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• Address+pos-set

Address+pos-set-equiv

Basic equivalence relation for address+pos-set structures.

Definitions and Theorems

Function: address+pos-set-equiv$inline

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

Theorem: address+pos-set-equiv-is-an-equivalence

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

Theorem: address+pos-set-equiv-implies-equal-address+pos-set-fix-1

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

Theorem: address+pos-set-fix-under-address+pos-set-equiv

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

Theorem: equal-of-address+pos-set-fix-1-forward-to-address+pos-set-equiv

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

Theorem: equal-of-address+pos-set-fix-2-forward-to-address+pos-set-equiv

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

Theorem: address+pos-set-equiv-of-address+pos-set-fix-1-forward

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

Theorem: address+pos-set-equiv-of-address+pos-set-fix-2-forward

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