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