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• Std/alists/abstract

Keyval-alist-p

Generic typed list recognizer function.

Definitions and Theorems

Function: keyval-alist-p

(defun keyval-alist-p (x)
  (if (atom x)
      (keyval-alist-final-cdr-p x)
    (and (consp (car x))
         (keytype-p (caar x))
         (valtype-p (cdar x))
         (keyval-alist-p (cdr x)))))

Theorem: keytype-p-of-caar-when-keyval-alist-p-when-keytype-p-nil

(defthm keytype-p-of-caar-when-keyval-alist-p-when-keytype-p-nil
  (implies (and (keytype-p nil) (keyval-alist-p x))
           (keytype-p (caar x)))
  :rule-classes :rewrite)

Theorem: valtype-p-of-cdar-when-keyval-alist-p-when-valtype-p-nil

(defthm valtype-p-of-cdar-when-keyval-alist-p-when-valtype-p-nil
  (implies (and (valtype-p nil) (keyval-alist-p x))
           (valtype-p (cdar x)))
  :rule-classes :rewrite)

Theorem: keytype-p-of-caar-when-keyval-alist-p-when-not-keytype-p-nil-and-not-negated

(defthm
 keytype-p-of-caar-when-keyval-alist-p-when-not-keytype-p-nil-and-not-negated
 (implies (and (not (keytype-p nil))
               (keyval-alist-p x))
          (iff (keytype-p (caar x)) (consp x)))
 :rule-classes :rewrite)

Theorem: keytype-p-of-caar-when-keyval-alist-p-when-not-keytype-p-nil-and-negated

(defthm
 keytype-p-of-caar-when-keyval-alist-p-when-not-keytype-p-nil-and-negated
 (implies (and (not (keytype-p nil))
               (keyval-alist-p x))
          (iff (non-keytype-p (caar x))
               (not (consp x))))
 :rule-classes :rewrite)

Theorem: valtype-p-of-cdar-when-keyval-alist-p-when-not-valtype-p-nil-and-not-negated

(defthm
 valtype-p-of-cdar-when-keyval-alist-p-when-not-valtype-p-nil-and-not-negated
 (implies (and (not (valtype-p nil))
               (keyval-alist-p x))
          (iff (valtype-p (cdar x)) (consp x)))
 :rule-classes :rewrite)

Theorem: valtype-p-of-cdar-when-keyval-alist-p-when-not-valtype-p-nil-and-negated

(defthm
 valtype-p-of-cdar-when-keyval-alist-p-when-not-valtype-p-nil-and-negated
 (implies (and (not (valtype-p nil))
               (keyval-alist-p x))
          (iff (non-valtype-p (cdar x))
               (not (consp x))))
 :rule-classes :rewrite)

Theorem: keytype-p-of-caar-when-keyval-alist-p-when-unknown-nil

(defthm keytype-p-of-caar-when-keyval-alist-p-when-unknown-nil
  (implies (keyval-alist-p x)
           (iff (keytype-p (caar x))
                (or (consp x) (keytype-p nil))))
  :rule-classes :rewrite)

Theorem: keytype-p-of-caar-when-keyval-alist-p-when-unknown-nil-negated

(defthm
     keytype-p-of-caar-when-keyval-alist-p-when-unknown-nil-negated
  (implies (keyval-alist-p x)
           (iff (non-keytype-p (caar x))
                (and (not (consp x))
                     (non-keytype-p nil))))
  :rule-classes :rewrite)

Theorem: valtype-p-of-cdar-when-keyval-alist-p-when-unknown-nil

(defthm valtype-p-of-cdar-when-keyval-alist-p-when-unknown-nil
  (implies (keyval-alist-p x)
           (iff (valtype-p (cdar x))
                (or (consp x) (valtype-p nil))))
  :rule-classes :rewrite)

Theorem: valtype-p-of-cdar-when-keyval-alist-p-when-unknown-nil-negated

(defthm
     valtype-p-of-cdar-when-keyval-alist-p-when-unknown-nil-negated
  (implies (keyval-alist-p x)
           (iff (non-valtype-p (cdar x))
                (and (not (consp x))
                     (non-valtype-p nil))))
  :rule-classes :rewrite)

