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• Pointer

Pointer-equiv

Basic equivalence relation for pointer structures.

Definitions and Theorems

Function: pointer-equiv$inline

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

Theorem: pointer-equiv-is-an-equivalence

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

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

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

Theorem: pointer-fix-under-pointer-equiv

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

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

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

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

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

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

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

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

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