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• Omaps

List-notin

Check if none of the keys in a list is in an omap.

Signature

(list-notin keys map) → yes/no

Arguments

keys — Guard (true-listp keys).

map — Guard (mapp map).

Returns

yes/no — Type (booleanp yes/no).

Definitions and Theorems

Function: list-notin

(defun list-notin (keys map)
  (declare (xargs :guard (and (true-listp keys) (mapp map))))
  (let ((__function__ 'list-notin))
    (declare (ignorable __function__))
    (or (endp keys)
        (and (not (assoc (car keys) map))
             (list-notin (cdr keys) map)))))

Theorem: booleanp-of-list-notin

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

Theorem: not-assoc-map-when-member-of-list-notin

(defthm not-assoc-map-when-member-of-list-notin
  (implies (and (list-notin keys map)
                (member-equal key keys))
           (not (assoc key map))))

Theorem: list-notin-of-list-fix-keys

(defthm list-notin-of-list-fix-keys
  (equal (list-notin (list-fix keys) map)
         (list-notin keys map)))

Theorem: list-notin-list-equiv-congruence-on-keys

(defthm list-notin-list-equiv-congruence-on-keys
  (implies (list-equiv keys keys-equiv)
           (equal (list-notin keys map)
                  (list-notin keys-equiv map)))
  :rule-classes :congruence)

Theorem: list-notin-of-mfix-map

(defthm list-notin-of-mfix-map
  (equal (list-notin keys (mfix map))
         (list-notin keys map)))

Theorem: list-notin-mequiv-congruence-on-map

(defthm list-notin-mequiv-congruence-on-map
  (implies (mequiv map map-equiv)
           (equal (list-notin keys map)
                  (list-notin keys map-equiv)))
  :rule-classes :congruence)