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Cgraph

Cgraph-p

Recognizer for cgraph.

Signature
(cgraph-p x) → *

Definitions and Theorems

Function: cgraph-p

(defun cgraph-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'cgraph-p))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (fgl-object-p (caar x))
           (cgraph-edgelist-p (cdar x))
           (cgraph-p (cdr x))))))

Theorem: cgraph-p-of-revappend

(defthm cgraph-p-of-revappend
  (equal (cgraph-p (revappend x y))
         (and (cgraph-p (list-fix x))
              (cgraph-p y)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-remove

(defthm cgraph-p-of-remove
  (implies (cgraph-p x)
           (cgraph-p (remove a x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-last

(defthm cgraph-p-of-last
  (implies (cgraph-p (double-rewrite x))
           (cgraph-p (last x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-nthcdr

(defthm cgraph-p-of-nthcdr
  (implies (cgraph-p (double-rewrite x))
           (cgraph-p (nthcdr n x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-butlast

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

Theorem: cgraph-p-of-update-nth

(defthm cgraph-p-of-update-nth
  (implies (cgraph-p (double-rewrite x))
           (iff (cgraph-p (update-nth n y x))
                (and (and (consp y)
                          (fgl-object-p (car y))
                          (cgraph-edgelist-p (cdr y)))
                     (or (<= (nfix n) (len x))
                         (and (consp nil)
                              (fgl-object-p (car nil))
                              (cgraph-edgelist-p (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-repeat

(defthm cgraph-p-of-repeat
  (iff (cgraph-p (acl2::repeat n x))
       (or (and (consp x)
                (fgl-object-p (car x))
                (cgraph-edgelist-p (cdr x)))
           (zp n)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-take

(defthm cgraph-p-of-take
  (implies (cgraph-p (double-rewrite x))
           (iff (cgraph-p (take n x))
                (or (and (consp nil)
                         (fgl-object-p (car nil))
                         (cgraph-edgelist-p (cdr nil)))
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-union-equal

(defthm cgraph-p-of-union-equal
  (equal (cgraph-p (union-equal x y))
         (and (cgraph-p (list-fix x))
              (cgraph-p (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-intersection-equal-2

(defthm cgraph-p-of-intersection-equal-2
  (implies (cgraph-p (double-rewrite y))
           (cgraph-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-intersection-equal-1

(defthm cgraph-p-of-intersection-equal-1
  (implies (cgraph-p (double-rewrite x))
           (cgraph-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-set-difference-equal

(defthm cgraph-p-of-set-difference-equal
  (implies (cgraph-p x)
           (cgraph-p (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-when-subsetp-equal

(defthm cgraph-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y) (cgraph-p y))
                (equal (cgraph-p x) (true-listp x)))
       (implies (and (cgraph-p y) (subsetp-equal x y))
                (equal (cgraph-p x) (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-rcons

(defthm cgraph-p-of-rcons
  (iff (cgraph-p (acl2::rcons a x))
       (and (and (consp a)
                 (fgl-object-p (car a))
                 (cgraph-edgelist-p (cdr a)))
            (cgraph-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-append

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

Theorem: cgraph-p-of-rev

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

Theorem: cgraph-p-of-duplicated-members

(defthm cgraph-p-of-duplicated-members
  (implies (cgraph-p x)
           (cgraph-p (acl2::duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-difference

(defthm cgraph-p-of-difference
  (implies (cgraph-p x)
           (cgraph-p (set::difference x y)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-intersect-2

(defthm cgraph-p-of-intersect-2
  (implies (cgraph-p y)
           (cgraph-p (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-intersect-1

(defthm cgraph-p-of-intersect-1
  (implies (cgraph-p x)
           (cgraph-p (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-union

(defthm cgraph-p-of-union
  (iff (cgraph-p (set::union x y))
       (and (cgraph-p (set::sfix x))
            (cgraph-p (set::sfix y))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-mergesort

(defthm cgraph-p-of-mergesort
  (iff (cgraph-p (set::mergesort x))
       (cgraph-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-delete

(defthm cgraph-p-of-delete
  (implies (cgraph-p x)
           (cgraph-p (set::delete k x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-insert

(defthm cgraph-p-of-insert
  (iff (cgraph-p (set::insert a x))
       (and (cgraph-p (set::sfix x))
            (and (consp a)
                 (fgl-object-p (car a))
                 (cgraph-edgelist-p (cdr a)))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-sfix

(defthm cgraph-p-of-sfix
  (iff (cgraph-p (set::sfix x))
       (or (cgraph-p x) (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-list-fix

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

Theorem: true-listp-when-cgraph-p-compound-recognizer

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

Theorem: cgraph-p-when-not-consp

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

Theorem: cgraph-p-of-cdr-when-cgraph-p

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

Theorem: cgraph-p-of-cons

(defthm cgraph-p-of-cons
  (equal (cgraph-p (cons a x))
         (and (and (consp a)
                   (fgl-object-p (car a))
                   (cgraph-edgelist-p (cdr a)))
              (cgraph-p x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-p-of-remove-assoc

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

Theorem: cgraph-p-of-put-assoc

(defthm cgraph-p-of-put-assoc
  (implies (and (cgraph-p x))
           (iff (cgraph-p (put-assoc-equal name acl2::val x))
                (and (fgl-object-p name)
                     (cgraph-edgelist-p acl2::val))))
  :rule-classes ((:rewrite)))

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

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

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

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

Theorem: cgraph-p-of-hons-acons

(defthm cgraph-p-of-hons-acons
  (equal (cgraph-p (hons-acons a n x))
         (and (fgl-object-p a)
              (cgraph-edgelist-p n)
              (cgraph-p x)))
  :rule-classes ((:rewrite)))

Theorem: cgraph-edgelist-p-of-cdr-of-hons-assoc-equal-when-cgraph-p

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

Theorem: alistp-when-cgraph-p-rewrite

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

Theorem: alistp-when-cgraph-p

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

Theorem: cgraph-edgelist-p-of-cdar-when-cgraph-p

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

Theorem: fgl-object-p-of-caar-when-cgraph-p

(defthm fgl-object-p-of-caar-when-cgraph-p
  (implies (cgraph-p x)
           (iff (fgl-object-p (caar x))
                (or (consp x) (fgl-object-p nil))))
  :rule-classes ((:rewrite)))