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• Vl-defaultskew-item-list

Vl-defaultskew-item-list-fix

(vl-defaultskew-item-list-fix x) is a usual fty list fixing function.

Signature

(vl-defaultskew-item-list-fix x) → fty::newx

Arguments

x — Guard (vl-defaultskew-item-list-p x).

Returns

fty::newx — Type (vl-defaultskew-item-list-p fty::newx).

In the logic, we apply vl-defaultskew-item-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: vl-defaultskew-item-list-fix$inline

(defun vl-defaultskew-item-list-fix$inline (x)
  (declare (xargs :guard (vl-defaultskew-item-list-p x)))
  (let ((__function__ 'vl-defaultskew-item-list-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             x
           (cons (vl-defaultskew-item-fix (car x))
                 (vl-defaultskew-item-list-fix (cdr x))))
         :exec x)))

Theorem: vl-defaultskew-item-list-p-of-vl-defaultskew-item-list-fix

(defthm vl-defaultskew-item-list-p-of-vl-defaultskew-item-list-fix
  (b* ((fty::newx (vl-defaultskew-item-list-fix$inline x)))
    (vl-defaultskew-item-list-p fty::newx))
  :rule-classes :rewrite)

Theorem: vl-defaultskew-item-list-fix-when-vl-defaultskew-item-list-p

(defthm vl-defaultskew-item-list-fix-when-vl-defaultskew-item-list-p
  (implies (vl-defaultskew-item-list-p x)
           (equal (vl-defaultskew-item-list-fix x)
                  x)))

Function: vl-defaultskew-item-list-equiv$inline

(defun vl-defaultskew-item-list-equiv$inline (acl2::x acl2::y)
 (declare (xargs :guard (and (vl-defaultskew-item-list-p acl2::x)
                             (vl-defaultskew-item-list-p acl2::y))))
 (equal (vl-defaultskew-item-list-fix acl2::x)
        (vl-defaultskew-item-list-fix acl2::y)))

Theorem: vl-defaultskew-item-list-equiv-is-an-equivalence

(defthm vl-defaultskew-item-list-equiv-is-an-equivalence
  (and (booleanp (vl-defaultskew-item-list-equiv x y))
       (vl-defaultskew-item-list-equiv x x)
       (implies (vl-defaultskew-item-list-equiv x y)
                (vl-defaultskew-item-list-equiv y x))
       (implies (and (vl-defaultskew-item-list-equiv x y)
                     (vl-defaultskew-item-list-equiv y z))
                (vl-defaultskew-item-list-equiv x z)))
  :rule-classes (:equivalence))

Theorem: vl-defaultskew-item-list-equiv-implies-equal-vl-defaultskew-item-list-fix-1

(defthm
 vl-defaultskew-item-list-equiv-implies-equal-vl-defaultskew-item-list-fix-1
 (implies (vl-defaultskew-item-list-equiv acl2::x x-equiv)
          (equal (vl-defaultskew-item-list-fix acl2::x)
                 (vl-defaultskew-item-list-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: vl-defaultskew-item-list-fix-under-vl-defaultskew-item-list-equiv

(defthm
  vl-defaultskew-item-list-fix-under-vl-defaultskew-item-list-equiv
  (vl-defaultskew-item-list-equiv
       (vl-defaultskew-item-list-fix acl2::x)
       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-vl-defaultskew-item-list-fix-1-forward-to-vl-defaultskew-item-list-equiv

(defthm
 equal-of-vl-defaultskew-item-list-fix-1-forward-to-vl-defaultskew-item-list-equiv
 (implies (equal (vl-defaultskew-item-list-fix acl2::x)
                 acl2::y)
          (vl-defaultskew-item-list-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-vl-defaultskew-item-list-fix-2-forward-to-vl-defaultskew-item-list-equiv

(defthm
 equal-of-vl-defaultskew-item-list-fix-2-forward-to-vl-defaultskew-item-list-equiv
 (implies (equal acl2::x
                 (vl-defaultskew-item-list-fix acl2::y))
          (vl-defaultskew-item-list-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: vl-defaultskew-item-list-equiv-of-vl-defaultskew-item-list-fix-1-forward

(defthm
 vl-defaultskew-item-list-equiv-of-vl-defaultskew-item-list-fix-1-forward
 (implies (vl-defaultskew-item-list-equiv
               (vl-defaultskew-item-list-fix acl2::x)
               acl2::y)
          (vl-defaultskew-item-list-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: vl-defaultskew-item-list-equiv-of-vl-defaultskew-item-list-fix-2-forward

(defthm
 vl-defaultskew-item-list-equiv-of-vl-defaultskew-item-list-fix-2-forward
 (implies (vl-defaultskew-item-list-equiv
               acl2::x
               (vl-defaultskew-item-list-fix acl2::y))
          (vl-defaultskew-item-list-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: car-of-vl-defaultskew-item-list-fix-x-under-vl-defaultskew-item-equiv

(defthm
 car-of-vl-defaultskew-item-list-fix-x-under-vl-defaultskew-item-equiv
 (vl-defaultskew-item-equiv
      (car (vl-defaultskew-item-list-fix acl2::x))
      (car acl2::x)))

Theorem: car-vl-defaultskew-item-list-equiv-congruence-on-x-under-vl-defaultskew-item-equiv

(defthm
 car-vl-defaultskew-item-list-equiv-congruence-on-x-under-vl-defaultskew-item-equiv
 (implies (vl-defaultskew-item-list-equiv acl2::x x-equiv)
          (vl-defaultskew-item-equiv (car acl2::x)
                                     (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-vl-defaultskew-item-list-fix-x-under-vl-defaultskew-item-list-equiv

