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• Visibility

Visibility-equiv

Basic equivalence relation for visibility structures.

Definitions and Theorems

Function: visibility-equiv$inline

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

Theorem: visibility-equiv-is-an-equivalence

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

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

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

Theorem: visibility-fix-under-visibility-equiv

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

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

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

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

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

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

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

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

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