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• Filepath-plexeme-list-alist

Filepath-plexeme-list-alistp

Recognizer for filepath-plexeme-list-alist.

Signature

(filepath-plexeme-list-alistp x) → *

Definitions and Theorems

Function: filepath-plexeme-list-alistp

(defun filepath-plexeme-list-alistp (x)
  (declare (xargs :guard t))
  (if (atom x)
      (eq x nil)
    (and (consp (car x))
         (filepathp (caar x))
         (plexeme-listp (cdar x))
         (filepath-plexeme-list-alistp (cdr x)))))

Theorem: filepath-plexeme-list-alistp-of-revappend

(defthm filepath-plexeme-list-alistp-of-revappend
  (equal (filepath-plexeme-list-alistp (revappend acl2::x acl2::y))
         (and (filepath-plexeme-list-alistp (list-fix acl2::x))
              (filepath-plexeme-list-alistp acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-remove

(defthm filepath-plexeme-list-alistp-of-remove
  (implies (filepath-plexeme-list-alistp acl2::x)
           (filepath-plexeme-list-alistp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-last

(defthm filepath-plexeme-list-alistp-of-last
  (implies (filepath-plexeme-list-alistp (double-rewrite acl2::x))
           (filepath-plexeme-list-alistp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-nthcdr

(defthm filepath-plexeme-list-alistp-of-nthcdr
  (implies (filepath-plexeme-list-alistp (double-rewrite acl2::x))
           (filepath-plexeme-list-alistp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-butlast

(defthm filepath-plexeme-list-alistp-of-butlast
  (implies (filepath-plexeme-list-alistp (double-rewrite acl2::x))
           (filepath-plexeme-list-alistp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-update-nth

(defthm filepath-plexeme-list-alistp-of-update-nth
  (implies (filepath-plexeme-list-alistp (double-rewrite acl2::x))
           (iff (filepath-plexeme-list-alistp
                     (update-nth acl2::n acl2::y acl2::x))
                (and (and (consp acl2::y)
                          (filepathp (car acl2::y))
                          (plexeme-listp (cdr acl2::y)))
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (and (consp nil)
                              (filepathp (car nil))
                              (plexeme-listp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-repeat

(defthm filepath-plexeme-list-alistp-of-repeat
  (iff (filepath-plexeme-list-alistp (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (filepathp (car acl2::x))
                (plexeme-listp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-take

(defthm filepath-plexeme-list-alistp-of-take
 (implies (filepath-plexeme-list-alistp (double-rewrite acl2::x))
          (iff (filepath-plexeme-list-alistp (take acl2::n acl2::x))
               (or (and (consp nil)
                        (filepathp (car nil))
                        (plexeme-listp (cdr nil)))
                   (<= (nfix acl2::n) (len acl2::x)))))
 :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-union-equal

(defthm filepath-plexeme-list-alistp-of-union-equal
 (equal
      (filepath-plexeme-list-alistp (union-equal acl2::x acl2::y))
      (and (filepath-plexeme-list-alistp (list-fix acl2::x))
           (filepath-plexeme-list-alistp (double-rewrite acl2::y))))
 :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-intersection-equal-2

(defthm filepath-plexeme-list-alistp-of-intersection-equal-2
  (implies (filepath-plexeme-list-alistp (double-rewrite acl2::y))
           (filepath-plexeme-list-alistp
                (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-intersection-equal-1

(defthm filepath-plexeme-list-alistp-of-intersection-equal-1
  (implies (filepath-plexeme-list-alistp (double-rewrite acl2::x))
           (filepath-plexeme-list-alistp
                (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-set-difference-equal

(defthm filepath-plexeme-list-alistp-of-set-difference-equal
  (implies (filepath-plexeme-list-alistp acl2::x)
           (filepath-plexeme-list-alistp
                (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-when-subsetp-equal

(defthm filepath-plexeme-list-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (filepath-plexeme-list-alistp acl2::y))
                (equal (filepath-plexeme-list-alistp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (filepath-plexeme-list-alistp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (filepath-plexeme-list-alistp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-rcons

(defthm filepath-plexeme-list-alistp-of-rcons
  (iff (filepath-plexeme-list-alistp (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (filepathp (car acl2::a))
                 (plexeme-listp (cdr acl2::a)))
            (filepath-plexeme-list-alistp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-append

(defthm filepath-plexeme-list-alistp-of-append
  (equal (filepath-plexeme-list-alistp (append acl2::a acl2::b))
         (and (filepath-plexeme-list-alistp (list-fix acl2::a))
              (filepath-plexeme-list-alistp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-rev

(defthm filepath-plexeme-list-alistp-of-rev
  (equal (filepath-plexeme-list-alistp (rev acl2::x))
         (filepath-plexeme-list-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-duplicated-members

(defthm filepath-plexeme-list-alistp-of-duplicated-members
  (implies
       (filepath-plexeme-list-alistp acl2::x)
       (filepath-plexeme-list-alistp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-difference

(defthm filepath-plexeme-list-alistp-of-difference
  (implies
       (filepath-plexeme-list-alistp acl2::x)
       (filepath-plexeme-list-alistp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-intersect-2

(defthm filepath-plexeme-list-alistp-of-intersect-2
  (implies
       (filepath-plexeme-list-alistp acl2::y)
       (filepath-plexeme-list-alistp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-intersect-1

