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• Sllong-integer-list

Sllong-integer-list-fix

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

Signature

(sllong-integer-list-fix x) → fty::newx

Arguments

x — Guard (sllong-integer-listp x).

Returns

fty::newx — Type (sllong-integer-listp fty::newx).

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

(defun sllong-integer-list-fix$inline (x)
  (declare (xargs :guard (sllong-integer-listp x)))
  (mbe :logic
       (if (atom x)
           nil
         (cons (sllong-integer-fix (car x))
               (sllong-integer-list-fix (cdr x))))
       :exec x))

Theorem: sllong-integer-listp-of-sllong-integer-list-fix

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

Theorem: sllong-integer-list-fix-when-sllong-integer-listp

(defthm sllong-integer-list-fix-when-sllong-integer-listp
  (implies (sllong-integer-listp x)
           (equal (sllong-integer-list-fix x) x)))

Function: sllong-integer-list-equiv$inline

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

Theorem: sllong-integer-list-equiv-is-an-equivalence

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

Theorem: sllong-integer-list-equiv-implies-equal-sllong-integer-list-fix-1

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

Theorem: sllong-integer-list-fix-under-sllong-integer-list-equiv

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

Theorem: equal-of-sllong-integer-list-fix-1-forward-to-sllong-integer-list-equiv

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

Theorem: equal-of-sllong-integer-list-fix-2-forward-to-sllong-integer-list-equiv

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

Theorem: sllong-integer-list-equiv-of-sllong-integer-list-fix-1-forward

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

Theorem: sllong-integer-list-equiv-of-sllong-integer-list-fix-2-forward

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

Theorem: car-of-sllong-integer-list-fix-x-under-sllong-integer-equiv

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

Theorem: car-sllong-integer-list-equiv-congruence-on-x-under-sllong-integer-equiv

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

Theorem: cdr-of-sllong-integer-list-fix-x-under-sllong-integer-list-equiv

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

Theorem: cdr-sllong-integer-list-equiv-congruence-on-x-under-sllong-integer-list-equiv

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

Theorem: cons-of-sllong-integer-fix-x-under-sllong-integer-list-equiv

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

Theorem: cons-sllong-integer-equiv-congruence-on-x-under-sllong-integer-list-equiv

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

Theorem: cons-of-sllong-integer-list-fix-y-under-sllong-integer-list-equiv

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

Theorem: cons-sllong-integer-list-equiv-congruence-on-y-under-sllong-integer-list-equiv

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

Theorem: consp-of-sllong-integer-list-fix

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

Theorem: sllong-integer-list-fix-under-iff

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

Theorem: sllong-integer-list-fix-of-cons

(defthm sllong-integer-list-fix-of-cons
  (equal (sllong-integer-list-fix (cons a x))
         (cons (sllong-integer-fix a)
               (sllong-integer-list-fix x))))

Theorem: len-of-sllong-integer-list-fix

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

Theorem: sllong-integer-list-fix-of-append

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

Theorem: sllong-integer-list-fix-of-repeat

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

Theorem: list-equiv-refines-sllong-integer-list-equiv

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

Theorem: nth-of-sllong-integer-list-fix

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

Theorem: sllong-integer-list-equiv-implies-sllong-integer-list-equiv-append-1

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

Theorem: sllong-integer-list-equiv-implies-sllong-integer-list-equiv-append-2

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

Theorem: sllong-integer-list-equiv-implies-sllong-integer-list-equiv-nthcdr-2

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

Theorem: sllong-integer-list-equiv-implies-sllong-integer-list-equiv-take-2

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

Theorem: sllong-integer-list-fix$inline-of-sllong-integer-list-fix-x

(defthm sllong-integer-list-fix$inline-of-sllong-integer-list-fix-x
 (equal (sllong-integer-list-fix$inline (sllong-integer-list-fix x))
        (sllong-integer-list-fix$inline x)))

Theorem: sllong-integer-list-fix$inline-sllong-integer-list-equiv-congruence-on-x

(defthm
 sllong-integer-list-fix$inline-sllong-integer-list-equiv-congruence-on-x
 (implies (sllong-integer-list-equiv x x-equiv)
          (equal (sllong-integer-list-fix$inline x)
                 (sllong-integer-list-fix$inline x-equiv)))
 :rule-classes :congruence)