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