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