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Funenvp

Funenvp-basics

Basic theorems about funenvp, generated by std::deflist.

Definitions and Theorems

Theorem: funenvp-of-cons

(defthm funenvp-of-cons
  (equal (funenvp (cons acl2::a acl2::x))
         (and (funscopep acl2::a)
              (funenvp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenvp-of-cdr-when-funenvp

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

Theorem: funenvp-when-not-consp

(defthm funenvp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (funenvp acl2::x) (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funscopep-of-car-when-funenvp

(defthm funscopep-of-car-when-funenvp
  (implies (funenvp acl2::x)
           (funscopep (car acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-funenvp-compound-recognizer

(defthm true-listp-when-funenvp-compound-recognizer
  (implies (funenvp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: funenvp-of-list-fix

(defthm funenvp-of-list-fix
  (implies (funenvp acl2::x)
           (funenvp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenvp-of-sfix

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

Theorem: funenvp-of-insert

(defthm funenvp-of-insert
  (iff (funenvp (insert acl2::a acl2::x))
       (and (funenvp (sfix acl2::x))
            (funscopep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: funenvp-of-delete

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

Theorem: funenvp-of-mergesort

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

Theorem: funenvp-of-union

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

Theorem: funenvp-of-intersect-1

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

Theorem: funenvp-of-intersect-2

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

Theorem: funenvp-of-difference

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

Theorem: funenvp-of-duplicated-members

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

Theorem: funenvp-of-rev

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

Theorem: funenvp-of-append

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

Theorem: funenvp-of-rcons

(defthm funenvp-of-rcons
  (iff (funenvp (rcons acl2::a acl2::x))
       (and (funscopep acl2::a)
            (funenvp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: funscopep-when-member-equal-of-funenvp

(defthm funscopep-when-member-equal-of-funenvp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (funenvp acl2::x))
                (funscopep acl2::a))
       (implies (and (funenvp acl2::x)
                     (member-equal acl2::a acl2::x))
                (funscopep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: funenvp-when-subsetp-equal

(defthm funenvp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (funenvp acl2::y))
                (equal (funenvp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (funenvp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (funenvp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: funenvp-of-set-difference-equal

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

Theorem: funenvp-of-intersection-equal-1

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

Theorem: funenvp-of-intersection-equal-2

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

Theorem: funenvp-of-union-equal

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

Theorem: funenvp-of-take

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

Theorem: funenvp-of-repeat

(defthm funenvp-of-repeat
  (iff (funenvp (repeat acl2::n acl2::x))
       (or (funscopep acl2::x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: funscopep-of-nth-when-funenvp

(defthm funscopep-of-nth-when-funenvp
  (implies (funenvp acl2::x)
           (funscopep (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: funenvp-of-update-nth

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

Theorem: funenvp-of-butlast

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

Theorem: funenvp-of-nthcdr

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

Theorem: funenvp-of-last

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

Theorem: funenvp-of-remove

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

Theorem: funenvp-of-revappend

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