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• Lhs-p

Lhs-p-basics

Basic theorems about lhs-p, generated by std::deflist.

Definitions and Theorems

Theorem: lhs-p-of-cons

(defthm lhs-p-of-cons
  (equal (lhs-p (cons acl2::a x))
         (and (lhrange-p acl2::a) (lhs-p x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-cdr-when-lhs-p

(defthm lhs-p-of-cdr-when-lhs-p
  (implies (lhs-p (double-rewrite x))
           (lhs-p (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-when-not-consp

(defthm lhs-p-when-not-consp
  (implies (not (consp x))
           (equal (lhs-p x) (not x)))
  :rule-classes ((:rewrite)))

Theorem: lhrange-p-of-car-when-lhs-p

(defthm lhrange-p-of-car-when-lhs-p
  (implies (lhs-p x)
           (iff (lhrange-p (car x))
                (or (consp x) (lhrange-p nil))))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-lhs-p-compound-recognizer

(defthm true-listp-when-lhs-p-compound-recognizer
  (implies (lhs-p x) (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: lhs-p-of-list-fix

(defthm lhs-p-of-list-fix
  (implies (lhs-p x) (lhs-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-rev

(defthm lhs-p-of-rev
  (equal (lhs-p (rev x))
         (lhs-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-repeat

(defthm lhs-p-of-repeat
  (iff (lhs-p (repeat acl2::n x))
       (or (lhrange-p x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-append

(defthm lhs-p-of-append
  (equal (lhs-p (append acl2::a acl2::b))
         (and (lhs-p (list-fix acl2::a))
              (lhs-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-rcons

(defthm lhs-p-of-rcons
  (iff (lhs-p (acl2::rcons acl2::a x))
       (and (lhrange-p acl2::a)
            (lhs-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: lhrange-p-when-member-equal-of-lhs-p

(defthm lhrange-p-when-member-equal-of-lhs-p
  (and (implies (and (member-equal acl2::a x) (lhs-p x))
                (lhrange-p acl2::a))
       (implies (and (lhs-p x) (member-equal acl2::a x))
                (lhrange-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-when-subsetp-equal

(defthm lhs-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y) (lhs-p y))
                (equal (lhs-p x) (true-listp x)))
       (implies (and (lhs-p y) (subsetp-equal x y))
                (equal (lhs-p x) (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-set-difference-equal

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

Theorem: lhs-p-of-intersection-equal-1

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

Theorem: lhs-p-of-intersection-equal-2

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

Theorem: lhs-p-of-union-equal

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

Theorem: lhs-p-of-update-nth

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

Theorem: lhs-p-of-butlast

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