		Search-engine friendly clone of the ACL2 documentation.

Maybe-nat-equiv

(maybe-natp-equiv x y) is an equivalence relation for maybe-natps, i.e., equality up to maybe-natp-fix.

Performance note. In the execution, this is just an inlined call of eql.

Definitions and Theorems

Function: maybe-nat-equiv$inline

(defun maybe-nat-equiv$inline (x y)
  (declare (xargs :guard (and (maybe-natp x) (maybe-natp y))))
  (eql (maybe-natp-fix x)
       (maybe-natp-fix y)))

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

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

Theorem: maybe-nat-equiv-implies-equal-maybe-natp-fix-1

(defthm maybe-nat-equiv-implies-equal-maybe-natp-fix-1
  (implies (maybe-nat-equiv x x-equiv)
           (equal (maybe-natp-fix x)
                  (maybe-natp-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: maybe-natp-fix-under-maybe-nat-equiv

(defthm maybe-natp-fix-under-maybe-nat-equiv
  (maybe-nat-equiv (maybe-natp-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))