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Lhspairs

Lhspairs-p

Recognizer for lhspairs.

Signature
(lhspairs-p x) → *

Definitions and Theorems

Function: lhspairs-p

(defun lhspairs-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'lhspairs-p))
    (declare (ignorable __function__))
    (if (atom x)
        t
      (and (consp (car x))
           (lhs-p (caar x))
           (lhs-p (cdar x))
           (lhspairs-p (cdr x))))))

Theorem: lhspairs-p-of-butlast

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

Theorem: lhspairs-p-of-update-nth

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

Theorem: lhspairs-p-of-union-equal

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

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

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

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

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

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

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

Theorem: lhspairs-p-set-equiv-congruence

(defthm lhspairs-p-set-equiv-congruence
  (implies (set-equiv x y)
           (equal (lhspairs-p x) (lhspairs-p y)))
  :rule-classes :congruence)

Theorem: lhspairs-p-when-subsetp-equal

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

Theorem: lhspairs-p-of-rcons

(defthm lhspairs-p-of-rcons
  (iff (lhspairs-p (acl2::rcons acl2::a x))
       (and (and (consp acl2::a)
                 (lhs-p (car acl2::a))
                 (lhs-p (cdr acl2::a)))
            (lhspairs-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: lhspairs-p-of-repeat

(defthm lhspairs-p-of-repeat
  (iff (lhspairs-p (repeat acl2::n x))
       (or (and (consp x)
                (lhs-p (car x))
                (lhs-p (cdr x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: lhspairs-p-of-rev

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

Theorem: lhspairs-p-of-list-fix

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

Theorem: lhspairs-p-of-append

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

Theorem: lhspairs-p-when-not-consp

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

Theorem: lhspairs-p-of-cdr-when-lhspairs-p

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

Theorem: lhspairs-p-of-cons

(defthm lhspairs-p-of-cons
  (equal (lhspairs-p (cons acl2::a x))
         (and (and (consp acl2::a)
                   (lhs-p (car acl2::a))
                   (lhs-p (cdr acl2::a)))
              (lhspairs-p x)))
  :rule-classes ((:rewrite)))

Theorem: lhspairs-p-of-fast-alist-clean

(defthm lhspairs-p-of-fast-alist-clean
  (implies (lhspairs-p x)
           (lhspairs-p (fast-alist-clean x)))
  :rule-classes ((:rewrite)))

Theorem: lhspairs-p-of-hons-shrink-alist

(defthm lhspairs-p-of-hons-shrink-alist
  (implies (and (lhspairs-p x) (lhspairs-p y))
           (lhspairs-p (hons-shrink-alist x y)))
  :rule-classes ((:rewrite)))

Theorem: lhspairs-p-of-hons-acons

(defthm lhspairs-p-of-hons-acons
  (equal (lhspairs-p (hons-acons acl2::a acl2::n x))
         (and (lhs-p acl2::a)
              (lhs-p acl2::n)
              (lhspairs-p x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-cdr-of-hons-assoc-equal-when-lhspairs-p

(defthm lhs-p-of-cdr-of-hons-assoc-equal-when-lhspairs-p
  (implies (lhspairs-p x)
           (iff (lhs-p (cdr (hons-assoc-equal acl2::k x)))
                (or (hons-assoc-equal acl2::k x)
                    (lhs-p nil))))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-cdar-when-lhspairs-p

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

Theorem: lhs-p-of-caar-when-lhspairs-p

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