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• Maybe-nat-list

Maybe-nat-list-fix

(maybe-nat-list-fix x) is a usual fty list fixing function.

Signature

(maybe-nat-list-fix x) → fty::newx

Arguments

x — Guard (maybe-nat-list-p x).

Returns

fty::newx — Type (maybe-nat-list-p fty::newx).

In the logic, we apply maybe-natp-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: maybe-nat-list-fix$inline

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

Theorem: maybe-nat-list-p-of-maybe-nat-list-fix

(defthm maybe-nat-list-p-of-maybe-nat-list-fix
        (b* ((fty::newx (maybe-nat-list-fix$inline x)))
            (maybe-nat-list-p fty::newx))
        :rule-classes :rewrite)

Theorem: maybe-nat-list-fix-when-maybe-nat-list-p

(defthm maybe-nat-list-fix-when-maybe-nat-list-p
        (implies (maybe-nat-list-p x)
                 (equal (maybe-nat-list-fix x) x)))

Function: maybe-nat-list-equiv$inline

(defun maybe-nat-list-equiv$inline
       (acl2::x acl2::y)
       (declare (xargs :guard (and (maybe-nat-list-p acl2::x)
                                   (maybe-nat-list-p acl2::y))))
       (equal (maybe-nat-list-fix acl2::x)
              (maybe-nat-list-fix acl2::y)))

Theorem: maybe-nat-list-equiv-is-an-equivalence

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

Theorem: maybe-nat-list-equiv-implies-equal-maybe-nat-list-fix-1

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

Theorem: maybe-nat-list-fix-under-maybe-nat-list-equiv

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

Theorem: equal-of-maybe-nat-list-fix-1-forward-to-maybe-nat-list-equiv

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

Theorem: equal-of-maybe-nat-list-fix-2-forward-to-maybe-nat-list-equiv

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

Theorem: maybe-nat-list-equiv-of-maybe-nat-list-fix-1-forward

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

Theorem: maybe-nat-list-equiv-of-maybe-nat-list-fix-2-forward

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

Theorem: car-of-maybe-nat-list-fix-x-under-maybe-nat-equiv

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

Theorem: car-maybe-nat-list-equiv-congruence-on-x-under-maybe-nat-equiv

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

Theorem: cdr-of-maybe-nat-list-fix-x-under-maybe-nat-list-equiv

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

Theorem: cdr-maybe-nat-list-equiv-congruence-on-x-under-maybe-nat-list-equiv

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

Theorem: cons-of-maybe-natp-fix-x-under-maybe-nat-list-equiv

(defthm
     cons-of-maybe-natp-fix-x-under-maybe-nat-list-equiv
     (maybe-nat-list-equiv (cons (maybe-natp-fix acl2::x) acl2::y)
                           (cons acl2::x acl2::y)))

Theorem: cons-maybe-nat-equiv-congruence-on-x-under-maybe-nat-list-equiv

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

Theorem: cons-of-maybe-nat-list-fix-y-under-maybe-nat-list-equiv

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

Theorem: cons-maybe-nat-list-equiv-congruence-on-y-under-maybe-nat-list-equiv

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

Theorem: consp-of-maybe-nat-list-fix

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

Theorem: maybe-nat-list-fix-of-cons

(defthm maybe-nat-list-fix-of-cons
        (equal (maybe-nat-list-fix (cons a x))
               (cons (maybe-natp-fix a)
                     (maybe-nat-list-fix x))))

Theorem: len-of-maybe-nat-list-fix

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

Theorem: maybe-nat-list-fix-of-append

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

Theorem: maybe-nat-list-fix-of-repeat

(defthm maybe-nat-list-fix-of-repeat
        (equal (maybe-nat-list-fix (repeat acl2::n acl2::x))
               (repeat acl2::n (maybe-natp-fix acl2::x))))

Theorem: nth-of-maybe-nat-list-fix

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

Theorem: maybe-nat-list-equiv-implies-maybe-nat-list-equiv-append-1

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

Theorem: maybe-nat-list-equiv-implies-maybe-nat-list-equiv-append-2

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

Theorem: maybe-nat-list-equiv-implies-maybe-nat-list-equiv-nthcdr-2

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

Theorem: maybe-nat-list-equiv-implies-maybe-nat-list-equiv-take-2

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