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Transunit

Transunit-equiv

Basic equivalence relation for transunit structures.

Definitions and Theorems

Function: transunit-equiv$inline

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

Theorem: transunit-equiv-is-an-equivalence

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

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

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

Theorem: transunit-fix-under-transunit-equiv

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

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

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

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

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

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

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

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

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