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Pseudo-term-alist

Pseudo-term-alistp

Signature
(pseudo-term-alistp x) → *

Definitions and Theorems

Function: pseudo-term-alistp

(defun pseudo-term-alistp (x)
  (declare (xargs :guard t))
  (let ((acl2::__function__ 'pseudo-term-alistp))
    (declare (ignorable acl2::__function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (pseudo-termp (caar x))
           (pseudo-termp (cdar x))
           (pseudo-term-alistp (cdr x))))))

Theorem: pseudo-term-alistp-of-revappend

(defthm pseudo-term-alistp-of-revappend
  (equal (pseudo-term-alistp (revappend acl2::x acl2::y))
         (and (pseudo-term-alistp (acl2::list-fix acl2::x))
              (pseudo-term-alistp acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-remove

(defthm pseudo-term-alistp-of-remove
  (implies (pseudo-term-alistp acl2::x)
           (pseudo-term-alistp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-last

(defthm pseudo-term-alistp-of-last
  (implies (pseudo-term-alistp (double-rewrite acl2::x))
           (pseudo-term-alistp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-nthcdr

(defthm pseudo-term-alistp-of-nthcdr
  (implies (pseudo-term-alistp (double-rewrite acl2::x))
           (pseudo-term-alistp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-butlast

(defthm pseudo-term-alistp-of-butlast
  (implies (pseudo-term-alistp (double-rewrite acl2::x))
           (pseudo-term-alistp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-update-nth

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

Theorem: pseudo-term-alistp-of-repeat

(defthm pseudo-term-alistp-of-repeat
  (iff (pseudo-term-alistp (acl2::repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (pseudo-termp (car acl2::x))
                (pseudo-termp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-take

(defthm pseudo-term-alistp-of-take
  (implies (pseudo-term-alistp (double-rewrite acl2::x))
           (iff (pseudo-term-alistp (take acl2::n acl2::x))
                (or (and (consp nil)
                         (pseudo-termp (car nil))
                         (pseudo-termp (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-union-equal

(defthm pseudo-term-alistp-of-union-equal
  (equal (pseudo-term-alistp (union-equal acl2::x acl2::y))
         (and (pseudo-term-alistp (acl2::list-fix acl2::x))
              (pseudo-term-alistp (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-intersection-equal-2

(defthm pseudo-term-alistp-of-intersection-equal-2
 (implies (pseudo-term-alistp (double-rewrite acl2::y))
          (pseudo-term-alistp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-intersection-equal-1

(defthm pseudo-term-alistp-of-intersection-equal-1
 (implies (pseudo-term-alistp (double-rewrite acl2::x))
          (pseudo-term-alistp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-set-difference-equal

(defthm pseudo-term-alistp-of-set-difference-equal
  (implies
       (pseudo-term-alistp acl2::x)
       (pseudo-term-alistp (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-when-subsetp-equal

(defthm pseudo-term-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (pseudo-term-alistp acl2::y))
                (equal (pseudo-term-alistp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (pseudo-term-alistp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (pseudo-term-alistp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-rcons

(defthm pseudo-term-alistp-of-rcons
  (iff (pseudo-term-alistp (acl2::rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (pseudo-termp (car acl2::a))
                 (pseudo-termp (cdr acl2::a)))
            (pseudo-term-alistp (acl2::list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-append

(defthm pseudo-term-alistp-of-append
  (equal (pseudo-term-alistp (append acl2::a acl2::b))
         (and (pseudo-term-alistp (acl2::list-fix acl2::a))
              (pseudo-term-alistp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-rev

(defthm pseudo-term-alistp-of-rev
  (equal (pseudo-term-alistp (acl2::rev acl2::x))
         (pseudo-term-alistp (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-duplicated-members

(defthm pseudo-term-alistp-of-duplicated-members
  (implies (pseudo-term-alistp acl2::x)
           (pseudo-term-alistp (acl2::duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-difference

(defthm pseudo-term-alistp-of-difference
  (implies (pseudo-term-alistp acl2::x)
           (pseudo-term-alistp (set::difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-intersect-2

(defthm pseudo-term-alistp-of-intersect-2
  (implies (pseudo-term-alistp acl2::y)
           (pseudo-term-alistp (set::intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-intersect-1

(defthm pseudo-term-alistp-of-intersect-1
  (implies (pseudo-term-alistp acl2::x)
           (pseudo-term-alistp (set::intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-union

