Basic equivalence relation for 4veclistlist structures.
Function:
(defun 4veclistlist-equiv$inline (x y) (declare (xargs :guard (and (4veclistlist-p x) (4veclistlist-p y)))) (equal (4veclistlist-fix x) (4veclistlist-fix y)))
Theorem:
(defthm 4veclistlist-equiv-is-an-equivalence (and (booleanp (4veclistlist-equiv x y)) (4veclistlist-equiv x x) (implies (4veclistlist-equiv x y) (4veclistlist-equiv y x)) (implies (and (4veclistlist-equiv x y) (4veclistlist-equiv y z)) (4veclistlist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm 4veclistlist-equiv-implies-equal-4veclistlist-fix-1 (implies (4veclistlist-equiv x x-equiv) (equal (4veclistlist-fix x) (4veclistlist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm 4veclistlist-fix-under-4veclistlist-equiv (4veclistlist-equiv (4veclistlist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-4veclistlist-fix-1-forward-to-4veclistlist-equiv (implies (equal (4veclistlist-fix x) y) (4veclistlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-4veclistlist-fix-2-forward-to-4veclistlist-equiv (implies (equal x (4veclistlist-fix y)) (4veclistlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm 4veclistlist-equiv-of-4veclistlist-fix-1-forward (implies (4veclistlist-equiv (4veclistlist-fix x) y) (4veclistlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm 4veclistlist-equiv-of-4veclistlist-fix-2-forward (implies (4veclistlist-equiv x (4veclistlist-fix y)) (4veclistlist-equiv x y)) :rule-classes :forward-chaining)