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Svex-s4env

Svex-s4env-p

Recognizer for svex-s4env.

Signature
(svex-s4env-p x) → *

Definitions and Theorems

Function: svex-s4env-p

(defun svex-s4env-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'svex-s4env-p))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (svar-p (caar x))
           (s4vec-p (cdar x))
           (svex-s4env-p (cdr x))))))

Theorem: svex-s4env-p-of-repeat

(defthm svex-s4env-p-of-repeat
  (iff (svex-s4env-p (repeat acl2::n x))
       (or (and (consp x)
                (svar-p (car x))
                (s4vec-p (cdr x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-rev

(defthm svex-s4env-p-of-rev
  (equal (svex-s4env-p (rev x))
         (svex-s4env-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-list-fix

(defthm svex-s4env-p-of-list-fix
  (implies (svex-s4env-p x)
           (svex-s4env-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-svex-s4env-p-compound-recognizer

(defthm true-listp-when-svex-s4env-p-compound-recognizer
  (implies (svex-s4env-p x)
           (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: svex-s4env-p-when-not-consp

(defthm svex-s4env-p-when-not-consp
  (implies (not (consp x))
           (equal (svex-s4env-p x) (not x)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-cdr-when-svex-s4env-p

(defthm svex-s4env-p-of-cdr-when-svex-s4env-p
  (implies (svex-s4env-p (double-rewrite x))
           (svex-s4env-p (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-cons

(defthm svex-s4env-p-of-cons
  (equal (svex-s4env-p (cons acl2::a x))
         (and (and (consp acl2::a)
                   (svar-p (car acl2::a))
                   (s4vec-p (cdr acl2::a)))
              (svex-s4env-p x)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-remove-assoc

(defthm svex-s4env-p-of-remove-assoc
  (implies (svex-s4env-p x)
           (svex-s4env-p (remove-assoc-equal acl2::name x)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-put-assoc

(defthm svex-s4env-p-of-put-assoc
  (implies
       (and (svex-s4env-p x))
       (iff (svex-s4env-p (put-assoc-equal acl2::name acl2::val x))
            (and (svar-p acl2::name)
                 (s4vec-p acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-fast-alist-clean

(defthm svex-s4env-p-of-fast-alist-clean
  (implies (svex-s4env-p x)
           (svex-s4env-p (fast-alist-clean x)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-hons-shrink-alist

(defthm svex-s4env-p-of-hons-shrink-alist
  (implies (and (svex-s4env-p x) (svex-s4env-p y))
           (svex-s4env-p (hons-shrink-alist x y)))
  :rule-classes ((:rewrite)))

Theorem: svex-s4env-p-of-hons-acons

(defthm svex-s4env-p-of-hons-acons
  (equal (svex-s4env-p (hons-acons acl2::a acl2::n x))
         (and (svar-p acl2::a)
              (s4vec-p acl2::n)
              (svex-s4env-p x)))
  :rule-classes ((:rewrite)))

Theorem: s4vec-p-of-cdr-of-hons-assoc-equal-when-svex-s4env-p

(defthm s4vec-p-of-cdr-of-hons-assoc-equal-when-svex-s4env-p
  (implies (svex-s4env-p x)
           (iff (s4vec-p (cdr (hons-assoc-equal acl2::k x)))
                (or (hons-assoc-equal acl2::k x)
                    (s4vec-p nil))))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-svex-s4env-p-rewrite

(defthm alistp-when-svex-s4env-p-rewrite
  (implies (svex-s4env-p x) (alistp x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-svex-s4env-p

(defthm alistp-when-svex-s4env-p
  (implies (svex-s4env-p x) (alistp x))
  :rule-classes :tau-system)

Theorem: s4vec-p-of-cdar-when-svex-s4env-p

(defthm s4vec-p-of-cdar-when-svex-s4env-p
  (implies (svex-s4env-p x)
           (iff (s4vec-p (cdar x))
                (or (consp x) (s4vec-p nil))))
  :rule-classes ((:rewrite)))

Theorem: svar-p-of-caar-when-svex-s4env-p

(defthm svar-p-of-caar-when-svex-s4env-p
  (implies (svex-s4env-p x)
           (iff (svar-p (caar x))
                (or (consp x) (svar-p nil))))
  :rule-classes ((:rewrite)))