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• Object

Object-equiv

Basic equivalence relation for object structures.

Definitions and Theorems

Function: object-equiv$inline

(defun object-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (objectp acl2::x)
                              (objectp acl2::y))))
  (equal (object-fix acl2::x)
         (object-fix acl2::y)))

Theorem: object-equiv-is-an-equivalence

(defthm object-equiv-is-an-equivalence
  (and (booleanp (object-equiv x y))
       (object-equiv x x)
       (implies (object-equiv x y)
                (object-equiv y x))
       (implies (and (object-equiv x y)
                     (object-equiv y z))
                (object-equiv x z)))
  :rule-classes (:equivalence))

Theorem: object-equiv-implies-equal-object-fix-1

(defthm object-equiv-implies-equal-object-fix-1
  (implies (object-equiv acl2::x x-equiv)
           (equal (object-fix acl2::x)
                  (object-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: object-fix-under-object-equiv

(defthm object-fix-under-object-equiv
  (object-equiv (object-fix acl2::x)
                acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-object-fix-1-forward-to-object-equiv

(defthm equal-of-object-fix-1-forward-to-object-equiv
  (implies (equal (object-fix acl2::x) acl2::y)
           (object-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-object-fix-2-forward-to-object-equiv

(defthm equal-of-object-fix-2-forward-to-object-equiv
  (implies (equal acl2::x (object-fix acl2::y))
           (object-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: object-equiv-of-object-fix-1-forward

(defthm object-equiv-of-object-fix-1-forward
  (implies (object-equiv (object-fix acl2::x)
                         acl2::y)
           (object-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: object-equiv-of-object-fix-2-forward

(defthm object-equiv-of-object-fix-2-forward
  (implies (object-equiv acl2::x (object-fix acl2::y))
           (object-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)