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3vec

3vec-equiv

Equivalence up to 3vec-fix.

Definitions and Theorems

Function: 3vec-equiv$inline

(defun 3vec-equiv$inline (x y)
  (declare (xargs :guard (and (3vec-p! x) (3vec-p! y))))
  (equal (3vec-fix x) (3vec-fix y)))

Theorem: 3vec-equiv-is-an-equivalence

(defthm 3vec-equiv-is-an-equivalence
  (and (booleanp (3vec-equiv x y))
       (3vec-equiv x x)
       (implies (3vec-equiv x y)
                (3vec-equiv y x))
       (implies (and (3vec-equiv x y) (3vec-equiv y z))
                (3vec-equiv x z)))
  :rule-classes (:equivalence))

Theorem: 3vec-equiv-implies-equal-3vec-fix-1

(defthm 3vec-equiv-implies-equal-3vec-fix-1
  (implies (3vec-equiv x x-equiv)
           (equal (3vec-fix x) (3vec-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: 3vec-fix-under-3vec-equiv

(defthm 3vec-fix-under-3vec-equiv
  (3vec-equiv (3vec-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-3vec-fix-1-forward-to-3vec-equiv

(defthm equal-of-3vec-fix-1-forward-to-3vec-equiv
  (implies (equal (3vec-fix x) y)
           (3vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-3vec-fix-2-forward-to-3vec-equiv

(defthm equal-of-3vec-fix-2-forward-to-3vec-equiv
  (implies (equal x (3vec-fix y))
           (3vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 3vec-equiv-of-3vec-fix-1-forward

(defthm 3vec-equiv-of-3vec-fix-1-forward
  (implies (3vec-equiv (3vec-fix x) y)
           (3vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 3vec-equiv-of-3vec-fix-2-forward

(defthm 3vec-equiv-of-3vec-fix-2-forward
  (implies (3vec-equiv x (3vec-fix y))
           (3vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 4vec-equiv-refines-3vec-equiv

(defthm 4vec-equiv-refines-3vec-equiv
  (implies (4vec-equiv x y)
           (3vec-equiv x y))
  :rule-classes (:refinement))