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