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