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Atc-symbol-varinfo-alist

Atc-symbol-varinfo-alistp

Recognizer for atc-symbol-varinfo-alist.

Signature
(atc-symbol-varinfo-alistp x) → *

Definitions and Theorems

Function: atc-symbol-varinfo-alistp

(defun atc-symbol-varinfo-alistp (x)
  (declare (xargs :guard t))
  (let ((__function__ 'atc-symbol-varinfo-alistp))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (symbolp (caar x))
           (atc-var-infop (cdar x))
           (atc-symbol-varinfo-alistp (cdr x))))))

Theorem: atc-symbol-varinfo-alistp-of-append

(defthm atc-symbol-varinfo-alistp-of-append
  (equal (atc-symbol-varinfo-alistp (append acl2::a acl2::b))
         (and (atc-symbol-varinfo-alistp (list-fix acl2::a))
              (atc-symbol-varinfo-alistp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-rev

(defthm atc-symbol-varinfo-alistp-of-rev
  (equal (atc-symbol-varinfo-alistp (rev acl2::x))
         (atc-symbol-varinfo-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-list-fix

(defthm atc-symbol-varinfo-alistp-of-list-fix
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (atc-symbol-varinfo-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-atc-symbol-varinfo-alistp-compound-recognizer

(defthm
      true-listp-when-atc-symbol-varinfo-alistp-compound-recognizer
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: atc-symbol-varinfo-alistp-when-not-consp

(defthm atc-symbol-varinfo-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (atc-symbol-varinfo-alistp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-cdr-when-atc-symbol-varinfo-alistp

(defthm
    atc-symbol-varinfo-alistp-of-cdr-when-atc-symbol-varinfo-alistp
  (implies (atc-symbol-varinfo-alistp (double-rewrite acl2::x))
           (atc-symbol-varinfo-alistp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-cons

(defthm atc-symbol-varinfo-alistp-of-cons
  (equal (atc-symbol-varinfo-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (symbolp (car acl2::a))
                   (atc-var-infop (cdr acl2::a)))
              (atc-symbol-varinfo-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-remove-assoc

(defthm atc-symbol-varinfo-alistp-of-remove-assoc
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (atc-symbol-varinfo-alistp
                (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-put-assoc

(defthm atc-symbol-varinfo-alistp-of-put-assoc
  (implies (and (atc-symbol-varinfo-alistp acl2::x))
           (iff (atc-symbol-varinfo-alistp
                     (put-assoc-equal acl2::name acl2::val acl2::x))
                (and (symbolp acl2::name)
                     (atc-var-infop acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-fast-alist-clean

(defthm atc-symbol-varinfo-alistp-of-fast-alist-clean
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (atc-symbol-varinfo-alistp (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-hons-shrink-alist

(defthm atc-symbol-varinfo-alistp-of-hons-shrink-alist
 (implies
    (and (atc-symbol-varinfo-alistp acl2::x)
         (atc-symbol-varinfo-alistp acl2::y))
    (atc-symbol-varinfo-alistp (hons-shrink-alist acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: atc-symbol-varinfo-alistp-of-hons-acons

(defthm atc-symbol-varinfo-alistp-of-hons-acons
 (equal
    (atc-symbol-varinfo-alistp (hons-acons acl2::a acl2::n acl2::x))
    (and (symbolp acl2::a)
         (atc-var-infop acl2::n)
         (atc-symbol-varinfo-alistp acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: atc-var-infop-of-cdr-of-hons-assoc-equal-when-atc-symbol-varinfo-alistp

(defthm
 atc-var-infop-of-cdr-of-hons-assoc-equal-when-atc-symbol-varinfo-alistp
 (implies
      (atc-symbol-varinfo-alistp acl2::x)
      (iff (atc-var-infop (cdr (hons-assoc-equal acl2::k acl2::x)))
           (hons-assoc-equal acl2::k acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-atc-symbol-varinfo-alistp-rewrite

(defthm alistp-when-atc-symbol-varinfo-alistp-rewrite
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-atc-symbol-varinfo-alistp

(defthm alistp-when-atc-symbol-varinfo-alistp
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: atc-var-infop-of-cdar-when-atc-symbol-varinfo-alistp

(defthm atc-var-infop-of-cdar-when-atc-symbol-varinfo-alistp
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (iff (atc-var-infop (cdar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: symbolp-of-caar-when-atc-symbol-varinfo-alistp

(defthm symbolp-of-caar-when-atc-symbol-varinfo-alistp
  (implies (atc-symbol-varinfo-alistp acl2::x)
           (symbolp (caar acl2::x)))
  :rule-classes ((:rewrite)))