(defthm filepath-plexeme-list-alistp-of-intersect-1
  (implies
       (filepath-plexeme-list-alistp acl2::x)
       (filepath-plexeme-list-alistp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-union

(defthm filepath-plexeme-list-alistp-of-union
  (iff (filepath-plexeme-list-alistp (union acl2::x acl2::y))
       (and (filepath-plexeme-list-alistp (sfix acl2::x))
            (filepath-plexeme-list-alistp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-mergesort

(defthm filepath-plexeme-list-alistp-of-mergesort
  (iff (filepath-plexeme-list-alistp (mergesort acl2::x))
       (filepath-plexeme-list-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-delete

(defthm filepath-plexeme-list-alistp-of-delete
  (implies (filepath-plexeme-list-alistp acl2::x)
           (filepath-plexeme-list-alistp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-insert

(defthm filepath-plexeme-list-alistp-of-insert
  (iff (filepath-plexeme-list-alistp (insert acl2::a acl2::x))
       (and (filepath-plexeme-list-alistp (sfix acl2::x))
            (and (consp acl2::a)
                 (filepathp (car acl2::a))
                 (plexeme-listp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-sfix

(defthm filepath-plexeme-list-alistp-of-sfix
  (iff (filepath-plexeme-list-alistp (sfix acl2::x))
       (or (filepath-plexeme-list-alistp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-list-fix

(defthm filepath-plexeme-list-alistp-of-list-fix
  (implies (filepath-plexeme-list-alistp acl2::x)
           (filepath-plexeme-list-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-filepath-plexeme-list-alistp-compound-recognizer

(defthm
   true-listp-when-filepath-plexeme-list-alistp-compound-recognizer
  (implies (filepath-plexeme-list-alistp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: filepath-plexeme-list-alistp-when-not-consp

(defthm filepath-plexeme-list-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (filepath-plexeme-list-alistp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-cdr-when-filepath-plexeme-list-alistp

(defthm
 filepath-plexeme-list-alistp-of-cdr-when-filepath-plexeme-list-alistp
 (implies (filepath-plexeme-list-alistp (double-rewrite acl2::x))
          (filepath-plexeme-list-alistp (cdr acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-cons

(defthm filepath-plexeme-list-alistp-of-cons
  (equal (filepath-plexeme-list-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (filepathp (car acl2::a))
                   (plexeme-listp (cdr acl2::a)))
              (filepath-plexeme-list-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-remove-assoc

(defthm filepath-plexeme-list-alistp-of-remove-assoc
  (implies (filepath-plexeme-list-alistp acl2::x)
           (filepath-plexeme-list-alistp
                (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-listp-of-cdr-of-assoc-when-filepath-plexeme-list-alistp

(defthm
    plexeme-listp-of-cdr-of-assoc-when-filepath-plexeme-list-alistp
  (implies (filepath-plexeme-list-alistp acl2::x)
           (plexeme-listp (cdr (assoc-equal acl2::k acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-put-assoc

(defthm filepath-plexeme-list-alistp-of-put-assoc
  (implies (and (filepath-plexeme-list-alistp acl2::x))
           (iff (filepath-plexeme-list-alistp
                     (put-assoc-equal acl2::name acl2::val acl2::x))
                (and (filepathp acl2::name)
                     (plexeme-listp acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-fast-alist-clean

(defthm filepath-plexeme-list-alistp-of-fast-alist-clean
 (implies (filepath-plexeme-list-alistp acl2::x)
          (filepath-plexeme-list-alistp (fast-alist-clean acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-hons-shrink-alist

(defthm filepath-plexeme-list-alistp-of-hons-shrink-alist
  (implies (and (filepath-plexeme-list-alistp acl2::x)
                (filepath-plexeme-list-alistp acl2::y))
           (filepath-plexeme-list-alistp
                (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: filepath-plexeme-list-alistp-of-hons-acons

(defthm filepath-plexeme-list-alistp-of-hons-acons
  (equal (filepath-plexeme-list-alistp
              (hons-acons acl2::a acl2::n acl2::x))
         (and (filepathp acl2::a)
              (plexeme-listp acl2::n)
              (filepath-plexeme-list-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-listp-of-cdr-of-hons-assoc-equal-when-filepath-plexeme-list-alistp

(defthm
 plexeme-listp-of-cdr-of-hons-assoc-equal-when-filepath-plexeme-list-alistp
 (implies (filepath-plexeme-list-alistp acl2::x)
          (plexeme-listp (cdr (hons-assoc-equal acl2::k acl2::x))))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-filepath-plexeme-list-alistp-rewrite

(defthm alistp-when-filepath-plexeme-list-alistp-rewrite
  (implies (filepath-plexeme-list-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-filepath-plexeme-list-alistp

(defthm alistp-when-filepath-plexeme-list-alistp
  (implies (filepath-plexeme-list-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: filepathp-of-caar-when-filepath-plexeme-list-alistp

(defthm filepathp-of-caar-when-filepath-plexeme-list-alistp
  (implies (filepath-plexeme-list-alistp acl2::x)
           (iff (filepathp (caar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-listp-of-cdar-when-filepath-plexeme-list-alistp

(defthm plexeme-listp-of-cdar-when-filepath-plexeme-list-alistp
  (implies (filepath-plexeme-list-alistp acl2::x)
           (plexeme-listp (cdar acl2::x)))
  :rule-classes ((:rewrite)))