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Driver

Driver-equiv

Basic equivalence relation for driver structures.

Definitions and Theorems

Function: driver-equiv$inline

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

Theorem: driver-equiv-is-an-equivalence

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

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

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

Theorem: driver-fix-under-driver-equiv

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

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

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

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

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

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

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

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

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