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Heap

Heapp

Recognizer for heap.

Signature
(heapp x) → *

Definitions and Theorems

Function: heapp

(defun heapp (x)
  (declare (xargs :guard t))
  (if (atom x)
      (null x)
    (and (consp (car x))
         (addressp (caar x))
         (valuep (cdar x))
         (or (null (cdr x))
             (and (consp (cdr x))
                  (consp (cadr x))
                  (acl2::fast-<< (caar x) (caadr x))
                  (heapp (cdr x)))))))

Theorem: booleanp-of-heapp

(defthm booleanp-of-heapp
  (booleanp (heapp x)))

Theorem: mapp-when-heapp

(defthm mapp-when-heapp
  (implies (heapp x) (omap::mapp x))
  :rule-classes (:rewrite :forward-chaining))

Theorem: heapp-of-tail

(defthm heapp-of-tail
  (implies (heapp x)
           (heapp (omap::tail x))))

Theorem: addressp-of-head-key-when-heapp

(defthm addressp-of-head-key-when-heapp
  (implies (and (heapp x) (not (omap::emptyp x)))
           (addressp (mv-nth 0 (omap::head x)))))

Theorem: valuep-of-head-val-when-heapp

(defthm valuep-of-head-val-when-heapp
  (implies (and (heapp x) (not (omap::emptyp x)))
           (valuep (mv-nth 1 (omap::head x)))))

Theorem: heapp-of-update

(defthm heapp-of-update
  (implies (and (heapp x) (addressp k) (valuep v))
           (heapp (omap::update k v x))))

Theorem: heapp-of-update*

(defthm heapp-of-update*
  (implies (and (heapp x) (heapp y))
           (heapp (omap::update* x y))))

Theorem: heapp-of-delete

(defthm heapp-of-delete
  (implies (heapp x)
           (heapp (omap::delete k x))))

Theorem: heapp-of-delete*

(defthm heapp-of-delete*
  (implies (heapp x)
           (heapp (omap::delete* k x))))

Theorem: addressp-when-assoc-heapp-binds-free-x

(defthm addressp-when-assoc-heapp-binds-free-x
  (implies (and (omap::assoc k x) (heapp x))
           (addressp k)))

Theorem: addressp-of-car-of-assoc-heapp

(defthm addressp-of-car-of-assoc-heapp
  (implies (and (heapp x) (omap::assoc k x))
           (addressp (car (omap::assoc k x)))))

Theorem: valuep-of-cdr-of-assoc-heapp

(defthm valuep-of-cdr-of-assoc-heapp
  (implies (and (heapp x) (omap::assoc k x))
           (valuep (cdr (omap::assoc k x)))))

Theorem: valuep-of-lookup-when-heapp

(defthm valuep-of-lookup-when-heapp
  (implies (and (heapp x) (omap::assoc k x))
           (valuep (omap::lookup k x))))