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• Pseudo-var

Pseudo-var-fix

Signature

(pseudo-var-fix x) → new-x

Arguments

x — Guard (pseudo-var-p x).

Returns

new-x — Type (pseudo-var-p new-x).

Definitions and Theorems

Function: pseudo-var-fix

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

Theorem: pseudo-var-p-of-pseudo-var-fix

(defthm pseudo-var-p-of-pseudo-var-fix
  (b* ((new-x (pseudo-var-fix x)))
    (pseudo-var-p new-x))
  :rule-classes :type-prescription)

Theorem: pseudo-var-fix-when-pseudo-var-p

(defthm pseudo-var-fix-when-pseudo-var-p
  (implies (pseudo-var-p x)
           (equal (pseudo-var-fix x) x)))

Function: pseudo-var-equiv$inline

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

Theorem: pseudo-var-equiv-is-an-equivalence

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

Theorem: pseudo-var-equiv-implies-equal-pseudo-var-fix-1

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

Theorem: pseudo-var-fix-under-pseudo-var-equiv

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

Theorem: pseudo-var-fix-of-pseudo-var-fix-x

(defthm pseudo-var-fix-of-pseudo-var-fix-x
  (equal (pseudo-var-fix (pseudo-var-fix x))
         (pseudo-var-fix x)))

Theorem: pseudo-var-fix-pseudo-var-equiv-congruence-on-x

(defthm pseudo-var-fix-pseudo-var-equiv-congruence-on-x
  (implies (pseudo-var-equiv x x-equiv)
           (equal (pseudo-var-fix x)
                  (pseudo-var-fix x-equiv)))
  :rule-classes :congruence)