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Address-pos-map

Address-pos-mapp

Recognizer for address-pos-map.

Signature
(address-pos-mapp x) → *

Definitions and Theorems

Function: address-pos-mapp

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

Theorem: booleanp-of-address-pos-mapp

(defthm booleanp-of-address-pos-mapp
  (booleanp (address-pos-mapp x)))

Theorem: mapp-when-address-pos-mapp

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

Theorem: address-pos-mapp-of-tail

(defthm address-pos-mapp-of-tail
  (implies (address-pos-mapp x)
           (address-pos-mapp (omap::tail x))))

Theorem: addressp-of-head-key-when-address-pos-mapp

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

Theorem: posp-of-head-val-when-address-pos-mapp

(defthm posp-of-head-val-when-address-pos-mapp
  (implies (and (address-pos-mapp x)
                (not (omap::emptyp x)))
           (posp (mv-nth 1 (omap::head x)))))

Theorem: address-pos-mapp-of-update

(defthm address-pos-mapp-of-update
  (implies (and (address-pos-mapp x)
                (addressp k)
                (posp v))
           (address-pos-mapp (omap::update k v x))))

Theorem: address-pos-mapp-of-update*

(defthm address-pos-mapp-of-update*
  (implies (and (address-pos-mapp x)
                (address-pos-mapp y))
           (address-pos-mapp (omap::update* x y))))

Theorem: address-pos-mapp-of-delete

(defthm address-pos-mapp-of-delete
  (implies (address-pos-mapp x)
           (address-pos-mapp (omap::delete k x))))

Theorem: address-pos-mapp-of-delete*

(defthm address-pos-mapp-of-delete*
  (implies (address-pos-mapp x)
           (address-pos-mapp (omap::delete* k x))))

Theorem: addressp-when-assoc-address-pos-mapp-binds-free-x

(defthm addressp-when-assoc-address-pos-mapp-binds-free-x
  (implies (and (omap::assoc k x)
                (address-pos-mapp x))
           (addressp k)))

Theorem: addressp-of-car-of-assoc-address-pos-mapp

(defthm addressp-of-car-of-assoc-address-pos-mapp
  (implies (and (address-pos-mapp x)
                (omap::assoc k x))
           (addressp (car (omap::assoc k x)))))

Theorem: posp-of-cdr-of-assoc-address-pos-mapp

(defthm posp-of-cdr-of-assoc-address-pos-mapp
  (implies (and (address-pos-mapp x)
                (omap::assoc k x))
           (posp (cdr (omap::assoc k x)))))

Theorem: posp-of-lookup-when-address-pos-mapp

(defthm posp-of-lookup-when-address-pos-mapp
  (implies (and (address-pos-mapp x)
                (omap::assoc k x))
           (posp (omap::lookup k x))))