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Maybe-integer-equiv

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

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

Definitions and Theorems

Function: maybe-integer-equiv$inline

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

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

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

Theorem: maybe-integer-equiv-implies-equal-maybe-integerp-fix-1

(defthm maybe-integer-equiv-implies-equal-maybe-integerp-fix-1
  (implies (maybe-integer-equiv x x-equiv)
           (equal (maybe-integerp-fix x)
                  (maybe-integerp-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: maybe-integerp-fix-under-maybe-integer-equiv

(defthm maybe-integerp-fix-under-maybe-integer-equiv
  (maybe-integer-equiv (maybe-integerp-fix x)
                       x)
  :rule-classes (:rewrite :rewrite-quoted-constant))