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Word-list

Word-list-fix

(word-list-fix x) is a usual ACL2::fty list fixing function.

Signature
(word-list-fix x) → fty::newx
Arguments: x — Guard (word-listp x).
Returns: fty::newx — Type (word-listp fty::newx).

In the logic, we apply word-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: word-list-fix$inline

(defun word-list-fix$inline (x)
  (declare (xargs :guard (word-listp x)))
  (let ((acl2::__function__ 'word-list-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (word-fix (car x))
                 (word-list-fix (cdr x))))
         :exec x)))

Theorem: word-listp-of-word-list-fix

(defthm word-listp-of-word-list-fix
  (b* ((fty::newx (word-list-fix$inline x)))
    (word-listp fty::newx))
  :rule-classes :rewrite)

Theorem: word-list-fix-when-word-listp

(defthm word-list-fix-when-word-listp
  (implies (word-listp x)
           (equal (word-list-fix x) x)))

Function: word-list-equiv$inline

(defun word-list-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (word-listp acl2::x)
                              (word-listp acl2::y))))
  (equal (word-list-fix acl2::x)
         (word-list-fix acl2::y)))

Theorem: word-list-equiv-is-an-equivalence

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

Theorem: word-list-equiv-implies-equal-word-list-fix-1

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

Theorem: word-list-fix-under-word-list-equiv

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

Theorem: equal-of-word-list-fix-1-forward-to-word-list-equiv

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

Theorem: equal-of-word-list-fix-2-forward-to-word-list-equiv

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

Theorem: word-list-equiv-of-word-list-fix-1-forward

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

Theorem: word-list-equiv-of-word-list-fix-2-forward

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

Theorem: car-of-word-list-fix-x-under-word-equiv

(defthm car-of-word-list-fix-x-under-word-equiv
  (word-equiv (car (word-list-fix acl2::x))
              (car acl2::x)))

Theorem: car-word-list-equiv-congruence-on-x-under-word-equiv

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

Theorem: cdr-of-word-list-fix-x-under-word-list-equiv

(defthm cdr-of-word-list-fix-x-under-word-list-equiv
  (word-list-equiv (cdr (word-list-fix acl2::x))
                   (cdr acl2::x)))

Theorem: cdr-word-list-equiv-congruence-on-x-under-word-list-equiv

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

Theorem: cons-of-word-fix-x-under-word-list-equiv

(defthm cons-of-word-fix-x-under-word-list-equiv
  (word-list-equiv (cons (word-fix acl2::x) acl2::y)
                   (cons acl2::x acl2::y)))

Theorem: cons-word-equiv-congruence-on-x-under-word-list-equiv

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

Theorem: cons-of-word-list-fix-y-under-word-list-equiv

(defthm cons-of-word-list-fix-y-under-word-list-equiv
  (word-list-equiv (cons acl2::x (word-list-fix acl2::y))
                   (cons acl2::x acl2::y)))

Theorem: cons-word-list-equiv-congruence-on-y-under-word-list-equiv

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

Theorem: consp-of-word-list-fix

(defthm consp-of-word-list-fix
  (equal (consp (word-list-fix acl2::x))
         (consp acl2::x)))

Theorem: word-list-fix-under-iff

(defthm word-list-fix-under-iff
  (iff (word-list-fix acl2::x)
       (consp acl2::x)))

Theorem: word-list-fix-of-cons

(defthm word-list-fix-of-cons
  (equal (word-list-fix (cons a x))
         (cons (word-fix a) (word-list-fix x))))

Theorem: len-of-word-list-fix

(defthm len-of-word-list-fix
  (equal (len (word-list-fix acl2::x))
         (len acl2::x)))

Theorem: word-list-fix-of-append

(defthm word-list-fix-of-append
  (equal (word-list-fix (append std::a std::b))
         (append (word-list-fix std::a)
                 (word-list-fix std::b))))

Theorem: word-list-fix-of-repeat

(defthm word-list-fix-of-repeat
  (equal (word-list-fix (acl2::repeat acl2::n acl2::x))
         (acl2::repeat acl2::n (word-fix acl2::x))))

Theorem: list-equiv-refines-word-list-equiv

(defthm list-equiv-refines-word-list-equiv
  (implies (acl2::list-equiv acl2::x acl2::y)
           (word-list-equiv acl2::x acl2::y))
  :rule-classes :refinement)

Theorem: nth-of-word-list-fix

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

Theorem: word-list-equiv-implies-word-list-equiv-append-1

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

Theorem: word-list-equiv-implies-word-list-equiv-append-2

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

Theorem: word-list-equiv-implies-word-list-equiv-nthcdr-2

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

Theorem: word-list-equiv-implies-word-list-equiv-take-2

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