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Segment-driverlist

Segment-driverlist-equiv

Basic equivalence relation for segment-driverlist structures.

Definitions and Theorems

Function: segment-driverlist-equiv$inline

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

Theorem: segment-driverlist-equiv-is-an-equivalence

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

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

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

Theorem: segment-driverlist-fix-under-segment-driverlist-equiv

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

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

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

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

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

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

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

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

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