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• Label

Label-equiv

Basic equivalence relation for label structures.

Definitions and Theorems

Function: label-equiv$inline

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

Theorem: label-equiv-is-an-equivalence

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

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

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

Theorem: label-fix-under-label-equiv

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

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

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

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

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

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

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

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

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