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