		Search-engine friendly clone of the ACL2 documentation.

A4vec

A4vec-fix

Fixing function for a4vec structures.

Signature
(a4vec-fix x) → new-x
Arguments: x — Guard (a4vec-p x).
Returns: new-x — Type (a4vec-p new-x).

Definitions and Theorems

Function: a4vec-fix$inline

(defun a4vec-fix$inline (x)
  (declare (xargs :guard (a4vec-p x)))
  (let ((__function__ 'a4vec-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((upper (list-fix (car x)))
              (lower (list-fix (cdr x))))
           (cons upper lower))
         :exec x)))

Theorem: a4vec-p-of-a4vec-fix

(defthm a4vec-p-of-a4vec-fix
  (b* ((new-x (a4vec-fix$inline x)))
    (a4vec-p new-x))
  :rule-classes :rewrite)

Theorem: a4vec-fix-when-a4vec-p

(defthm a4vec-fix-when-a4vec-p
  (implies (a4vec-p x)
           (equal (a4vec-fix x) x)))

Function: a4vec-equiv$inline

(defun a4vec-equiv$inline (x y)
  (declare (xargs :guard (and (a4vec-p x) (a4vec-p y))))
  (equal (a4vec-fix x) (a4vec-fix y)))

Theorem: a4vec-equiv-is-an-equivalence

(defthm a4vec-equiv-is-an-equivalence
  (and (booleanp (a4vec-equiv x y))
       (a4vec-equiv x x)
       (implies (a4vec-equiv x y)
                (a4vec-equiv y x))
       (implies (and (a4vec-equiv x y)
                     (a4vec-equiv y z))
                (a4vec-equiv x z)))
  :rule-classes (:equivalence))

Theorem: a4vec-equiv-implies-equal-a4vec-fix-1

(defthm a4vec-equiv-implies-equal-a4vec-fix-1
  (implies (a4vec-equiv x x-equiv)
           (equal (a4vec-fix x)
                  (a4vec-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: a4vec-fix-under-a4vec-equiv

(defthm a4vec-fix-under-a4vec-equiv
  (a4vec-equiv (a4vec-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-a4vec-fix-1-forward-to-a4vec-equiv

(defthm equal-of-a4vec-fix-1-forward-to-a4vec-equiv
  (implies (equal (a4vec-fix x) y)
           (a4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-a4vec-fix-2-forward-to-a4vec-equiv

(defthm equal-of-a4vec-fix-2-forward-to-a4vec-equiv
  (implies (equal x (a4vec-fix y))
           (a4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: a4vec-equiv-of-a4vec-fix-1-forward

(defthm a4vec-equiv-of-a4vec-fix-1-forward
  (implies (a4vec-equiv (a4vec-fix x) y)
           (a4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: a4vec-equiv-of-a4vec-fix-2-forward

(defthm a4vec-equiv-of-a4vec-fix-2-forward
  (implies (a4vec-equiv x (a4vec-fix y))
           (a4vec-equiv x y))
  :rule-classes :forward-chaining)