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Axi-map

Axi-map-p

Recognizer for axi-map.

Signature
(axi-map-p x) → *

Definitions and Theorems

Function: axi-map-p

(defun axi-map-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'axi-map-p))
    (declare (ignorable __function__))
    (if (atom x)
        t
      (and (consp (car x))
           (axi-term-p (caar x))
           (axi-lit-p (cdar x))
           (axi-map-p (cdr x))))))

Theorem: axi-map-p-of-rev

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

Theorem: axi-map-p-of-list-fix

(defthm axi-map-p-of-list-fix
  (equal (axi-map-p (acl2::list-fix x))
         (axi-map-p x))
  :rule-classes ((:rewrite)))

Theorem: axi-map-p-of-append

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

Theorem: axi-map-p-when-not-consp

(defthm axi-map-p-when-not-consp
  (implies (not (consp x)) (axi-map-p x))
  :rule-classes ((:rewrite)))

Theorem: axi-map-p-of-cdr-when-axi-map-p

(defthm axi-map-p-of-cdr-when-axi-map-p
  (implies (axi-map-p (double-rewrite x))
           (axi-map-p (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: axi-map-p-of-cons

(defthm axi-map-p-of-cons
  (equal (axi-map-p (cons acl2::a x))
         (and (and (consp acl2::a)
                   (axi-term-p (car acl2::a))
                   (axi-lit-p (cdr acl2::a)))
              (axi-map-p x)))
  :rule-classes ((:rewrite)))

Theorem: axi-map-p-of-fast-alist-clean

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

Theorem: axi-map-p-of-hons-shrink-alist

(defthm axi-map-p-of-hons-shrink-alist
  (implies (and (axi-map-p x) (axi-map-p acl2::y))
           (axi-map-p (hons-shrink-alist x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: axi-map-p-of-hons-acons

(defthm axi-map-p-of-hons-acons
  (equal (axi-map-p (hons-acons acl2::a acl2::n x))
         (and (axi-term-p acl2::a)
              (axi-lit-p acl2::n)
              (axi-map-p x)))
  :rule-classes ((:rewrite)))

Theorem: axi-lit-p-of-cdr-of-hons-assoc-equal-when-axi-map-p

(defthm axi-lit-p-of-cdr-of-hons-assoc-equal-when-axi-map-p
  (implies (axi-map-p x)
           (iff (axi-lit-p (cdr (hons-assoc-equal acl2::k x)))
                (or (hons-assoc-equal acl2::k x)
                    (axi-lit-p nil))))
  :rule-classes ((:rewrite)))

Theorem: axi-lit-p-of-cdar-when-axi-map-p

(defthm axi-lit-p-of-cdar-when-axi-map-p
  (implies (axi-map-p x)
           (iff (axi-lit-p (cdar x))
                (or (consp x) (axi-lit-p nil))))
  :rule-classes ((:rewrite)))

Theorem: axi-term-p-of-caar-when-axi-map-p

(defthm axi-term-p-of-caar-when-axi-map-p
  (implies (axi-map-p x)
           (iff (axi-term-p (caar x))
                (or (consp x) (axi-term-p nil))))
  :rule-classes ((:rewrite)))