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• Enumer

Enumer-equiv

Basic equivalence relation for enumer structures.

Definitions and Theorems

Function: enumer-equiv$inline

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

Theorem: enumer-equiv-is-an-equivalence

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

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

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

Theorem: enumer-fix-under-enumer-equiv

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

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

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

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

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

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

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

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

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