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• Funenv-nofunp

Funenv-nofunp-basics

Basic theorems about funenv-nofunp, generated by std::deflist.

Definitions and Theorems

Theorem: funenv-nofunp-of-cons

(defthm funenv-nofunp-of-cons
  (equal (funenv-nofunp (cons acl2::a acl2::x))
         (and (funscope-nofunp acl2::a)
              (funenv-nofunp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-cdr-when-funenv-nofunp

(defthm funenv-nofunp-of-cdr-when-funenv-nofunp
  (implies (funenv-nofunp (double-rewrite acl2::x))
           (funenv-nofunp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-when-not-consp

(defthm funenv-nofunp-when-not-consp
  (implies (not (consp acl2::x))
           (funenv-nofunp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: funscope-nofunp-of-car-when-funenv-nofunp

(defthm funscope-nofunp-of-car-when-funenv-nofunp
  (implies (funenv-nofunp acl2::x)
           (funscope-nofunp (car acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-append

(defthm funenv-nofunp-of-append
  (equal (funenv-nofunp (append acl2::a acl2::b))
         (and (funenv-nofunp acl2::a)
              (funenv-nofunp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-list-fix

(defthm funenv-nofunp-of-list-fix
  (equal (funenv-nofunp (list-fix acl2::x))
         (funenv-nofunp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-sfix

(defthm funenv-nofunp-of-sfix
  (iff (funenv-nofunp (sfix acl2::x))
       (or (funenv-nofunp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-insert

(defthm funenv-nofunp-of-insert
  (iff (funenv-nofunp (insert acl2::a acl2::x))
       (and (funenv-nofunp (sfix acl2::x))
            (funscope-nofunp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-delete

(defthm funenv-nofunp-of-delete
  (implies (funenv-nofunp acl2::x)
           (funenv-nofunp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-mergesort

(defthm funenv-nofunp-of-mergesort
  (iff (funenv-nofunp (mergesort acl2::x))
       (funenv-nofunp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-union

(defthm funenv-nofunp-of-union
  (iff (funenv-nofunp (union acl2::x acl2::y))
       (and (funenv-nofunp (sfix acl2::x))
            (funenv-nofunp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-intersect-1

(defthm funenv-nofunp-of-intersect-1
  (implies (funenv-nofunp acl2::x)
           (funenv-nofunp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-intersect-2

(defthm funenv-nofunp-of-intersect-2
  (implies (funenv-nofunp acl2::y)
           (funenv-nofunp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-difference

(defthm funenv-nofunp-of-difference
  (implies (funenv-nofunp acl2::x)
           (funenv-nofunp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-duplicated-members

(defthm funenv-nofunp-of-duplicated-members
  (implies (funenv-nofunp acl2::x)
           (funenv-nofunp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-rev

(defthm funenv-nofunp-of-rev
  (equal (funenv-nofunp (rev acl2::x))
         (funenv-nofunp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-rcons

(defthm funenv-nofunp-of-rcons
  (iff (funenv-nofunp (rcons acl2::a acl2::x))
       (and (funscope-nofunp acl2::a)
            (funenv-nofunp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: funscope-nofunp-when-member-equal-of-funenv-nofunp

(defthm funscope-nofunp-when-member-equal-of-funenv-nofunp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (funenv-nofunp acl2::x))
                (funscope-nofunp acl2::a))
       (implies (and (funenv-nofunp acl2::x)
                     (member-equal acl2::a acl2::x))
                (funscope-nofunp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-when-subsetp-equal

(defthm funenv-nofunp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (funenv-nofunp acl2::y))
                (funenv-nofunp acl2::x))
       (implies (and (funenv-nofunp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (funenv-nofunp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-set-equiv-congruence

(defthm funenv-nofunp-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (funenv-nofunp acl2::x)
                  (funenv-nofunp acl2::y)))
  :rule-classes :congruence)

Theorem: funenv-nofunp-of-set-difference-equal

(defthm funenv-nofunp-of-set-difference-equal
  (implies (funenv-nofunp acl2::x)
           (funenv-nofunp (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-intersection-equal-1

(defthm funenv-nofunp-of-intersection-equal-1
  (implies (funenv-nofunp (double-rewrite acl2::x))
           (funenv-nofunp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-intersection-equal-2

(defthm funenv-nofunp-of-intersection-equal-2
  (implies (funenv-nofunp (double-rewrite acl2::y))
           (funenv-nofunp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-union-equal

(defthm funenv-nofunp-of-union-equal
  (equal (funenv-nofunp (union-equal acl2::x acl2::y))
         (and (funenv-nofunp (list-fix acl2::x))
              (funenv-nofunp (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-take

(defthm funenv-nofunp-of-take
  (implies (funenv-nofunp (double-rewrite acl2::x))
           (iff (funenv-nofunp (take acl2::n acl2::x))
                (or (funscope-nofunp nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-repeat

(defthm funenv-nofunp-of-repeat
  (iff (funenv-nofunp (repeat acl2::n acl2::x))
       (or (funscope-nofunp acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: funscope-nofunp-of-nth-when-funenv-nofunp

(defthm funscope-nofunp-of-nth-when-funenv-nofunp
  (implies (funenv-nofunp acl2::x)
           (funscope-nofunp (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-update-nth

(defthm funenv-nofunp-of-update-nth
  (implies (funenv-nofunp (double-rewrite acl2::x))
           (iff (funenv-nofunp (update-nth acl2::n acl2::y acl2::x))
                (and (funscope-nofunp acl2::y)
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (funscope-nofunp nil)))))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-butlast

(defthm funenv-nofunp-of-butlast
  (implies (funenv-nofunp (double-rewrite acl2::x))
           (funenv-nofunp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-nthcdr

(defthm funenv-nofunp-of-nthcdr
  (implies (funenv-nofunp (double-rewrite acl2::x))
           (funenv-nofunp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-last

(defthm funenv-nofunp-of-last
  (implies (funenv-nofunp (double-rewrite acl2::x))
           (funenv-nofunp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-remove

(defthm funenv-nofunp-of-remove
  (implies (funenv-nofunp acl2::x)
           (funenv-nofunp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenv-nofunp-of-revappend

(defthm funenv-nofunp-of-revappend
  (equal (funenv-nofunp (revappend acl2::x acl2::y))
         (and (funenv-nofunp (list-fix acl2::x))
              (funenv-nofunp acl2::y)))
  :rule-classes ((:rewrite)))