		Search-engine friendly clone of the ACL2 documentation.

4vec

4vec-equiv

Equivalence relation for 4vecs.

Definitions and Theorems

Function: 4vec-equiv$inline

(defun 4vec-equiv$inline (x y)
  (declare (xargs :guard (and (4vec-p x) (4vec-p y))))
  (equal (4vec-fix x) (4vec-fix y)))

Theorem: 4vec-equiv-is-an-equivalence

(defthm 4vec-equiv-is-an-equivalence
  (and (booleanp (4vec-equiv x y))
       (4vec-equiv x x)
       (implies (4vec-equiv x y)
                (4vec-equiv y x))
       (implies (and (4vec-equiv x y) (4vec-equiv y z))
                (4vec-equiv x z)))
  :rule-classes (:equivalence))

Theorem: 4vec-equiv-implies-equal-4vec-fix-1

(defthm 4vec-equiv-implies-equal-4vec-fix-1
  (implies (4vec-equiv x x-equiv)
           (equal (4vec-fix x) (4vec-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: 4vec-fix-under-4vec-equiv

(defthm 4vec-fix-under-4vec-equiv
  (4vec-equiv (4vec-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-4vec-fix-1-forward-to-4vec-equiv

(defthm equal-of-4vec-fix-1-forward-to-4vec-equiv
  (implies (equal (4vec-fix x) y)
           (4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-4vec-fix-2-forward-to-4vec-equiv

(defthm equal-of-4vec-fix-2-forward-to-4vec-equiv
  (implies (equal x (4vec-fix y))
           (4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 4vec-equiv-of-4vec-fix-1-forward

(defthm 4vec-equiv-of-4vec-fix-1-forward
  (implies (4vec-equiv (4vec-fix x) y)
           (4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 4vec-equiv-of-4vec-fix-2-forward

(defthm 4vec-equiv-of-4vec-fix-2-forward
  (implies (4vec-equiv x (4vec-fix y))
           (4vec-equiv x y))
  :rule-classes :forward-chaining)