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