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• Lambda-listp

Lambda-listp-basics

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

Definitions and Theorems

Theorem: lambda-listp-of-cons

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

Theorem: lambda-listp-of-cdr-when-lambda-listp

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

Theorem: lambda-listp-when-not-consp

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

Theorem: lambdap-of-car-when-lambda-listp

(defthm lambdap-of-car-when-lambda-listp
  (implies (lambda-listp x wrld)
           (iff (lambdap (car x) wrld) (consp x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-lambda-listp

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

Theorem: lambda-listp-of-list-fix

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

Theorem: lambda-listp-of-sfix

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

Theorem: lambda-listp-of-insert

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

Theorem: lambda-listp-of-delete

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

Theorem: lambda-listp-of-mergesort

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

Theorem: lambda-listp-of-union

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

Theorem: lambda-listp-of-intersect-1

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

Theorem: lambda-listp-of-intersect-2

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

Theorem: lambda-listp-of-difference

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

Theorem: lambda-listp-of-duplicated-members

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

Theorem: lambda-listp-of-rev

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

Theorem: lambda-listp-of-append

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

Theorem: lambda-listp-of-rcons

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

Theorem: lambdap-when-member-equal-of-lambda-listp

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

Theorem: lambda-listp-when-subsetp-equal

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

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

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

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

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

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

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

Theorem: lambda-listp-of-union-equal

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

Theorem: lambda-listp-of-take

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

Theorem: lambda-listp-of-repeat

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

Theorem: lambdap-of-nth-when-lambda-listp

(defthm lambdap-of-nth-when-lambda-listp
  (implies (lambda-listp x wrld)
           (iff (lambdap (nth n x) wrld)
                (< (nfix n) (len x))))
  :rule-classes ((:rewrite)))

Theorem: lambda-listp-of-update-nth

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

Theorem: lambda-listp-of-butlast

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

Theorem: lambda-listp-of-nthcdr

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

Theorem: lambda-listp-of-last

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

Theorem: lambda-listp-of-remove

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

Theorem: lambda-listp-of-revappend

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