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• Modalist

Modalist-p

Recognizer for modalist.

Signature

(modalist-p x) → *

Definitions and Theorems

Function: modalist-p

(defun modalist-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'modalist-p))
    (declare (ignorable __function__))
    (if (atom x)
        t
      (and (consp (car x))
           (modname-p (caar x))
           (module-p (cdar x))
           (modalist-p (cdr x))))))

Theorem: modalist-p-of-butlast

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

Theorem: modalist-p-of-take

(defthm modalist-p-of-take
  (implies (modalist-p (double-rewrite x))
           (iff (modalist-p (take acl2::n x))
                (or (and (consp nil)
                         (modname-p (car nil))
                         (module-p (cdr nil)))
                    (<= (nfix acl2::n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: modalist-p-of-repeat

(defthm modalist-p-of-repeat
  (iff (modalist-p (repeat acl2::n x))
       (or (and (consp x)
                (modname-p (car x))
                (module-p (cdr x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: modalist-p-of-rev

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

Theorem: modalist-p-of-list-fix

(defthm modalist-p-of-list-fix
  (equal (modalist-p (list-fix x))
         (modalist-p x))
  :rule-classes ((:rewrite)))

Theorem: modalist-p-of-append

(defthm modalist-p-of-append
  (equal (modalist-p (append acl2::a acl2::b))
         (and (modalist-p acl2::a)
              (modalist-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: modalist-p-when-not-consp

(defthm modalist-p-when-not-consp
  (implies (not (consp x)) (modalist-p x))
  :rule-classes ((:rewrite)))

Theorem: modalist-p-of-cdr-when-modalist-p

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

Theorem: modalist-p-of-cons

(defthm modalist-p-of-cons
  (equal (modalist-p (cons acl2::a x))
         (and (and (consp acl2::a)
                   (modname-p (car acl2::a))
                   (module-p (cdr acl2::a)))
              (modalist-p x)))
  :rule-classes ((:rewrite)))

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

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

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

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

Theorem: modalist-p-of-hons-acons

(defthm modalist-p-of-hons-acons
  (equal (modalist-p (hons-acons acl2::a acl2::n x))
         (and (modname-p acl2::a)
              (module-p acl2::n)
              (modalist-p x)))
  :rule-classes ((:rewrite)))

Theorem: module-p-of-cdr-of-hons-assoc-equal-when-modalist-p

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

Theorem: module-p-of-cdar-when-modalist-p

(defthm module-p-of-cdar-when-modalist-p
  (implies (modalist-p x)
           (iff (module-p (cdar x))
                (or (consp x) (module-p nil))))
  :rule-classes ((:rewrite)))

Theorem: modname-p-of-caar-when-modalist-p

(defthm modname-p-of-caar-when-modalist-p
  (implies (modalist-p x)
           (iff (modname-p (caar x))
                (or (consp x) (modname-p nil))))
  :rule-classes ((:rewrite)))