This is an ordinary defenum.
Function:
(defun env$-field-p (x) (declare (xargs :guard t)) (or (eq x ':alist) (eq x ':bitarr) (eq x ':obj-alist)))
Theorem: type-when-env$-field-p
(defthm type-when-env$-field-p (implies (env$-field-p x) (if (symbolp x) (if (not (equal x 't)) (not (equal x 'nil)) 'nil) 'nil)) :rule-classes :compound-recognizer)