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Dec-digit-char-list

Dec-digit-char-list-fix

(dec-digit-char-list-fix x) is a usual ACL2::fty list fixing function.

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

In the logic, we apply dec-digit-char-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: dec-digit-char-list-fix$inline

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

Theorem: dec-digit-char-listp-of-dec-digit-char-list-fix

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

Theorem: dec-digit-char-list-fix-when-dec-digit-char-listp

(defthm dec-digit-char-list-fix-when-dec-digit-char-listp
  (implies (dec-digit-char-listp x)
           (equal (dec-digit-char-list-fix x) x)))

Function: dec-digit-char-list-equiv$inline

(defun dec-digit-char-list-equiv$inline (x y)
  (declare (xargs :guard (and (dec-digit-char-listp x)
                              (dec-digit-char-listp y))))
  (equal (dec-digit-char-list-fix x)
         (dec-digit-char-list-fix y)))

Theorem: dec-digit-char-list-equiv-is-an-equivalence

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

Theorem: dec-digit-char-list-equiv-implies-equal-dec-digit-char-list-fix-1

(defthm
  dec-digit-char-list-equiv-implies-equal-dec-digit-char-list-fix-1
  (implies (dec-digit-char-list-equiv x x-equiv)
           (equal (dec-digit-char-list-fix x)
                  (dec-digit-char-list-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: dec-digit-char-list-fix-under-dec-digit-char-list-equiv

(defthm dec-digit-char-list-fix-under-dec-digit-char-list-equiv
  (dec-digit-char-list-equiv (dec-digit-char-list-fix x)
                             x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-dec-digit-char-list-fix-1-forward-to-dec-digit-char-list-equiv

(defthm
 equal-of-dec-digit-char-list-fix-1-forward-to-dec-digit-char-list-equiv
 (implies (equal (dec-digit-char-list-fix x) y)
          (dec-digit-char-list-equiv x y))
 :rule-classes :forward-chaining)

Theorem: equal-of-dec-digit-char-list-fix-2-forward-to-dec-digit-char-list-equiv

(defthm
 equal-of-dec-digit-char-list-fix-2-forward-to-dec-digit-char-list-equiv
 (implies (equal x (dec-digit-char-list-fix y))
          (dec-digit-char-list-equiv x y))
 :rule-classes :forward-chaining)

Theorem: dec-digit-char-list-equiv-of-dec-digit-char-list-fix-1-forward

(defthm
     dec-digit-char-list-equiv-of-dec-digit-char-list-fix-1-forward
  (implies (dec-digit-char-list-equiv (dec-digit-char-list-fix x)
                                      y)
           (dec-digit-char-list-equiv x y))
  :rule-classes :forward-chaining)

Theorem: dec-digit-char-list-equiv-of-dec-digit-char-list-fix-2-forward

(defthm
     dec-digit-char-list-equiv-of-dec-digit-char-list-fix-2-forward
  (implies (dec-digit-char-list-equiv x (dec-digit-char-list-fix y))
           (dec-digit-char-list-equiv x y))
  :rule-classes :forward-chaining)

Theorem: car-of-dec-digit-char-list-fix-x-under-dec-digit-char-equiv

(defthm car-of-dec-digit-char-list-fix-x-under-dec-digit-char-equiv
  (dec-digit-char-equiv (car (dec-digit-char-list-fix x))
                        (car x)))

Theorem: car-dec-digit-char-list-equiv-congruence-on-x-under-dec-digit-char-equiv

(defthm
 car-dec-digit-char-list-equiv-congruence-on-x-under-dec-digit-char-equiv
 (implies (dec-digit-char-list-equiv x x-equiv)
          (dec-digit-char-equiv (car x)
                                (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-dec-digit-char-list-fix-x-under-dec-digit-char-list-equiv

(defthm
   cdr-of-dec-digit-char-list-fix-x-under-dec-digit-char-list-equiv
  (dec-digit-char-list-equiv (cdr (dec-digit-char-list-fix x))
                             (cdr x)))

Theorem: cdr-dec-digit-char-list-equiv-congruence-on-x-under-dec-digit-char-list-equiv

(defthm
 cdr-dec-digit-char-list-equiv-congruence-on-x-under-dec-digit-char-list-equiv
 (implies (dec-digit-char-list-equiv x x-equiv)
          (dec-digit-char-list-equiv (cdr x)
                                     (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-dec-digit-char-fix-x-under-dec-digit-char-list-equiv

