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Driverlist

Driverlist-equiv

Basic equivalence relation for driverlist structures.

Definitions and Theorems

Function: driverlist-equiv$inline

(defun driverlist-equiv$inline (x y)
  (declare (xargs :guard (and (driverlist-p x)
                              (driverlist-p y))))
  (equal (driverlist-fix x)
         (driverlist-fix y)))

Theorem: driverlist-equiv-is-an-equivalence

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

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

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

Theorem: driverlist-fix-under-driverlist-equiv

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

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

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

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

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

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

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

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

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