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    • Schar-list

    Schar-list-fix

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

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

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

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

    Theorem: schar-listp-of-schar-list-fix

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

    Theorem: schar-list-fix-when-schar-listp

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

    Function: schar-list-equiv$inline

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

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

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

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

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

    Theorem: schar-list-fix-under-schar-list-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Theorem: consp-of-schar-list-fix

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

    Theorem: schar-list-fix-under-iff

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

    Theorem: schar-list-fix-of-cons

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

    Theorem: len-of-schar-list-fix

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

    Theorem: schar-list-fix-of-append

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

    Theorem: schar-list-fix-of-repeat

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

    Theorem: list-equiv-refines-schar-list-equiv

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

    Theorem: nth-of-schar-list-fix

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

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

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

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

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

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

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

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

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