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• Backref

Backref-equiv

Basic equivalence relation for backref structures.

Definitions and Theorems

Function: backref-equiv$inline

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

Theorem: backref-equiv-is-an-equivalence

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

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

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

Theorem: backref-fix-under-backref-equiv

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

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

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

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

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

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

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

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

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