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Maybe-vex-p

Signature
(maybe-vex-p x) → *

Definitions and Theorems

Function: maybe-vex-p

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

Function: maybe-vex-fix

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

Function: maybe-vex-equiv$inline

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

Theorem: maybe-vex-equiv-is-an-equivalence

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

Theorem: maybe-vex-equiv-implies-equal-maybe-vex-fix-1

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

Theorem: maybe-vex-fix-under-maybe-vex-equiv

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