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Maybe-pos-equiv

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

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

Definitions and Theorems

Function: maybe-pos-equiv$inline

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

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

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

Theorem: maybe-pos-equiv-implies-equal-maybe-posp-fix-1

(defthm maybe-pos-equiv-implies-equal-maybe-posp-fix-1
  (implies (maybe-pos-equiv x x-equiv)
           (equal (maybe-posp-fix x)
                  (maybe-posp-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: maybe-posp-fix-under-maybe-pos-equiv

(defthm maybe-posp-fix-under-maybe-pos-equiv
  (maybe-pos-equiv (maybe-posp-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))