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• Omaps

Emptyp

Check if an omap is empty.

Signature

(emptyp map) → yes/no

Arguments

map — Guard (mapp map).

Returns

yes/no — Type (booleanp yes/no).

This is similar to set::emptyp for osets.

Definitions and Theorems

Function: emptyp

(defun emptyp (map)
  (declare (xargs :guard (mapp map)))
  (let ((__function__ 'emptyp))
    (declare (ignorable __function__))
    (null (mfix map))))

Theorem: booleanp-of-emptyp

(defthm booleanp-of-emptyp
  (b* ((yes/no (emptyp map)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: mapp-when-not-emptyp

(defthm mapp-when-not-emptyp
  (implies (not (emptyp map)) (mapp map)))

Theorem: mfix-when-emptyp

(defthm mfix-when-emptyp
  (implies (emptyp x)
           (equal (mfix x) nil)))

Theorem: mapp-non-nil-implies-not-emptyp

(defthm mapp-non-nil-implies-not-emptyp
  (implies (and (mapp map) map)
           (not (emptyp map))))

Theorem: acl2-count-head-when-not-emptyp

(defthm acl2-count-head-when-not-emptyp
  (implies (not (emptyp map))
           (< (+ (acl2-count (car (car map)))
                 (acl2-count (cdr (car map))))
              (acl2-count map))))

Theorem: emptyp-of-mfix-map

(defthm emptyp-of-mfix-map
  (equal (emptyp (mfix map))
         (emptyp map)))

Theorem: emptyp-mequiv-congruence-on-map

(defthm emptyp-mequiv-congruence-on-map
  (implies (mequiv map map-equiv)
           (equal (emptyp map) (emptyp map-equiv)))
  :rule-classes :congruence)