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• Sd-problemlist-p

Sd-problemlist-p-basics

Basic theorems about sd-problemlist-p, generated by deflist.

Definitions and Theorems

Theorem: sd-problemlist-p-of-cons

(defthm sd-problemlist-p-of-cons
  (equal (sd-problemlist-p (cons acl2::a acl2::x))
         (and (sd-problem-p acl2::a)
              (sd-problemlist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-cdr-when-sd-problemlist-p

(defthm sd-problemlist-p-of-cdr-when-sd-problemlist-p
  (implies (sd-problemlist-p (double-rewrite acl2::x))
           (sd-problemlist-p (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-when-not-consp

(defthm sd-problemlist-p-when-not-consp
  (implies (not (consp acl2::x))
           (sd-problemlist-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: sd-problem-p-of-car-when-sd-problemlist-p

(defthm sd-problem-p-of-car-when-sd-problemlist-p
  (implies (sd-problemlist-p acl2::x)
           (iff (sd-problem-p (car acl2::x))
                (or (consp acl2::x)
                    (sd-problem-p nil))))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-append

(defthm sd-problemlist-p-of-append
  (equal (sd-problemlist-p (append acl2::a acl2::b))
         (and (sd-problemlist-p acl2::a)
              (sd-problemlist-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-list-fix

(defthm sd-problemlist-p-of-list-fix
  (equal (sd-problemlist-p (list-fix acl2::x))
         (sd-problemlist-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-sfix

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

Theorem: sd-problemlist-p-of-insert

(defthm sd-problemlist-p-of-insert
  (iff (sd-problemlist-p (insert acl2::a acl2::x))
       (and (sd-problemlist-p (sfix acl2::x))
            (sd-problem-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-delete

(defthm sd-problemlist-p-of-delete
  (implies (sd-problemlist-p acl2::x)
           (sd-problemlist-p (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-mergesort

(defthm sd-problemlist-p-of-mergesort
  (iff (sd-problemlist-p (mergesort acl2::x))
       (sd-problemlist-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-union

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

Theorem: sd-problemlist-p-of-intersect-1

(defthm sd-problemlist-p-of-intersect-1
  (implies (sd-problemlist-p acl2::x)
           (sd-problemlist-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-intersect-2

(defthm sd-problemlist-p-of-intersect-2
  (implies (sd-problemlist-p acl2::y)
           (sd-problemlist-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-difference

(defthm sd-problemlist-p-of-difference
  (implies (sd-problemlist-p acl2::x)
           (sd-problemlist-p (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-duplicated-members

(defthm sd-problemlist-p-of-duplicated-members
  (implies (sd-problemlist-p acl2::x)
           (sd-problemlist-p (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-rev

(defthm sd-problemlist-p-of-rev
  (equal (sd-problemlist-p (rev acl2::x))
         (sd-problemlist-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-rcons

(defthm sd-problemlist-p-of-rcons
  (iff (sd-problemlist-p (acl2::rcons acl2::a acl2::x))
       (and (sd-problem-p acl2::a)
            (sd-problemlist-p (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sd-problem-p-when-member-equal-of-sd-problemlist-p

(defthm sd-problem-p-when-member-equal-of-sd-problemlist-p
  (and (implies (and (member-equal acl2::a acl2::x)
                     (sd-problemlist-p acl2::x))
                (sd-problem-p acl2::a))
       (implies (and (sd-problemlist-p acl2::x)
                     (member-equal acl2::a acl2::x))
                (sd-problem-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-when-subsetp-equal

(defthm sd-problemlist-p-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (sd-problemlist-p acl2::y))
                (sd-problemlist-p acl2::x))
       (implies (and (sd-problemlist-p acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (sd-problemlist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-set-equiv-congruence

(defthm sd-problemlist-p-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (sd-problemlist-p acl2::x)
                  (sd-problemlist-p acl2::y)))
  :rule-classes :congruence)

Theorem: sd-problemlist-p-of-set-difference-equal

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

Theorem: sd-problemlist-p-of-intersection-equal-1

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

Theorem: sd-problemlist-p-of-intersection-equal-2

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

Theorem: sd-problemlist-p-of-union-equal

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

Theorem: sd-problemlist-p-of-take

(defthm sd-problemlist-p-of-take
  (implies (sd-problemlist-p (double-rewrite acl2::x))
           (iff (sd-problemlist-p (take acl2::n acl2::x))
                (or (sd-problem-p nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-repeat

(defthm sd-problemlist-p-of-repeat
  (iff (sd-problemlist-p (repeat acl2::n acl2::x))
       (or (sd-problem-p acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: sd-problem-p-of-nth-when-sd-problemlist-p

(defthm sd-problem-p-of-nth-when-sd-problemlist-p
  (implies (and (sd-problemlist-p acl2::x)
                (< (nfix acl2::n) (len acl2::x)))
           (sd-problem-p (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-update-nth

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

Theorem: sd-problemlist-p-of-butlast

(defthm sd-problemlist-p-of-butlast
  (implies (sd-problemlist-p (double-rewrite acl2::x))
           (sd-problemlist-p (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-nthcdr

(defthm sd-problemlist-p-of-nthcdr
  (implies (sd-problemlist-p (double-rewrite acl2::x))
           (sd-problemlist-p (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-last

(defthm sd-problemlist-p-of-last
  (implies (sd-problemlist-p (double-rewrite acl2::x))
           (sd-problemlist-p (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-remove

(defthm sd-problemlist-p-of-remove
  (implies (sd-problemlist-p acl2::x)
           (sd-problemlist-p (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sd-problemlist-p-of-revappend

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