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Segment-driver

Segment-driver-equiv

Basic equivalence relation for segment-driver structures.

Definitions and Theorems

Function: segment-driver-equiv$inline

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

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

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

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

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

Theorem: segment-driver-fix-under-segment-driver-equiv

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

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

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

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

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

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

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

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

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