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Pseudo-term-subst

Pseudo-term-subst-p

Signature
(pseudo-term-subst-p x) → *

Definitions and Theorems

Function: pseudo-term-subst-p

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

Theorem: true-listp-when-pseudo-term-subst-p-compound-recognizer

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

Theorem: pseudo-term-subst-p-when-not-consp

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

Theorem: pseudo-term-subst-p-of-cdr-when-pseudo-term-subst-p

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

Theorem: pseudo-term-subst-p-of-cons

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

Theorem: pseudo-term-subst-p-of-remove-assoc

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

Theorem: pseudo-term-subst-p-of-put-assoc

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

Theorem: pseudo-term-subst-p-of-fast-alist-clean

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

Theorem: pseudo-term-subst-p-of-hons-shrink-alist

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

Theorem: pseudo-term-subst-p-of-hons-acons

(defthm pseudo-term-subst-p-of-hons-acons
  (equal (pseudo-term-subst-p (hons-acons a n x))
         (and (pseudo-var-p a)
              (pseudo-termp n)
              (pseudo-term-subst-p x)))
  :rule-classes ((:rewrite)))

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

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

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

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

Theorem: alistp-when-pseudo-term-subst-p

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

Theorem: pseudo-termp-of-cdar-when-pseudo-term-subst-p

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

Theorem: pseudo-var-p-of-caar-when-pseudo-term-subst-p

(defthm pseudo-var-p-of-caar-when-pseudo-term-subst-p
  (implies (pseudo-term-subst-p x)
           (iff (pseudo-var-p (caar x))
                (or (consp x) (pseudo-var-p nil))))
  :rule-classes ((:rewrite)))