(defthm
 cdr-of-vl-defaultskew-item-list-fix-x-under-vl-defaultskew-item-list-equiv
 (vl-defaultskew-item-list-equiv
      (cdr (vl-defaultskew-item-list-fix acl2::x))
      (cdr acl2::x)))

Theorem: cdr-vl-defaultskew-item-list-equiv-congruence-on-x-under-vl-defaultskew-item-list-equiv

(defthm
 cdr-vl-defaultskew-item-list-equiv-congruence-on-x-under-vl-defaultskew-item-list-equiv
 (implies (vl-defaultskew-item-list-equiv acl2::x x-equiv)
          (vl-defaultskew-item-list-equiv (cdr acl2::x)
                                          (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-vl-defaultskew-item-fix-x-under-vl-defaultskew-item-list-equiv

(defthm
 cons-of-vl-defaultskew-item-fix-x-under-vl-defaultskew-item-list-equiv
 (vl-defaultskew-item-list-equiv
      (cons (vl-defaultskew-item-fix acl2::x)
            acl2::y)
      (cons acl2::x acl2::y)))

Theorem: cons-vl-defaultskew-item-equiv-congruence-on-x-under-vl-defaultskew-item-list-equiv

(defthm
 cons-vl-defaultskew-item-equiv-congruence-on-x-under-vl-defaultskew-item-list-equiv
 (implies (vl-defaultskew-item-equiv acl2::x x-equiv)
          (vl-defaultskew-item-list-equiv (cons acl2::x acl2::y)
                                          (cons x-equiv acl2::y)))
 :rule-classes :congruence)

Theorem: cons-of-vl-defaultskew-item-list-fix-y-under-vl-defaultskew-item-list-equiv

(defthm
 cons-of-vl-defaultskew-item-list-fix-y-under-vl-defaultskew-item-list-equiv
 (vl-defaultskew-item-list-equiv
      (cons acl2::x
            (vl-defaultskew-item-list-fix acl2::y))
      (cons acl2::x acl2::y)))

Theorem: cons-vl-defaultskew-item-list-equiv-congruence-on-y-under-vl-defaultskew-item-list-equiv

(defthm
 cons-vl-defaultskew-item-list-equiv-congruence-on-y-under-vl-defaultskew-item-list-equiv
 (implies (vl-defaultskew-item-list-equiv acl2::y y-equiv)
          (vl-defaultskew-item-list-equiv (cons acl2::x acl2::y)
                                          (cons acl2::x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-vl-defaultskew-item-list-fix

(defthm consp-of-vl-defaultskew-item-list-fix
  (equal (consp (vl-defaultskew-item-list-fix acl2::x))
         (consp acl2::x)))

Theorem: vl-defaultskew-item-list-fix-of-cons

(defthm vl-defaultskew-item-list-fix-of-cons
  (equal (vl-defaultskew-item-list-fix (cons a x))
         (cons (vl-defaultskew-item-fix a)
               (vl-defaultskew-item-list-fix x))))

Theorem: len-of-vl-defaultskew-item-list-fix

(defthm len-of-vl-defaultskew-item-list-fix
  (equal (len (vl-defaultskew-item-list-fix acl2::x))
         (len acl2::x)))

Theorem: vl-defaultskew-item-list-fix-of-append

(defthm vl-defaultskew-item-list-fix-of-append
  (equal (vl-defaultskew-item-list-fix (append std::a std::b))
         (append (vl-defaultskew-item-list-fix std::a)
                 (vl-defaultskew-item-list-fix std::b))))

Theorem: vl-defaultskew-item-list-fix-of-repeat

(defthm vl-defaultskew-item-list-fix-of-repeat
  (equal (vl-defaultskew-item-list-fix (repeat acl2::n acl2::x))
         (repeat acl2::n
                 (vl-defaultskew-item-fix acl2::x))))

Theorem: nth-of-vl-defaultskew-item-list-fix

(defthm nth-of-vl-defaultskew-item-list-fix
  (equal (nth acl2::n
              (vl-defaultskew-item-list-fix acl2::x))
         (if (< (nfix acl2::n) (len acl2::x))
             (vl-defaultskew-item-fix (nth acl2::n acl2::x))
           nil)))

Theorem: vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-append-1

(defthm
 vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-append-1
 (implies
     (vl-defaultskew-item-list-equiv acl2::x fty::x-equiv)
     (vl-defaultskew-item-list-equiv (append acl2::x acl2::y)
                                     (append fty::x-equiv acl2::y)))
 :rule-classes (:congruence))

Theorem: vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-append-2

(defthm
 vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-append-2
 (implies
     (vl-defaultskew-item-list-equiv acl2::y fty::y-equiv)
     (vl-defaultskew-item-list-equiv (append acl2::x acl2::y)
                                     (append acl2::x fty::y-equiv)))
 :rule-classes (:congruence))

Theorem: vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-nthcdr-2

(defthm
 vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-nthcdr-2
 (implies (vl-defaultskew-item-list-equiv acl2::l l-equiv)
          (vl-defaultskew-item-list-equiv (nthcdr acl2::n acl2::l)
                                          (nthcdr acl2::n l-equiv)))
 :rule-classes (:congruence))

Theorem: vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-take-2

(defthm
 vl-defaultskew-item-list-equiv-implies-vl-defaultskew-item-list-equiv-take-2
 (implies (vl-defaultskew-item-list-equiv acl2::l l-equiv)
          (vl-defaultskew-item-list-equiv (take acl2::n acl2::l)
                                          (take acl2::n l-equiv)))
 :rule-classes (:congruence))