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• Name-set

Name-set-equiv

Basic equivalence relation for name-set structures.

Definitions and Theorems

Function: name-set-equiv$inline

(defun name-set-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (name-setp acl2::x)
                              (name-setp acl2::y))))
  (equal (name-set-fix acl2::x)
         (name-set-fix acl2::y)))

Theorem: name-set-equiv-is-an-equivalence

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

Theorem: name-set-equiv-implies-equal-name-set-fix-1

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

Theorem: name-set-fix-under-name-set-equiv

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

Theorem: equal-of-name-set-fix-1-forward-to-name-set-equiv

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

Theorem: equal-of-name-set-fix-2-forward-to-name-set-equiv

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

Theorem: name-set-equiv-of-name-set-fix-1-forward

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

Theorem: name-set-equiv-of-name-set-fix-2-forward

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