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• Certificate

Certificate-equiv

Basic equivalence relation for certificate structures.

Definitions and Theorems

Function: certificate-equiv$inline

(defun certificate-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (certificatep acl2::x)
                              (certificatep acl2::y))))
  (equal (certificate-fix acl2::x)
         (certificate-fix acl2::y)))

Theorem: certificate-equiv-is-an-equivalence

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

Theorem: certificate-equiv-implies-equal-certificate-fix-1

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

Theorem: certificate-fix-under-certificate-equiv

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

Theorem: equal-of-certificate-fix-1-forward-to-certificate-equiv

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

Theorem: equal-of-certificate-fix-2-forward-to-certificate-equiv

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

Theorem: certificate-equiv-of-certificate-fix-1-forward

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

Theorem: certificate-equiv-of-certificate-fix-2-forward

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