  (implies (vl-defines-p acl2::x)
           (vl-defines-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-intersect-2

(defthm vl-defines-p-of-intersect-2
  (implies (vl-defines-p acl2::y)
           (vl-defines-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-difference

(defthm vl-defines-p-of-difference
  (implies (vl-defines-p acl2::x)
           (vl-defines-p (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-duplicated-members

(defthm vl-defines-p-of-duplicated-members
  (implies (vl-defines-p acl2::x)
           (vl-defines-p (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-rev

(defthm vl-defines-p-of-rev
  (equal (vl-defines-p (rev acl2::x))
         (vl-defines-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-rcons

(defthm vl-defines-p-of-rcons
  (iff (vl-defines-p (acl2::rcons acl2::a acl2::x))
       (and (vl-define-p acl2::a)
            (vl-defines-p (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-define-p-when-member-equal-of-vl-defines-p

(defthm vl-define-p-when-member-equal-of-vl-defines-p
  (and (implies (and (member-equal acl2::a acl2::x)
                     (vl-defines-p acl2::x))
                (vl-define-p acl2::a))
       (implies (and (vl-defines-p acl2::x)
                     (member-equal acl2::a acl2::x))
                (vl-define-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-when-subsetp-equal

(defthm vl-defines-p-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (vl-defines-p acl2::y))
                (vl-defines-p acl2::x))
       (implies (and (vl-defines-p acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (vl-defines-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-set-equiv-congruence

(defthm vl-defines-p-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (vl-defines-p acl2::x)
                  (vl-defines-p acl2::y)))
  :rule-classes :congruence)

Theorem: vl-defines-p-of-set-difference-equal

(defthm vl-defines-p-of-set-difference-equal
  (implies (vl-defines-p acl2::x)
           (vl-defines-p (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-intersection-equal-1

(defthm vl-defines-p-of-intersection-equal-1
  (implies (vl-defines-p (double-rewrite acl2::x))
           (vl-defines-p (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-intersection-equal-2

(defthm vl-defines-p-of-intersection-equal-2
  (implies (vl-defines-p (double-rewrite acl2::y))
           (vl-defines-p (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-union-equal

(defthm vl-defines-p-of-union-equal
  (equal (vl-defines-p (union-equal acl2::x acl2::y))
         (and (vl-defines-p (list-fix acl2::x))
              (vl-defines-p (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-take

(defthm vl-defines-p-of-take
  (implies (vl-defines-p (double-rewrite acl2::x))
           (iff (vl-defines-p (take acl2::n acl2::x))
                (or (vl-define-p nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-repeat

(defthm vl-defines-p-of-repeat
  (iff (vl-defines-p (repeat acl2::n acl2::x))
       (or (vl-define-p acl2::x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: vl-define-p-of-nth-when-vl-defines-p

(defthm vl-define-p-of-nth-when-vl-defines-p
  (implies (and (vl-defines-p acl2::x)
                (< (nfix acl2::n) (len acl2::x)))
           (vl-define-p (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-update-nth

(defthm vl-defines-p-of-update-nth
  (implies (vl-defines-p (double-rewrite acl2::x))
           (iff (vl-defines-p (update-nth acl2::n acl2::y acl2::x))
                (and (vl-define-p acl2::y)
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (vl-define-p nil)))))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-butlast

(defthm vl-defines-p-of-butlast
  (implies (vl-defines-p (double-rewrite acl2::x))
           (vl-defines-p (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-nthcdr

(defthm vl-defines-p-of-nthcdr
  (implies (vl-defines-p (double-rewrite acl2::x))
           (vl-defines-p (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-last

(defthm vl-defines-p-of-last
  (implies (vl-defines-p (double-rewrite acl2::x))
           (vl-defines-p (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-remove

(defthm vl-defines-p-of-remove
  (implies (vl-defines-p acl2::x)
           (vl-defines-p (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-defines-p-of-revappend

(defthm vl-defines-p-of-revappend
  (equal (vl-defines-p (revappend acl2::x acl2::y))
         (and (vl-defines-p (list-fix acl2::x))
              (vl-defines-p acl2::y)))
  :rule-classes ((:rewrite)))