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• Envmap

Envmap-p

Recognizer for envmap.

Signature

(envmap-p x) → *

Definitions and Theorems

Function: envmap-p

(defun envmap-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'envmap-p))
    (declare (ignorable __function__))
    (if (atom x)
        t
      (and (consp (car x))
           (pseudo-termp (caar x))
           (svex-alist-p (cdar x))
           (envmap-p (cdr x))))))

Theorem: envmap-p-of-update-nth

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

Theorem: envmap-p-of-take

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

Theorem: envmap-p-of-revappend

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

Theorem: envmap-p-of-remove

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

Theorem: envmap-p-of-nthcdr

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

Theorem: envmap-p-of-last

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

Theorem: envmap-p-of-union-equal

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

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

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

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

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

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

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

Theorem: envmap-p-set-equiv-congruence

(defthm envmap-p-set-equiv-congruence
  (implies (set-equiv x y)
           (equal (envmap-p x) (envmap-p y)))
  :rule-classes :congruence)

Theorem: envmap-p-when-subsetp-equal

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

Theorem: envmap-p-of-rcons

(defthm envmap-p-of-rcons
  (iff (envmap-p (acl2::rcons acl2::a x))
       (and (and (consp acl2::a)
                 (pseudo-termp (car acl2::a))
                 (svex-alist-p (cdr acl2::a)))
            (envmap-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: envmap-p-of-repeat

(defthm envmap-p-of-repeat
  (iff (envmap-p (repeat acl2::n x))
       (or (and (consp x)
                (pseudo-termp (car x))
                (svex-alist-p (cdr x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: envmap-p-of-rev

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

Theorem: envmap-p-of-list-fix

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

Theorem: envmap-p-of-append

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

Theorem: envmap-p-when-not-consp

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

Theorem: envmap-p-of-cdr-when-envmap-p

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

Theorem: envmap-p-of-cons

(defthm envmap-p-of-cons
  (equal (envmap-p (cons acl2::a x))
         (and (and (consp acl2::a)
                   (pseudo-termp (car acl2::a))
                   (svex-alist-p (cdr acl2::a)))
              (envmap-p x)))
  :rule-classes ((:rewrite)))

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

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

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

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

Theorem: envmap-p-of-hons-acons

(defthm envmap-p-of-hons-acons
  (equal (envmap-p (hons-acons acl2::a acl2::n x))
         (and (pseudo-termp acl2::a)
              (svex-alist-p acl2::n)
              (envmap-p x)))
  :rule-classes ((:rewrite)))

Theorem: svex-alist-p-of-cdr-of-hons-assoc-equal-when-envmap-p

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

Theorem: svex-alist-p-of-cdar-when-envmap-p

(defthm svex-alist-p-of-cdar-when-envmap-p
  (implies (envmap-p x)
           (iff (svex-alist-p (cdar x))
                (or (consp x) (svex-alist-p nil))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-termp-of-caar-when-envmap-p

(defthm pseudo-termp-of-caar-when-envmap-p
  (implies (envmap-p x)
           (iff (pseudo-termp (caar x))
                (or (consp x) (pseudo-termp nil))))
  :rule-classes ((:rewrite)))