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• Member

Member-equiv

Basic equivalence relation for member structures.

Definitions and Theorems

Function: member-equiv$inline

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

Theorem: member-equiv-is-an-equivalence

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

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

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

Theorem: member-fix-under-member-equiv

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

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

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

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

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

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

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

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

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