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Segment-driverlist

Segment-driverlist-fix

(segment-driverlist-fix x) is a usual fty list fixing function.

Signature
(segment-driverlist-fix x) → fty::newx
Arguments: x — Guard (segment-driverlist-p x).
Returns: fty::newx — Type (segment-driverlist-p fty::newx).

In the logic, we apply segment-driver-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: segment-driverlist-fix$inline

(defun segment-driverlist-fix$inline (x)
  (declare (xargs :guard (segment-driverlist-p x)))
  (let ((__function__ 'segment-driverlist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (segment-driver-fix (car x))
                 (segment-driverlist-fix (cdr x))))
         :exec x)))

Theorem: segment-driverlist-p-of-segment-driverlist-fix

(defthm segment-driverlist-p-of-segment-driverlist-fix
  (b* ((fty::newx (segment-driverlist-fix$inline x)))
    (segment-driverlist-p fty::newx))
  :rule-classes :rewrite)

Theorem: segment-driverlist-fix-when-segment-driverlist-p

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

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)

Theorem: car-of-segment-driverlist-fix-x-under-segment-driver-equiv

(defthm car-of-segment-driverlist-fix-x-under-segment-driver-equiv
  (segment-driver-equiv (car (segment-driverlist-fix x))
                        (car x)))

Theorem: car-segment-driverlist-equiv-congruence-on-x-under-segment-driver-equiv

(defthm
 car-segment-driverlist-equiv-congruence-on-x-under-segment-driver-equiv
 (implies (segment-driverlist-equiv x x-equiv)
          (segment-driver-equiv (car x)
                                (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-segment-driverlist-fix-x-under-segment-driverlist-equiv

(defthm
     cdr-of-segment-driverlist-fix-x-under-segment-driverlist-equiv
  (segment-driverlist-equiv (cdr (segment-driverlist-fix x))
                            (cdr x)))

Theorem: cdr-segment-driverlist-equiv-congruence-on-x-under-segment-driverlist-equiv

(defthm
 cdr-segment-driverlist-equiv-congruence-on-x-under-segment-driverlist-equiv
 (implies (segment-driverlist-equiv x x-equiv)
          (segment-driverlist-equiv (cdr x)
                                    (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-segment-driver-fix-x-under-segment-driverlist-equiv

(defthm cons-of-segment-driver-fix-x-under-segment-driverlist-equiv
  (segment-driverlist-equiv (cons (segment-driver-fix x) y)
                            (cons x y)))

Theorem: cons-segment-driver-equiv-congruence-on-x-under-segment-driverlist-equiv

(defthm
 cons-segment-driver-equiv-congruence-on-x-under-segment-driverlist-equiv
 (implies (segment-driver-equiv x x-equiv)
          (segment-driverlist-equiv (cons x y)
                                    (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-segment-driverlist-fix-y-under-segment-driverlist-equiv

(defthm
    cons-of-segment-driverlist-fix-y-under-segment-driverlist-equiv
  (segment-driverlist-equiv (cons x (segment-driverlist-fix y))
                            (cons x y)))

Theorem: cons-segment-driverlist-equiv-congruence-on-y-under-segment-driverlist-equiv

(defthm
 cons-segment-driverlist-equiv-congruence-on-y-under-segment-driverlist-equiv
 (implies (segment-driverlist-equiv y y-equiv)
          (segment-driverlist-equiv (cons x y)
                                    (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-segment-driverlist-fix

(defthm consp-of-segment-driverlist-fix
  (equal (consp (segment-driverlist-fix x))
         (consp x)))

Theorem: segment-driverlist-fix-under-iff

(defthm segment-driverlist-fix-under-iff
  (iff (segment-driverlist-fix x)
       (consp x)))

Theorem: segment-driverlist-fix-of-cons

(defthm segment-driverlist-fix-of-cons
  (equal (segment-driverlist-fix (cons a x))
         (cons (segment-driver-fix a)
               (segment-driverlist-fix x))))

Theorem: len-of-segment-driverlist-fix

(defthm len-of-segment-driverlist-fix
  (equal (len (segment-driverlist-fix x))
         (len x)))

Theorem: segment-driverlist-fix-of-append

(defthm segment-driverlist-fix-of-append
  (equal (segment-driverlist-fix (append std::a std::b))
         (append (segment-driverlist-fix std::a)
                 (segment-driverlist-fix std::b))))

Theorem: segment-driverlist-fix-of-repeat

(defthm segment-driverlist-fix-of-repeat
  (equal (segment-driverlist-fix (repeat acl2::n x))
         (repeat acl2::n (segment-driver-fix x))))

Theorem: list-equiv-refines-segment-driverlist-equiv

(defthm list-equiv-refines-segment-driverlist-equiv
  (implies (list-equiv x y)
           (segment-driverlist-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-segment-driverlist-fix

(defthm nth-of-segment-driverlist-fix
  (equal (nth acl2::n (segment-driverlist-fix x))
         (if (< (nfix acl2::n) (len x))
             (segment-driver-fix (nth acl2::n x))
           nil)))

Theorem: segment-driverlist-equiv-implies-segment-driverlist-equiv-append-1

(defthm
 segment-driverlist-equiv-implies-segment-driverlist-equiv-append-1
 (implies (segment-driverlist-equiv x fty::x-equiv)
          (segment-driverlist-equiv (append x y)
                                    (append fty::x-equiv y)))
 :rule-classes (:congruence))

Theorem: segment-driverlist-equiv-implies-segment-driverlist-equiv-append-2

(defthm
 segment-driverlist-equiv-implies-segment-driverlist-equiv-append-2
 (implies (segment-driverlist-equiv y fty::y-equiv)
          (segment-driverlist-equiv (append x y)
                                    (append x fty::y-equiv)))
 :rule-classes (:congruence))

Theorem: segment-driverlist-equiv-implies-segment-driverlist-equiv-nthcdr-2

(defthm
 segment-driverlist-equiv-implies-segment-driverlist-equiv-nthcdr-2
 (implies (segment-driverlist-equiv acl2::l l-equiv)
          (segment-driverlist-equiv (nthcdr acl2::n acl2::l)
                                    (nthcdr acl2::n l-equiv)))
 :rule-classes (:congruence))

Theorem: segment-driverlist-equiv-implies-segment-driverlist-equiv-take-2

(defthm
   segment-driverlist-equiv-implies-segment-driverlist-equiv-take-2
  (implies (segment-driverlist-equiv acl2::l l-equiv)
           (segment-driverlist-equiv (take acl2::n acl2::l)
                                     (take acl2::n l-equiv)))
  :rule-classes (:congruence))