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Segment-driver

Segment-driver-fix

Fixing function for segment-driver structures.

Signature
(segment-driver-fix x) → new-x
Arguments: x — Guard (segment-driver-p x).
Returns: new-x — Type (segment-driver-p new-x).

Definitions and Theorems

Function: segment-driver-fix$inline

(defun segment-driver-fix$inline (x)
  (declare (xargs :guard (segment-driver-p x)))
  (let ((__function__ 'segment-driver-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((lsb (nfix (std::prod-car (std::prod-car x))))
              (width (pos-fix (std::prod-cdr (std::prod-car x))))
              (value (svex-fix (std::prod-car (std::prod-cdr x))))
              (strength (nfix (std::prod-cdr (std::prod-cdr x)))))
           (std::prod-cons (std::prod-cons lsb width)
                           (std::prod-cons value strength)))
         :exec x)))

Theorem: segment-driver-p-of-segment-driver-fix

(defthm segment-driver-p-of-segment-driver-fix
  (b* ((new-x (segment-driver-fix$inline x)))
    (segment-driver-p new-x))
  :rule-classes :rewrite)

Theorem: segment-driver-fix-when-segment-driver-p

(defthm segment-driver-fix-when-segment-driver-p
  (implies (segment-driver-p x)
           (equal (segment-driver-fix x) x)))

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)