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• Maybe-backref

Maybe-backref-equiv

Basic equivalence relation for maybe-backref structures.

Definitions and Theorems

Function: maybe-backref-equiv$inline

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

Theorem: maybe-backref-equiv-is-an-equivalence

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

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

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

Theorem: maybe-backref-fix-under-maybe-backref-equiv

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

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

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

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

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

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

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

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

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