Basic equivalence relation for vl-directionlist structures.
Function:
(defun vl-directionlist-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (vl-directionlist-p acl2::x) (vl-directionlist-p acl2::y)))) (equal (vl-directionlist-fix acl2::x) (vl-directionlist-fix acl2::y)))
Theorem:
(defthm vl-directionlist-equiv-is-an-equivalence (and (booleanp (vl-directionlist-equiv x y)) (vl-directionlist-equiv x x) (implies (vl-directionlist-equiv x y) (vl-directionlist-equiv y x)) (implies (and (vl-directionlist-equiv x y) (vl-directionlist-equiv y z)) (vl-directionlist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm vl-directionlist-equiv-implies-equal-vl-directionlist-fix-1 (implies (vl-directionlist-equiv acl2::x x-equiv) (equal (vl-directionlist-fix acl2::x) (vl-directionlist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm vl-directionlist-fix-under-vl-directionlist-equiv (vl-directionlist-equiv (vl-directionlist-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-vl-directionlist-fix-1-forward-to-vl-directionlist-equiv (implies (equal (vl-directionlist-fix acl2::x) acl2::y) (vl-directionlist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-vl-directionlist-fix-2-forward-to-vl-directionlist-equiv (implies (equal acl2::x (vl-directionlist-fix acl2::y)) (vl-directionlist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm vl-directionlist-equiv-of-vl-directionlist-fix-1-forward (implies (vl-directionlist-equiv (vl-directionlist-fix acl2::x) acl2::y) (vl-directionlist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm vl-directionlist-equiv-of-vl-directionlist-fix-2-forward (implies (vl-directionlist-equiv acl2::x (vl-directionlist-fix acl2::y)) (vl-directionlist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)