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Term-bvars

Term-bvars-p

Recognizer for term-bvars.

Signature
(term-bvars-p x) → *

Definitions and Theorems

Function: term-bvars-p

(defun term-bvars-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'term-bvars-p))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (fgl-object-p (caar x))
           (natp (cdar x))
           (term-bvars-p (cdr x))))))

Theorem: term-bvars-p-of-rev

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

Theorem: term-bvars-p-of-list-fix

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

Theorem: true-listp-when-term-bvars-p-compound-recognizer

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

Theorem: term-bvars-p-when-not-consp

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

Theorem: term-bvars-p-of-cdr-when-term-bvars-p

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

Theorem: term-bvars-p-of-cons

(defthm term-bvars-p-of-cons
  (equal (term-bvars-p (cons a x))
         (and (and (consp a)
                   (fgl-object-p (car a))
                   (natp (cdr a)))
              (term-bvars-p x)))
  :rule-classes ((:rewrite)))

Theorem: term-bvars-p-of-remove-assoc

(defthm term-bvars-p-of-remove-assoc
  (implies (term-bvars-p x)
           (term-bvars-p (remove-assoc-equal name x)))
  :rule-classes ((:rewrite)))

Theorem: term-bvars-p-of-put-assoc

(defthm term-bvars-p-of-put-assoc
  (implies (and (term-bvars-p x))
           (iff (term-bvars-p (put-assoc-equal name acl2::val x))
                (and (fgl-object-p name)
                     (natp acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: term-bvars-p-of-fast-alist-clean

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

Theorem: term-bvars-p-of-hons-shrink-alist

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

Theorem: term-bvars-p-of-hons-acons

(defthm term-bvars-p-of-hons-acons
  (equal (term-bvars-p (hons-acons a n x))
         (and (fgl-object-p a)
              (natp n)
              (term-bvars-p x)))
  :rule-classes ((:rewrite)))

Theorem: natp-of-cdr-of-hons-assoc-equal-when-term-bvars-p

(defthm natp-of-cdr-of-hons-assoc-equal-when-term-bvars-p
  (implies (term-bvars-p x)
           (iff (natp (cdr (hons-assoc-equal k x)))
                (or (hons-assoc-equal k x) (natp nil))))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-term-bvars-p-rewrite

(defthm alistp-when-term-bvars-p-rewrite
  (implies (term-bvars-p x) (alistp x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-term-bvars-p

(defthm alistp-when-term-bvars-p
  (implies (term-bvars-p x) (alistp x))
  :rule-classes :tau-system)

Theorem: natp-of-cdar-when-term-bvars-p

(defthm natp-of-cdar-when-term-bvars-p
  (implies (term-bvars-p x)
           (iff (natp (cdar x))
                (or (consp x) (natp nil))))
  :rule-classes ((:rewrite)))

Theorem: fgl-object-p-of-caar-when-term-bvars-p

(defthm fgl-object-p-of-caar-when-term-bvars-p
  (implies (term-bvars-p x)
           (iff (fgl-object-p (caar x))
                (or (consp x) (fgl-object-p nil))))
  :rule-classes ((:rewrite)))