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• Termfn-listp

Termfn-listp-basics

Basic theorems about termfn-listp, generated by std::deflist.

Definitions and Theorems

Theorem: termfn-listp-of-cons

(defthm termfn-listp-of-cons
  (equal (termfn-listp (cons a x) wrld)
         (and (termfnp a wrld)
              (termfn-listp x wrld)))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-cdr-when-termfn-listp

(defthm termfn-listp-of-cdr-when-termfn-listp
  (implies (termfn-listp (double-rewrite x) wrld)
           (termfn-listp (cdr x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-when-not-consp

(defthm termfn-listp-when-not-consp
  (implies (not (consp x))
           (equal (termfn-listp x wrld) (not x)))
  :rule-classes ((:rewrite)))

Theorem: termfnp-of-car-when-termfn-listp

(defthm termfnp-of-car-when-termfn-listp
  (implies (termfn-listp x wrld)
           (iff (termfnp (car x) wrld)
                (or (consp x) (termfnp nil wrld))))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-termfn-listp

(defthm true-listp-when-termfn-listp
  (implies (termfn-listp x wrld)
           (true-listp x))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-list-fix

(defthm termfn-listp-of-list-fix
  (implies (termfn-listp x wrld)
           (termfn-listp (list-fix x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-sfix

(defthm termfn-listp-of-sfix
  (iff (termfn-listp (set::sfix x) wrld)
       (or (termfn-listp x wrld)
           (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-insert

(defthm termfn-listp-of-insert
  (iff (termfn-listp (set::insert a x) wrld)
       (and (termfn-listp (set::sfix x) wrld)
            (termfnp a wrld)))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-delete

(defthm termfn-listp-of-delete
  (implies (termfn-listp x wrld)
           (termfn-listp (set::delete k x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-mergesort

(defthm termfn-listp-of-mergesort
  (iff (termfn-listp (set::mergesort x) wrld)
       (termfn-listp (list-fix x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-union

(defthm termfn-listp-of-union
  (iff (termfn-listp (set::union x y) wrld)
       (and (termfn-listp (set::sfix x) wrld)
            (termfn-listp (set::sfix y) wrld)))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-intersect-1

(defthm termfn-listp-of-intersect-1
  (implies (termfn-listp x wrld)
           (termfn-listp (set::intersect x y)
                         wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-intersect-2

(defthm termfn-listp-of-intersect-2
  (implies (termfn-listp y wrld)
           (termfn-listp (set::intersect x y)
                         wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-difference

(defthm termfn-listp-of-difference
  (implies (termfn-listp x wrld)
           (termfn-listp (set::difference x y)
                         wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-duplicated-members

(defthm termfn-listp-of-duplicated-members
  (implies (termfn-listp x wrld)
           (termfn-listp (duplicated-members x)
                         wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-rev

(defthm termfn-listp-of-rev
  (equal (termfn-listp (rev x) wrld)
         (termfn-listp (list-fix x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-append

(defthm termfn-listp-of-append
  (equal (termfn-listp (append a b) wrld)
         (and (termfn-listp (list-fix a) wrld)
              (termfn-listp b wrld)))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-rcons

(defthm termfn-listp-of-rcons
  (iff (termfn-listp (rcons a x) wrld)
       (and (termfnp a wrld)
            (termfn-listp (list-fix x) wrld)))
  :rule-classes ((:rewrite)))

Theorem: termfnp-when-member-equal-of-termfn-listp

(defthm termfnp-when-member-equal-of-termfn-listp
  (and (implies (and (member-equal a x)
                     (termfn-listp x wrld))
                (termfnp a wrld))
       (implies (and (termfn-listp x wrld)
                     (member-equal a x))
                (termfnp a wrld)))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-when-subsetp-equal

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

Theorem: termfn-listp-of-set-difference-equal

(defthm termfn-listp-of-set-difference-equal
  (implies (termfn-listp x wrld)
           (termfn-listp (set-difference-equal x y)
                         wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-intersection-equal-1

(defthm termfn-listp-of-intersection-equal-1
  (implies (termfn-listp (double-rewrite x) wrld)
           (termfn-listp (intersection-equal x y)
                         wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-intersection-equal-2

(defthm termfn-listp-of-intersection-equal-2
  (implies (termfn-listp (double-rewrite y) wrld)
           (termfn-listp (intersection-equal x y)
                         wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-union-equal

(defthm termfn-listp-of-union-equal
  (equal (termfn-listp (union-equal x y) wrld)
         (and (termfn-listp (list-fix x) wrld)
              (termfn-listp (double-rewrite y) wrld)))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-take

(defthm termfn-listp-of-take
  (implies (termfn-listp (double-rewrite x) wrld)
           (iff (termfn-listp (take n x) wrld)
                (or (termfnp nil wrld)
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-repeat

(defthm termfn-listp-of-repeat
  (iff (termfn-listp (repeat n x) wrld)
       (or (termfnp x wrld) (zp n)))
  :rule-classes ((:rewrite)))

Theorem: termfnp-of-nth-when-termfn-listp

(defthm termfnp-of-nth-when-termfn-listp
  (implies (and (termfn-listp x wrld)
                (< (nfix n) (len x)))
           (termfnp (nth n x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-update-nth

(defthm termfn-listp-of-update-nth
  (implies (termfn-listp (double-rewrite x) wrld)
           (iff (termfn-listp (update-nth n y x) wrld)
                (and (termfnp y wrld)
                     (or (<= (nfix n) (len x))
                         (termfnp nil wrld)))))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-butlast

(defthm termfn-listp-of-butlast
  (implies (termfn-listp (double-rewrite x) wrld)
           (termfn-listp (butlast x n) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-nthcdr

(defthm termfn-listp-of-nthcdr
  (implies (termfn-listp (double-rewrite x) wrld)
           (termfn-listp (nthcdr n x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-last

(defthm termfn-listp-of-last
  (implies (termfn-listp (double-rewrite x) wrld)
           (termfn-listp (last x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-remove

(defthm termfn-listp-of-remove
  (implies (termfn-listp x wrld)
           (termfn-listp (remove a x) wrld))
  :rule-classes ((:rewrite)))

Theorem: termfn-listp-of-revappend

(defthm termfn-listp-of-revappend
  (equal (termfn-listp (revappend x y) wrld)
         (and (termfn-listp (list-fix x) wrld)
              (termfn-listp y wrld)))
  :rule-classes ((:rewrite)))