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Sllong-array

Sllong-array-fix

Fixing function for sllong-array structures.

Signature
(sllong-array-fix x) → new-x
Arguments: x — Guard (sllong-arrayp x).
Returns: new-x — Type (sllong-arrayp new-x).

Definitions and Theorems

Function: sllong-array-fix$inline

(defun sllong-array-fix$inline (x)
  (declare (xargs :guard (sllong-arrayp x)))
  (let ((__function__ 'sllong-array-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((elemtype (type-fix (std::da-nth 0 (cdr x))))
              (elements (sllong-list-fix (std::da-nth 1 (cdr x)))))
           (let ((elemtype (if (type-case elemtype :sllong)
                               elemtype
                             (type-sllong)))
                 (elements (if (consp elements)
                               elements
                             (list (sllong-from-integer 0)))))
             (cons :array (list elemtype elements))))
         :exec x)))

Theorem: sllong-arrayp-of-sllong-array-fix

(defthm sllong-arrayp-of-sllong-array-fix
  (b* ((new-x (sllong-array-fix$inline x)))
    (sllong-arrayp new-x))
  :rule-classes :rewrite)

Theorem: sllong-array-fix-when-sllong-arrayp

(defthm sllong-array-fix-when-sllong-arrayp
  (implies (sllong-arrayp x)
           (equal (sllong-array-fix x) x)))

Function: sllong-array-equiv$inline

(defun sllong-array-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sllong-arrayp acl2::x)
                              (sllong-arrayp acl2::y))))
  (equal (sllong-array-fix acl2::x)
         (sllong-array-fix acl2::y)))

Theorem: sllong-array-equiv-is-an-equivalence

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

Theorem: sllong-array-equiv-implies-equal-sllong-array-fix-1

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

Theorem: sllong-array-fix-under-sllong-array-equiv

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

Theorem: equal-of-sllong-array-fix-1-forward-to-sllong-array-equiv

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

Theorem: equal-of-sllong-array-fix-2-forward-to-sllong-array-equiv

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

Theorem: sllong-array-equiv-of-sllong-array-fix-1-forward

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

Theorem: sllong-array-equiv-of-sllong-array-fix-2-forward

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