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Nat-option-listp

Nat-option-listp-basics

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

Definitions and Theorems

Theorem: nat-option-listp-of-cons

(defthm nat-option-listp-of-cons
  (equal (nat-option-listp (cons a x))
         (and (nat-optionp a)
              (nat-option-listp x)))
  :rule-classes ((:rewrite)))

Theorem: nat-option-listp-of-cdr-when-nat-option-listp

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

Theorem: nat-option-listp-when-not-consp

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

Theorem: nat-optionp-of-car-when-nat-option-listp

(defthm nat-optionp-of-car-when-nat-option-listp
  (implies (nat-option-listp x)
           (iff (nat-optionp (car x))
                (or (consp x) (nat-optionp nil))))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-nat-option-listp-compound-recognizer

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

Theorem: nat-option-listp-of-list-fix

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

Theorem: nat-option-listp-of-rev

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