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• Expr-list-formalp

Expr-list-formalp-basics

Basic theorems about expr-list-formalp, generated by std::deflist.

Definitions and Theorems

Theorem: expr-list-formalp-of-cons

(defthm expr-list-formalp-of-cons
  (equal (expr-list-formalp (cons acl2::a acl2::x))
         (and (expr-formalp acl2::a)
              (expr-list-formalp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-cdr-when-expr-list-formalp

(defthm expr-list-formalp-of-cdr-when-expr-list-formalp
  (implies (expr-list-formalp (double-rewrite acl2::x))
           (expr-list-formalp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-when-not-consp

(defthm expr-list-formalp-when-not-consp
  (implies (not (consp acl2::x))
           (expr-list-formalp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: expr-formalp-of-car-when-expr-list-formalp

(defthm expr-formalp-of-car-when-expr-list-formalp
  (implies (expr-list-formalp acl2::x)
           (iff (expr-formalp (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-append

(defthm expr-list-formalp-of-append
  (equal (expr-list-formalp (append acl2::a acl2::b))
         (and (expr-list-formalp acl2::a)
              (expr-list-formalp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-list-fix

(defthm expr-list-formalp-of-list-fix
  (equal (expr-list-formalp (list-fix acl2::x))
         (expr-list-formalp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-sfix

(defthm expr-list-formalp-of-sfix
  (iff (expr-list-formalp (sfix acl2::x))
       (or (expr-list-formalp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-insert

(defthm expr-list-formalp-of-insert
  (iff (expr-list-formalp (insert acl2::a acl2::x))
       (and (expr-list-formalp (sfix acl2::x))
            (expr-formalp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-delete

(defthm expr-list-formalp-of-delete
  (implies (expr-list-formalp acl2::x)
           (expr-list-formalp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-mergesort

(defthm expr-list-formalp-of-mergesort
  (iff (expr-list-formalp (mergesort acl2::x))
       (expr-list-formalp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-union

(defthm expr-list-formalp-of-union
  (iff (expr-list-formalp (union acl2::x acl2::y))
       (and (expr-list-formalp (sfix acl2::x))
            (expr-list-formalp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-intersect-1

(defthm expr-list-formalp-of-intersect-1
  (implies (expr-list-formalp acl2::x)
           (expr-list-formalp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-intersect-2

(defthm expr-list-formalp-of-intersect-2
  (implies (expr-list-formalp acl2::y)
           (expr-list-formalp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-difference

(defthm expr-list-formalp-of-difference
  (implies (expr-list-formalp acl2::x)
           (expr-list-formalp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-duplicated-members

(defthm expr-list-formalp-of-duplicated-members
  (implies (expr-list-formalp acl2::x)
           (expr-list-formalp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-rev

(defthm expr-list-formalp-of-rev
  (equal (expr-list-formalp (rev acl2::x))
         (expr-list-formalp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-rcons

(defthm expr-list-formalp-of-rcons
  (iff (expr-list-formalp (rcons acl2::a acl2::x))
       (and (expr-formalp acl2::a)
            (expr-list-formalp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expr-formalp-when-member-equal-of-expr-list-formalp

(defthm expr-formalp-when-member-equal-of-expr-list-formalp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (expr-list-formalp acl2::x))
                (expr-formalp acl2::a))
       (implies (and (expr-list-formalp acl2::x)
                     (member-equal acl2::a acl2::x))
                (expr-formalp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-when-subsetp-equal

(defthm expr-list-formalp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (expr-list-formalp acl2::y))
                (expr-list-formalp acl2::x))
       (implies (and (expr-list-formalp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (expr-list-formalp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-set-equiv-congruence

(defthm expr-list-formalp-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (expr-list-formalp acl2::x)
                  (expr-list-formalp acl2::y)))
  :rule-classes :congruence)

Theorem: expr-list-formalp-of-set-difference-equal

(defthm expr-list-formalp-of-set-difference-equal
  (implies
       (expr-list-formalp acl2::x)
       (expr-list-formalp (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-intersection-equal-1

(defthm expr-list-formalp-of-intersection-equal-1
  (implies (expr-list-formalp (double-rewrite acl2::x))
           (expr-list-formalp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-intersection-equal-2

(defthm expr-list-formalp-of-intersection-equal-2
  (implies (expr-list-formalp (double-rewrite acl2::y))
           (expr-list-formalp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-union-equal

(defthm expr-list-formalp-of-union-equal
  (equal (expr-list-formalp (union-equal acl2::x acl2::y))
         (and (expr-list-formalp (list-fix acl2::x))
              (expr-list-formalp (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-take

(defthm expr-list-formalp-of-take
  (implies (expr-list-formalp (double-rewrite acl2::x))
           (iff (expr-list-formalp (take acl2::n acl2::x))
                (or (expr-formalp nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-repeat

(defthm expr-list-formalp-of-repeat
  (iff (expr-list-formalp (repeat acl2::n acl2::x))
       (or (expr-formalp acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: expr-formalp-of-nth-when-expr-list-formalp

(defthm expr-formalp-of-nth-when-expr-list-formalp
  (implies (expr-list-formalp acl2::x)
           (iff (expr-formalp (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-update-nth

(defthm expr-list-formalp-of-update-nth
  (implies
       (expr-list-formalp (double-rewrite acl2::x))
       (iff (expr-list-formalp (update-nth acl2::n acl2::y acl2::x))
            (and (expr-formalp acl2::y)
                 (or (<= (nfix acl2::n) (len acl2::x))
                     (expr-formalp nil)))))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-butlast

(defthm expr-list-formalp-of-butlast
  (implies (expr-list-formalp (double-rewrite acl2::x))
           (expr-list-formalp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-nthcdr

(defthm expr-list-formalp-of-nthcdr
  (implies (expr-list-formalp (double-rewrite acl2::x))
           (expr-list-formalp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-last

(defthm expr-list-formalp-of-last
  (implies (expr-list-formalp (double-rewrite acl2::x))
           (expr-list-formalp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-remove

(defthm expr-list-formalp-of-remove
  (implies (expr-list-formalp acl2::x)
           (expr-list-formalp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expr-list-formalp-of-revappend

(defthm expr-list-formalp-of-revappend
  (equal (expr-list-formalp (revappend acl2::x acl2::y))
         (and (expr-list-formalp (list-fix acl2::x))
              (expr-list-formalp acl2::y)))
  :rule-classes ((:rewrite)))