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• Maybe-integer

Maybe-integer-equiv

Basic equivalence relation for maybe-integer structures.

Definitions and Theorems

Function: maybe-integer-equiv$inline

(defun maybe-integer-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (maybe-integer-p acl2::x)
                              (maybe-integer-p acl2::y))))
  (equal (maybe-integer-fix acl2::x)
         (maybe-integer-fix acl2::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-integer-fix-1

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

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

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

Theorem: equal-of-maybe-integer-fix-1-forward-to-maybe-integer-equiv

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

Theorem: equal-of-maybe-integer-fix-2-forward-to-maybe-integer-equiv

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

Theorem: maybe-integer-equiv-of-maybe-integer-fix-1-forward

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

Theorem: maybe-integer-equiv-of-maybe-integer-fix-2-forward

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