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