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Omaps

List-in

Check if every key in a list is in an omap.

Signature
(list-in 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-in

(defun list-in (keys map)
  (declare (xargs :guard (and (true-listp keys) (mapp map))))
  (let ((__function__ 'list-in))
    (declare (ignorable __function__))
    (cond ((endp keys) t)
          (t (and (assoc (car keys) map)
                  (list-in (cdr keys) map))))))

Theorem: booleanp-of-list-in

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

Theorem: list-in-of-list-fix-keys

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

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

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

Theorem: list-in-of-mfix-map

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

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

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