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