(defthm pseudo-term-alistp-of-union
  (iff (pseudo-term-alistp (set::union acl2::x acl2::y))
       (and (pseudo-term-alistp (set::sfix acl2::x))
            (pseudo-term-alistp (set::sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-mergesort

(defthm pseudo-term-alistp-of-mergesort
  (iff (pseudo-term-alistp (set::mergesort acl2::x))
       (pseudo-term-alistp (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-delete

(defthm pseudo-term-alistp-of-delete
  (implies (pseudo-term-alistp acl2::x)
           (pseudo-term-alistp (set::delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-insert

(defthm pseudo-term-alistp-of-insert
  (iff (pseudo-term-alistp (set::insert acl2::a acl2::x))
       (and (pseudo-term-alistp (set::sfix acl2::x))
            (and (consp acl2::a)
                 (pseudo-termp (car acl2::a))
                 (pseudo-termp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-sfix

(defthm pseudo-term-alistp-of-sfix
  (iff (pseudo-term-alistp (set::sfix acl2::x))
       (or (pseudo-term-alistp acl2::x)
           (not (set::setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-list-fix

(defthm pseudo-term-alistp-of-list-fix
  (implies (pseudo-term-alistp acl2::x)
           (pseudo-term-alistp (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-pseudo-term-alistp-compound-recognizer

(defthm true-listp-when-pseudo-term-alistp-compound-recognizer
  (implies (pseudo-term-alistp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: pseudo-term-alistp-when-not-consp

(defthm pseudo-term-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (pseudo-term-alistp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-cdr-when-pseudo-term-alistp

(defthm pseudo-term-alistp-of-cdr-when-pseudo-term-alistp
  (implies (pseudo-term-alistp (double-rewrite acl2::x))
           (pseudo-term-alistp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-cons

(defthm pseudo-term-alistp-of-cons
  (equal (pseudo-term-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (pseudo-termp (car acl2::a))
                   (pseudo-termp (cdr acl2::a)))
              (pseudo-term-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-remove-assoc

(defthm pseudo-term-alistp-of-remove-assoc
  (implies
       (pseudo-term-alistp acl2::x)
       (pseudo-term-alistp (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-put-assoc

(defthm pseudo-term-alistp-of-put-assoc
  (implies (and (pseudo-term-alistp acl2::x))
           (iff (pseudo-term-alistp
                     (put-assoc-equal acl2::name acl2::val acl2::x))
                (and (pseudo-termp acl2::name)
                     (pseudo-termp acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-fast-alist-clean

(defthm pseudo-term-alistp-of-fast-alist-clean
  (implies (pseudo-term-alistp acl2::x)
           (pseudo-term-alistp (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-hons-shrink-alist

(defthm pseudo-term-alistp-of-hons-shrink-alist
  (implies (and (pseudo-term-alistp acl2::x)
                (pseudo-term-alistp acl2::y))
           (pseudo-term-alistp (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-term-alistp-of-hons-acons

(defthm pseudo-term-alistp-of-hons-acons
  (equal (pseudo-term-alistp (hons-acons acl2::a acl2::n acl2::x))
         (and (pseudo-termp acl2::a)
              (pseudo-termp acl2::n)
              (pseudo-term-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-termp-of-cdr-of-hons-assoc-equal-when-pseudo-term-alistp

(defthm
    pseudo-termp-of-cdr-of-hons-assoc-equal-when-pseudo-term-alistp
  (implies
       (pseudo-term-alistp acl2::x)
       (iff (pseudo-termp (cdr (hons-assoc-equal acl2::k acl2::x)))
            (or (hons-assoc-equal acl2::k acl2::x)
                (pseudo-termp nil))))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-pseudo-term-alistp-rewrite

(defthm alistp-when-pseudo-term-alistp-rewrite
  (implies (pseudo-term-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-pseudo-term-alistp

(defthm alistp-when-pseudo-term-alistp
  (implies (pseudo-term-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: pseudo-termp-of-cdar-when-pseudo-term-alistp

(defthm pseudo-termp-of-cdar-when-pseudo-term-alistp
  (implies (pseudo-term-alistp acl2::x)
           (iff (pseudo-termp (cdar acl2::x))
                (or (consp acl2::x)
                    (pseudo-termp nil))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-termp-of-caar-when-pseudo-term-alistp

(defthm pseudo-termp-of-caar-when-pseudo-term-alistp
  (implies (pseudo-term-alistp acl2::x)
           (iff (pseudo-termp (caar acl2::x))
                (or (consp acl2::x)
                    (pseudo-termp nil))))
  :rule-classes ((:rewrite)))