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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))