		Search-engine friendly clone of the ACL2 documentation.

Pos-list

Fixtype of lists of positive integers.

Definitions and Theorems

Theorem: pos-listp-of-cons

(defthm pos-listp-of-cons
        (equal (pos-listp (cons a x))
               (and (posp a) (pos-listp x)))
        :rule-classes ((:rewrite)))

Theorem: pos-listp-of-cdr-when-pos-listp

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

Theorem: pos-listp-when-not-consp

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

Theorem: posp-of-car-when-pos-listp

(defthm posp-of-car-when-pos-listp
        (implies (pos-listp x)
                 (iff (posp (car x)) (consp x)))
        :rule-classes ((:rewrite)))

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

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

Theorem: pos-listp-of-list-fix

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

Theorem: pos-listp-of-rev

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

Function: pos-list-fix$inline

(defun pos-list-fix$inline (x)
       (declare (xargs :guard (pos-listp x)))
       (let ((__function__ 'pos-list-fix))
            (declare (ignorable __function__))
            (mbe :logic (if (atom x)
                            nil
                            (cons (pos-fix (car x))
                                  (pos-list-fix (cdr x))))
                 :exec x)))

Theorem: pos-listp-of-pos-list-fix

(defthm pos-listp-of-pos-list-fix
        (b* ((fty::newx (pos-list-fix$inline x)))
            (pos-listp fty::newx))
        :rule-classes :rewrite)

Theorem: pos-list-fix-when-pos-listp

(defthm pos-list-fix-when-pos-listp
        (implies (pos-listp x)
                 (equal (pos-list-fix x) x)))

Function: pos-list-equiv$inline

(defun pos-list-equiv$inline (x y)
       (declare (xargs :guard (and (pos-listp x) (pos-listp y))))
       (equal (pos-list-fix x)
              (pos-list-fix y)))

Theorem: pos-list-equiv-is-an-equivalence

(defthm pos-list-equiv-is-an-equivalence
        (and (booleanp (pos-list-equiv x y))
             (pos-list-equiv x x)
             (implies (pos-list-equiv x y)
                      (pos-list-equiv y x))
             (implies (and (pos-list-equiv x y)
                           (pos-list-equiv y z))
                      (pos-list-equiv x z)))
        :rule-classes (:equivalence))

Theorem: pos-list-equiv-implies-equal-pos-list-fix-1

(defthm pos-list-equiv-implies-equal-pos-list-fix-1
        (implies (pos-list-equiv x x-equiv)
                 (equal (pos-list-fix x)
                        (pos-list-fix x-equiv)))
        :rule-classes (:congruence))

Theorem: pos-list-fix-under-pos-list-equiv

(defthm pos-list-fix-under-pos-list-equiv
        (pos-list-equiv (pos-list-fix x) x)
        :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-pos-list-fix-1-forward-to-pos-list-equiv

(defthm equal-of-pos-list-fix-1-forward-to-pos-list-equiv
        (implies (equal (pos-list-fix x) y)
                 (pos-list-equiv x y))
        :rule-classes :forward-chaining)

Theorem: equal-of-pos-list-fix-2-forward-to-pos-list-equiv

(defthm equal-of-pos-list-fix-2-forward-to-pos-list-equiv
        (implies (equal x (pos-list-fix y))
                 (pos-list-equiv x y))
        :rule-classes :forward-chaining)

Theorem: pos-list-equiv-of-pos-list-fix-1-forward

(defthm pos-list-equiv-of-pos-list-fix-1-forward
        (implies (pos-list-equiv (pos-list-fix x) y)
                 (pos-list-equiv x y))
        :rule-classes :forward-chaining)

Theorem: pos-list-equiv-of-pos-list-fix-2-forward

(defthm pos-list-equiv-of-pos-list-fix-2-forward
        (implies (pos-list-equiv x (pos-list-fix y))
                 (pos-list-equiv x y))
        :rule-classes :forward-chaining)

Theorem: car-of-pos-list-fix-x-under-pos-equiv

(defthm car-of-pos-list-fix-x-under-pos-equiv
        (pos-equiv (car (pos-list-fix x))
                   (car x)))

Theorem: car-pos-list-equiv-congruence-on-x-under-pos-equiv

(defthm car-pos-list-equiv-congruence-on-x-under-pos-equiv
        (implies (pos-list-equiv x x-equiv)
                 (pos-equiv (car x) (car x-equiv)))
        :rule-classes :congruence)

