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Maybe-3bits-p

Signature
(maybe-3bits-p x) → *

Definitions and Theorems

Function: maybe-3bits-p

(defun maybe-3bits-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'maybe-3bits-p))
    (declare (ignorable __function__))
    (or (not x) (3bits-p x))))

Function: maybe-3bits-fix

(defun maybe-3bits-fix (x)
  (declare (xargs :guard (maybe-3bits-p x)))
  (let ((__function__ 'maybe-3bits-fix))
    (declare (ignorable __function__))
    (mbe :logic (if (maybe-3bits-p x) x 'nil)
         :exec x)))

Function: maybe-3bits-equiv$inline

(defun maybe-3bits-equiv$inline (x y)
  (declare (xargs :guard (and (maybe-3bits-p x)
                              (maybe-3bits-p y))))
  (equal (maybe-3bits-fix x)
         (maybe-3bits-fix y)))

Theorem: maybe-3bits-equiv-is-an-equivalence

(defthm maybe-3bits-equiv-is-an-equivalence
  (and (booleanp (maybe-3bits-equiv x y))
       (maybe-3bits-equiv x x)
       (implies (maybe-3bits-equiv x y)
                (maybe-3bits-equiv y x))
       (implies (and (maybe-3bits-equiv x y)
                     (maybe-3bits-equiv y z))
                (maybe-3bits-equiv x z)))
  :rule-classes (:equivalence))

Theorem: maybe-3bits-equiv-implies-equal-maybe-3bits-fix-1

(defthm maybe-3bits-equiv-implies-equal-maybe-3bits-fix-1
  (implies (maybe-3bits-equiv x x-equiv)
           (equal (maybe-3bits-fix x)
                  (maybe-3bits-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: maybe-3bits-fix-under-maybe-3bits-equiv

(defthm maybe-3bits-fix-under-maybe-3bits-equiv
  (maybe-3bits-equiv (maybe-3bits-fix x)
                     x)
  :rule-classes (:rewrite :rewrite-quoted-constant))