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