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• Finalization

Finalization-equiv

Basic equivalence relation for finalization structures.

Definitions and Theorems

Function: finalization-equiv$inline

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

Theorem: finalization-equiv-is-an-equivalence

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

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

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

Theorem: finalization-fix-under-finalization-equiv

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

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

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

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

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

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

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

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

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