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Omaps

In*

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

Signature
(in* keys map) → yes/no
Arguments: keys — Guard (setp keys).; map — Guard (mapp map).
Returns: yes/no — Type (booleanp yes/no).

Definitions and Theorems

Function: in*

(defun in* (keys map)
  (declare (xargs :guard (and (setp keys) (mapp map))))
  (let ((__function__ 'in*))
    (declare (ignorable __function__))
    (cond ((set::emptyp keys) t)
          (t (and (assoc (set::head keys) map)
                  (in* (set::tail keys) map))))))

Theorem: booleanp-of-in*

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

Theorem: in*-when-left-emptyp

(defthm in*-when-left-emptyp
  (implies (set::emptyp keys)
           (in* keys map)))

Theorem: in*-when-rigth-emptyp

(defthm in*-when-rigth-emptyp
  (implies (emptyp map)
           (equal (in* keys map)
                  (set::emptyp keys))))

Theorem: in*-of-tail

(defthm in*-of-tail
  (implies (in* keys map)
           (in* (set::tail keys) map)))