Basic equivalence relation for s4veclist structures.
Function:
(defun s4veclist-equiv$inline (x y) (declare (xargs :guard (and (s4veclist-p x) (s4veclist-p y)))) (equal (s4veclist-fix x) (s4veclist-fix y)))
Theorem:
(defthm s4veclist-equiv-is-an-equivalence (and (booleanp (s4veclist-equiv x y)) (s4veclist-equiv x x) (implies (s4veclist-equiv x y) (s4veclist-equiv y x)) (implies (and (s4veclist-equiv x y) (s4veclist-equiv y z)) (s4veclist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm s4veclist-equiv-implies-equal-s4veclist-fix-1 (implies (s4veclist-equiv x x-equiv) (equal (s4veclist-fix x) (s4veclist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm s4veclist-fix-under-s4veclist-equiv (s4veclist-equiv (s4veclist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-s4veclist-fix-1-forward-to-s4veclist-equiv (implies (equal (s4veclist-fix x) y) (s4veclist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-s4veclist-fix-2-forward-to-s4veclist-equiv (implies (equal x (s4veclist-fix y)) (s4veclist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm s4veclist-equiv-of-s4veclist-fix-1-forward (implies (s4veclist-equiv (s4veclist-fix x) y) (s4veclist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm s4veclist-equiv-of-s4veclist-fix-2-forward (implies (s4veclist-equiv x (s4veclist-fix y)) (s4veclist-equiv x y)) :rule-classes :forward-chaining)