(defthm cons-of-dec-digit-char-fix-x-under-dec-digit-char-list-equiv
  (dec-digit-char-list-equiv (cons (dec-digit-char-fix x) y)
                             (cons x y)))

Theorem: cons-dec-digit-char-equiv-congruence-on-x-under-dec-digit-char-list-equiv

(defthm
 cons-dec-digit-char-equiv-congruence-on-x-under-dec-digit-char-list-equiv
 (implies (dec-digit-char-equiv x x-equiv)
          (dec-digit-char-list-equiv (cons x y)
                                     (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-dec-digit-char-list-fix-y-under-dec-digit-char-list-equiv

(defthm
  cons-of-dec-digit-char-list-fix-y-under-dec-digit-char-list-equiv
  (dec-digit-char-list-equiv (cons x (dec-digit-char-list-fix y))
                             (cons x y)))

Theorem: cons-dec-digit-char-list-equiv-congruence-on-y-under-dec-digit-char-list-equiv

(defthm
 cons-dec-digit-char-list-equiv-congruence-on-y-under-dec-digit-char-list-equiv
 (implies (dec-digit-char-list-equiv y y-equiv)
          (dec-digit-char-list-equiv (cons x y)
                                     (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-dec-digit-char-list-fix

(defthm consp-of-dec-digit-char-list-fix
  (equal (consp (dec-digit-char-list-fix x))
         (consp x)))

Theorem: dec-digit-char-list-fix-under-iff

(defthm dec-digit-char-list-fix-under-iff
  (iff (dec-digit-char-list-fix x)
       (consp x)))

Theorem: dec-digit-char-list-fix-of-cons

(defthm dec-digit-char-list-fix-of-cons
  (equal (dec-digit-char-list-fix (cons a x))
         (cons (dec-digit-char-fix a)
               (dec-digit-char-list-fix x))))

Theorem: len-of-dec-digit-char-list-fix

(defthm len-of-dec-digit-char-list-fix
  (equal (len (dec-digit-char-list-fix x))
         (len x)))

Theorem: dec-digit-char-list-fix-of-append

(defthm dec-digit-char-list-fix-of-append
  (equal (dec-digit-char-list-fix (append std::a std::b))
         (append (dec-digit-char-list-fix std::a)
                 (dec-digit-char-list-fix std::b))))

Theorem: dec-digit-char-list-fix-of-repeat

(defthm dec-digit-char-list-fix-of-repeat
  (equal (dec-digit-char-list-fix (repeat n x))
         (repeat n (dec-digit-char-fix x))))

Theorem: list-equiv-refines-dec-digit-char-list-equiv

(defthm list-equiv-refines-dec-digit-char-list-equiv
  (implies (list-equiv x y)
           (dec-digit-char-list-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-dec-digit-char-list-fix

(defthm nth-of-dec-digit-char-list-fix
  (equal (nth n (dec-digit-char-list-fix x))
         (if (< (nfix n) (len x))
             (dec-digit-char-fix (nth n x))
           nil)))

Theorem: dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-append-1

(defthm
 dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-append-1
 (implies (dec-digit-char-list-equiv x fty::x-equiv)
          (dec-digit-char-list-equiv (append x y)
                                     (append fty::x-equiv y)))
 :rule-classes (:congruence))

Theorem: dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-append-2

(defthm
 dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-append-2
 (implies (dec-digit-char-list-equiv y fty::y-equiv)
          (dec-digit-char-list-equiv (append x y)
                                     (append x fty::y-equiv)))
 :rule-classes (:congruence))

Theorem: dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-nthcdr-2

(defthm
 dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-nthcdr-2
 (implies (dec-digit-char-list-equiv l l-equiv)
          (dec-digit-char-list-equiv (nthcdr n l)
                                     (nthcdr n l-equiv)))
 :rule-classes (:congruence))

Theorem: dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-take-2

(defthm
 dec-digit-char-list-equiv-implies-dec-digit-char-list-equiv-take-2
 (implies (dec-digit-char-list-equiv l l-equiv)
          (dec-digit-char-list-equiv (take n l)
                                     (take n l-equiv)))
 :rule-classes (:congruence))