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• Smtlink-hint

Smtlink-hint-equiv

Basic equivalence relation for smtlink-hint structures.

Definitions and Theorems

Function: smtlink-hint-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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