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• Maybe-smtlink-hint

Maybe-smtlink-hint-equiv

Basic equivalence relation for maybe-smtlink-hint structures.

Definitions and Theorems

Function: maybe-smtlink-hint-equiv$inline

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

Theorem: maybe-smtlink-hint-equiv-is-an-equivalence

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

Theorem: maybe-smtlink-hint-equiv-implies-equal-maybe-smtlink-hint-fix-1

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

Theorem: maybe-smtlink-hint-fix-under-maybe-smtlink-hint-equiv

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

Theorem: equal-of-maybe-smtlink-hint-fix-1-forward-to-maybe-smtlink-hint-equiv

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

Theorem: equal-of-maybe-smtlink-hint-fix-2-forward-to-maybe-smtlink-hint-equiv

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

Theorem: maybe-smtlink-hint-equiv-of-maybe-smtlink-hint-fix-1-forward

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

Theorem: maybe-smtlink-hint-equiv-of-maybe-smtlink-hint-fix-2-forward

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