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Rex-p

Signature
(rex-p x) → *

Definitions and Theorems

Function: rex-p

(defun rex-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'rex-p))
    (declare (ignorable __function__))
    (or (not x)
        (member-equal x '(:w :not-w)))))

Function: rex-fix

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

Function: rex-equiv$inline

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

Theorem: rex-equiv-is-an-equivalence

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

Theorem: rex-equiv-implies-equal-rex-fix-1

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

Theorem: rex-fix-under-rex-equiv

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