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• Svdecomp-symenv

Svdecomp-symenv-p

Recognizer for svdecomp-symenv.

Signature

(svdecomp-symenv-p x) → *

Definitions and Theorems

Function: svdecomp-symenv-p

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

Theorem: svdecomp-symenv-p-of-update-nth

(defthm svdecomp-symenv-p-of-update-nth
  (implies (svdecomp-symenv-p (double-rewrite x))
           (iff (svdecomp-symenv-p (update-nth acl2::n y x))
                (and (and (consp y)
                          (svar-p (car y))
                          (pseudo-termp (cdr y)))
                     (or (<= (nfix acl2::n) (len x))
                         (and (consp nil)
                              (svar-p (car nil))
                              (pseudo-termp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-take

(defthm svdecomp-symenv-p-of-take
  (implies (svdecomp-symenv-p (double-rewrite x))
           (iff (svdecomp-symenv-p (take acl2::n x))
                (or (and (consp nil)
                         (svar-p (car nil))
                         (pseudo-termp (cdr nil)))
                    (<= (nfix acl2::n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-revappend

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

Theorem: svdecomp-symenv-p-of-remove

(defthm svdecomp-symenv-p-of-remove
  (implies (svdecomp-symenv-p x)
           (svdecomp-symenv-p (remove acl2::a x)))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-nthcdr

(defthm svdecomp-symenv-p-of-nthcdr
  (implies (svdecomp-symenv-p (double-rewrite x))
           (svdecomp-symenv-p (nthcdr acl2::n x)))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-last

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

Theorem: svdecomp-symenv-p-of-union-equal

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

Theorem: svdecomp-symenv-p-of-intersection-equal-2

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

Theorem: svdecomp-symenv-p-of-intersection-equal-1

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

Theorem: svdecomp-symenv-p-of-set-difference-equal

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

Theorem: svdecomp-symenv-p-when-subsetp-equal

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

Theorem: svdecomp-symenv-p-of-rcons

(defthm svdecomp-symenv-p-of-rcons
  (iff (svdecomp-symenv-p (acl2::rcons acl2::a x))
       (and (and (consp acl2::a)
                 (svar-p (car acl2::a))
                 (pseudo-termp (cdr acl2::a)))
            (svdecomp-symenv-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-append

(defthm svdecomp-symenv-p-of-append
  (equal (svdecomp-symenv-p (append acl2::a acl2::b))
         (and (svdecomp-symenv-p (list-fix acl2::a))
              (svdecomp-symenv-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-repeat

(defthm svdecomp-symenv-p-of-repeat
  (iff (svdecomp-symenv-p (repeat acl2::n x))
       (or (and (consp x)
                (svar-p (car x))
                (pseudo-termp (cdr x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-rev

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

Theorem: svdecomp-symenv-p-of-list-fix

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

Theorem: true-listp-when-svdecomp-symenv-p-compound-recognizer

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

Theorem: svdecomp-symenv-p-when-not-consp

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

Theorem: svdecomp-symenv-p-of-cdr-when-svdecomp-symenv-p

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

Theorem: svdecomp-symenv-p-of-cons

(defthm svdecomp-symenv-p-of-cons
  (equal (svdecomp-symenv-p (cons acl2::a x))
         (and (and (consp acl2::a)
                   (svar-p (car acl2::a))
                   (pseudo-termp (cdr acl2::a)))
              (svdecomp-symenv-p x)))
  :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-remove-assoc

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

Theorem: svdecomp-symenv-p-of-put-assoc

(defthm svdecomp-symenv-p-of-put-assoc
 (implies
   (and (svdecomp-symenv-p x))
   (iff (svdecomp-symenv-p (put-assoc-equal acl2::name acl2::val x))
        (and (svar-p acl2::name)
             (pseudo-termp acl2::val))))
 :rule-classes ((:rewrite)))

Theorem: svdecomp-symenv-p-of-fast-alist-clean

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

Theorem: svdecomp-symenv-p-of-hons-shrink-alist

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

Theorem: svdecomp-symenv-p-of-hons-acons

(defthm svdecomp-symenv-p-of-hons-acons
  (equal (svdecomp-symenv-p (hons-acons acl2::a acl2::n x))
         (and (svar-p acl2::a)
              (pseudo-termp acl2::n)
              (svdecomp-symenv-p x)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-termp-of-cdr-of-hons-assoc-equal-when-svdecomp-symenv-p

(defthm
     pseudo-termp-of-cdr-of-hons-assoc-equal-when-svdecomp-symenv-p
  (implies (svdecomp-symenv-p x)
           (iff (pseudo-termp (cdr (hons-assoc-equal acl2::k x)))
                (or (hons-assoc-equal acl2::k x)
                    (pseudo-termp nil))))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-svdecomp-symenv-p-rewrite

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

Theorem: alistp-when-svdecomp-symenv-p

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

Theorem: pseudo-termp-of-cdar-when-svdecomp-symenv-p

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

Theorem: svar-p-of-caar-when-svdecomp-symenv-p

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