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• Jcmember

Jcmember-equiv

Basic equivalence relation for jcmember structures.

Definitions and Theorems

Function: jcmember-equiv$inline

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

Theorem: jcmember-equiv-is-an-equivalence

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

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

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

Theorem: jcmember-fix-under-jcmember-equiv

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

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

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

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

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

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

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

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

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