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Alias-alist

Alias-alist-p

Signature
(alias-alist-p x) → *

Definitions and Theorems

Function: alias-alist-p

(defun alias-alist-p (x)
  (declare (xargs :guard t))
  (let ((acl2::__function__ 'alias-alist-p))
    (declare (ignorable acl2::__function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (sv::svar-p (caar x))
           (svex-p (cdar x))
           (alias-alist-p (cdr x))))))

Theorem: alias-alist-p-of-butlast

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

Theorem: alias-alist-p-of-append

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

Theorem: alias-alist-p-of-repeat

(defthm alias-alist-p-of-repeat
  (iff (alias-alist-p (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (sv::svar-p (car acl2::x))
                (svex-p (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: alias-alist-p-of-rev

(defthm alias-alist-p-of-rev
  (equal (alias-alist-p (rev acl2::x))
         (alias-alist-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: alias-alist-p-of-list-fix

(defthm alias-alist-p-of-list-fix
  (implies (alias-alist-p acl2::x)
           (alias-alist-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-alias-alist-p-compound-recognizer

(defthm true-listp-when-alias-alist-p-compound-recognizer
  (implies (alias-alist-p acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: alias-alist-p-when-not-consp

(defthm alias-alist-p-when-not-consp
  (implies (not (consp acl2::x))
           (equal (alias-alist-p acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: alias-alist-p-of-cdr-when-alias-alist-p

(defthm alias-alist-p-of-cdr-when-alias-alist-p
  (implies (alias-alist-p (double-rewrite acl2::x))
           (alias-alist-p (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: alias-alist-p-of-cons

(defthm alias-alist-p-of-cons
  (equal (alias-alist-p (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (sv::svar-p (car acl2::a))
                   (svex-p (cdr acl2::a)))
              (alias-alist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: alias-alist-p-of-remove-assoc

(defthm alias-alist-p-of-remove-assoc
  (implies (alias-alist-p acl2::x)
           (alias-alist-p (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: alias-alist-p-of-put-assoc

(defthm alias-alist-p-of-put-assoc
 (implies
  (and (alias-alist-p acl2::x))
  (iff
      (alias-alist-p (put-assoc-equal acl2::name acl2::val acl2::x))
      (and (sv::svar-p acl2::name)
           (svex-p acl2::val))))
 :rule-classes ((:rewrite)))

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

(defthm alias-alist-p-of-fast-alist-clean
  (implies (alias-alist-p acl2::x)
           (alias-alist-p (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

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

(defthm alias-alist-p-of-hons-shrink-alist
  (implies (and (alias-alist-p acl2::x)
                (alias-alist-p acl2::y))
           (alias-alist-p (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: alias-alist-p-of-hons-acons

(defthm alias-alist-p-of-hons-acons
  (equal (alias-alist-p (hons-acons acl2::a acl2::n acl2::x))
         (and (sv::svar-p acl2::a)
              (svex-p acl2::n)
              (alias-alist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: svex-p-of-cdr-of-hons-assoc-equal-when-alias-alist-p

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

Theorem: alistp-when-alias-alist-p-rewrite

(defthm alistp-when-alias-alist-p-rewrite
  (implies (alias-alist-p acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-alias-alist-p

(defthm alistp-when-alias-alist-p
  (implies (alias-alist-p acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: svex-p-of-cdar-when-alias-alist-p

(defthm svex-p-of-cdar-when-alias-alist-p
  (implies (alias-alist-p acl2::x)
           (iff (svex-p (cdar acl2::x))
                (or (consp acl2::x) (svex-p nil))))
  :rule-classes ((:rewrite)))

Theorem: svar-p-of-caar-when-alias-alist-p

(defthm svar-p-of-caar-when-alias-alist-p
  (implies (alias-alist-p acl2::x)
           (iff (sv::svar-p (caar acl2::x))
                (or (consp acl2::x) (sv::svar-p nil))))
  :rule-classes ((:rewrite)))