Theorem: cdr-of-pos-list-fix-x-under-pos-list-equiv

(defthm cdr-of-pos-list-fix-x-under-pos-list-equiv
        (pos-list-equiv (cdr (pos-list-fix x))
                        (cdr x)))

Theorem: cdr-pos-list-equiv-congruence-on-x-under-pos-list-equiv

(defthm cdr-pos-list-equiv-congruence-on-x-under-pos-list-equiv
        (implies (pos-list-equiv x x-equiv)
                 (pos-list-equiv (cdr x) (cdr x-equiv)))
        :rule-classes :congruence)

Theorem: cons-of-pos-fix-x-under-pos-list-equiv

(defthm cons-of-pos-fix-x-under-pos-list-equiv
        (pos-list-equiv (cons (pos-fix x) y)
                        (cons x y)))

Theorem: cons-pos-equiv-congruence-on-x-under-pos-list-equiv

(defthm cons-pos-equiv-congruence-on-x-under-pos-list-equiv
        (implies (pos-equiv x x-equiv)
                 (pos-list-equiv (cons x y)
                                 (cons x-equiv y)))
        :rule-classes :congruence)

Theorem: cons-of-pos-list-fix-y-under-pos-list-equiv

(defthm cons-of-pos-list-fix-y-under-pos-list-equiv
        (pos-list-equiv (cons x (pos-list-fix y))
                        (cons x y)))

Theorem: cons-pos-list-equiv-congruence-on-y-under-pos-list-equiv

(defthm cons-pos-list-equiv-congruence-on-y-under-pos-list-equiv
        (implies (pos-list-equiv y y-equiv)
                 (pos-list-equiv (cons x y)
                                 (cons x y-equiv)))
        :rule-classes :congruence)

Theorem: consp-of-pos-list-fix

(defthm consp-of-pos-list-fix
        (equal (consp (pos-list-fix x))
               (consp x)))

Theorem: pos-list-fix-under-iff

(defthm pos-list-fix-under-iff
        (iff (pos-list-fix x) (consp x)))

Theorem: pos-list-fix-of-cons

(defthm pos-list-fix-of-cons
        (equal (pos-list-fix (cons a x))
               (cons (pos-fix a) (pos-list-fix x))))

Theorem: len-of-pos-list-fix

(defthm len-of-pos-list-fix
        (equal (len (pos-list-fix x)) (len x)))

Theorem: pos-list-fix-of-append

(defthm pos-list-fix-of-append
        (equal (pos-list-fix (append std::a std::b))
               (append (pos-list-fix std::a)
                       (pos-list-fix std::b))))

Theorem: pos-list-fix-of-repeat

(defthm pos-list-fix-of-repeat
        (equal (pos-list-fix (repeat n x))
               (repeat n (pos-fix x))))

Theorem: list-equiv-refines-pos-list-equiv

(defthm list-equiv-refines-pos-list-equiv
        (implies (list-equiv x y)
                 (pos-list-equiv x y))
        :rule-classes :refinement)

Theorem: nth-of-pos-list-fix

(defthm nth-of-pos-list-fix
        (equal (nth n (pos-list-fix x))
               (if (< (nfix n) (len x))
                   (pos-fix (nth n x))
                   nil)))

Theorem: pos-list-equiv-implies-pos-list-equiv-append-1

(defthm pos-list-equiv-implies-pos-list-equiv-append-1
        (implies (pos-list-equiv x fty::x-equiv)
                 (pos-list-equiv (append x y)
                                 (append fty::x-equiv y)))
        :rule-classes (:congruence))

Theorem: pos-list-equiv-implies-pos-list-equiv-append-2

(defthm pos-list-equiv-implies-pos-list-equiv-append-2
        (implies (pos-list-equiv y fty::y-equiv)
                 (pos-list-equiv (append x y)
                                 (append x fty::y-equiv)))
        :rule-classes (:congruence))

Theorem: pos-list-equiv-implies-pos-list-equiv-nthcdr-2

(defthm pos-list-equiv-implies-pos-list-equiv-nthcdr-2
        (implies (pos-list-equiv l l-equiv)
                 (pos-list-equiv (nthcdr n l)
                                 (nthcdr n l-equiv)))
        :rule-classes (:congruence))

Theorem: pos-list-equiv-implies-pos-list-equiv-take-2

(defthm pos-list-equiv-implies-pos-list-equiv-take-2
        (implies (pos-list-equiv l l-equiv)
                 (pos-list-equiv (take n l)
                                 (take n l-equiv)))
        :rule-classes (:congruence))