Theorem: alistp-when-keyval-alist-p-tau

(defthm alistp-when-keyval-alist-p-tau
  (implies (and (keyval-alist-p x)
                (not (keyval-alist-final-cdr-p t)))
           (alistp x))
  :rule-classes nil)

Theorem: alistp-when-keyval-alist-p-rewrite

(defthm alistp-when-keyval-alist-p-rewrite
  (implies (and (keyval-alist-p x)
                (not (keyval-alist-final-cdr-p t)))
           (alistp x))
  :rule-classes :rewrite)

Theorem: valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-unknown-valtype-p-nil

(defthm
 valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-unknown-valtype-p-nil
 (implies (keyval-alist-p x)
          (iff (valtype-p (cdr (hons-assoc-equal k x)))
               (or (hons-assoc-equal k x)
                   (valtype-p nil))))
 :rule-classes :rewrite)

Theorem: valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-valtype-p-nil

(defthm
 valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-valtype-p-nil
 (implies (and (keyval-alist-p x) (valtype-p nil))
          (valtype-p (cdr (hons-assoc-equal k x))))
 :rule-classes :rewrite)

Theorem: valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-not-valtype-p-nil

(defthm
 valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-not-valtype-p-nil
 (implies (and (keyval-alist-p x)
               (not (valtype-p nil)))
          (iff (valtype-p (cdr (hons-assoc-equal k x)))
               (hons-assoc-equal k x)))
 :rule-classes :rewrite)

Theorem: valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-unknown-valtype-p-nil-negated

(defthm
 valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-unknown-valtype-p-nil-negated
 (implies (keyval-alist-p x)
          (iff (non-valtype-p (cdr (hons-assoc-equal k x)))
               (not (or (hons-assoc-equal k x)
                        (valtype-p nil)))))
 :rule-classes :rewrite)

Theorem: valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-not-valtype-p-nil-negated

(defthm
 valtype-p-of-cdr-of-hons-assoc-equal-when-keyval-alist-p-not-valtype-p-nil-negated
 (implies (and (keyval-alist-p x)
               (not (valtype-p nil)))
          (iff (non-valtype-p (cdr (hons-assoc-equal k x)))
               (not (hons-assoc-equal k x))))
 :rule-classes :rewrite)

Theorem: keyval-alist-p-of-hons-acons

(defthm keyval-alist-p-of-hons-acons
  (equal (keyval-alist-p (hons-acons a n x))
         (and (keytype-p a)
              (valtype-p n)
              (keyval-alist-p x)))
  :rule-classes :rewrite)

Theorem: keyval-alist-p-of-hons-shrink-alist

(defthm keyval-alist-p-of-hons-shrink-alist
  (implies (and (keyval-alist-p x)
                (keyval-alist-p y))
           (keyval-alist-p (hons-shrink-alist x y)))
  :rule-classes :rewrite)

Theorem: keyval-alist-p-of-fast-alist-clean

(defthm keyval-alist-p-of-fast-alist-clean
  (implies (keyval-alist-p x)
           (keyval-alist-p (fast-alist-clean x)))
  :rule-classes :rewrite)

Theorem: alistp-of-put-assoc

(defthm alistp-of-put-assoc
  (implies (and (keyval-alist-p x)
                (not (keyval-alist-final-cdr-p t)))
           (iff (keyval-alist-p (put-assoc-equal name val x))
                (and (keytype-p name) (valtype-p val))))
  :rule-classes :rewrite)

Theorem: valtype-p-of-cdr-of-assoc-when-keyval-alist-p-valtype-p-nil

(defthm valtype-p-of-cdr-of-assoc-when-keyval-alist-p-valtype-p-nil
  (implies (and (keyval-alist-p x) (valtype-p nil))
           (valtype-p (cdr (assoc-equal k x))))
  :rule-classes :rewrite)

Theorem: alistp-of-remove-assoc

(defthm alistp-of-remove-assoc
  (implies (keyval-alist-p x)
           (keyval-alist-p (remove-assoc-equal name x)))
  :rule-classes :rewrite)