		Search-engine friendly clone of the ACL2 documentation.

Bag

This is similar to the fixtype set::set of osets.

Definitions and Theorems

Function: bequiv$inline

(defun bequiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (bagp acl2::x) (bagp acl2::y))))
  (equal (bfix acl2::x) (bfix acl2::y)))

Theorem: bequiv-is-an-equivalence

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

Theorem: bequiv-implies-equal-bfix-1

(defthm bequiv-implies-equal-bfix-1
  (implies (bequiv acl2::x x-equiv)
           (equal (bfix acl2::x) (bfix x-equiv)))
  :rule-classes (:congruence))

Theorem: bfix-under-bequiv

(defthm bfix-under-bequiv
  (bequiv (bfix acl2::x) acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bfix-1-forward-to-bequiv

(defthm equal-of-bfix-1-forward-to-bequiv
  (implies (equal (bfix acl2::x) acl2::y)
           (bequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-bfix-2-forward-to-bequiv

(defthm equal-of-bfix-2-forward-to-bequiv
  (implies (equal acl2::x (bfix acl2::y))
           (bequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bequiv-of-bfix-1-forward

(defthm bequiv-of-bfix-1-forward
  (implies (bequiv (bfix acl2::x) acl2::y)
           (bequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bequiv-of-bfix-2-forward

(defthm bequiv-of-bfix-2-forward
  (implies (bequiv acl2::x (bfix acl2::y))
           (bequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)