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