		Search-engine friendly clone of the ACL2 documentation.

Lhbit

Lhbit-fix

Fixing function for lhbit structures.

Signature
(lhbit-fix x) → new-x
Arguments: x — Guard (lhbit-p x).
Returns: new-x — Type (lhbit-p new-x).

Definitions and Theorems

Function: lhbit-fix$inline

(defun lhbit-fix$inline (x)
  (declare (xargs :guard (lhbit-p x)))
  (let ((__function__ 'lhbit-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (lhbit-kind x)
           (:z (cons :z (list)))
           (:var (b* ((name (svar-fix (std::prod-car (cdr x))))
                      (idx (nfix (std::prod-cdr (cdr x)))))
                   (hons :var (std::prod-hons name idx)))))
         :exec x)))

Theorem: lhbit-p-of-lhbit-fix

(defthm lhbit-p-of-lhbit-fix
  (b* ((new-x (lhbit-fix$inline x)))
    (lhbit-p new-x))
  :rule-classes :rewrite)

Theorem: lhbit-fix-when-lhbit-p

(defthm lhbit-fix-when-lhbit-p
  (implies (lhbit-p x)
           (equal (lhbit-fix x) x)))

Function: lhbit-equiv$inline

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

Theorem: lhbit-equiv-is-an-equivalence

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

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

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

Theorem: lhbit-fix-under-lhbit-equiv

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

Theorem: equal-of-lhbit-fix-1-forward-to-lhbit-equiv

(defthm equal-of-lhbit-fix-1-forward-to-lhbit-equiv
  (implies (equal (lhbit-fix x) y)
           (lhbit-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-lhbit-fix-2-forward-to-lhbit-equiv

(defthm equal-of-lhbit-fix-2-forward-to-lhbit-equiv
  (implies (equal x (lhbit-fix y))
           (lhbit-equiv x y))
  :rule-classes :forward-chaining)

Theorem: lhbit-equiv-of-lhbit-fix-1-forward

(defthm lhbit-equiv-of-lhbit-fix-1-forward
  (implies (lhbit-equiv (lhbit-fix x) y)
           (lhbit-equiv x y))
  :rule-classes :forward-chaining)

Theorem: lhbit-equiv-of-lhbit-fix-2-forward

(defthm lhbit-equiv-of-lhbit-fix-2-forward
  (implies (lhbit-equiv x (lhbit-fix y))
           (lhbit-equiv x y))
  :rule-classes :forward-chaining)

Theorem: lhbit-kind$inline-of-lhbit-fix-x

(defthm lhbit-kind$inline-of-lhbit-fix-x
  (equal (lhbit-kind$inline (lhbit-fix x))
         (lhbit-kind$inline x)))

Theorem: lhbit-kind$inline-lhbit-equiv-congruence-on-x

(defthm lhbit-kind$inline-lhbit-equiv-congruence-on-x
  (implies (lhbit-equiv x x-equiv)
           (equal (lhbit-kind$inline x)
                  (lhbit-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-lhbit-fix

(defthm consp-of-lhbit-fix
  (consp (lhbit-fix x))
  :rule-classes :type-prescription)