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Integer-range-listp

Integer-range-list-fix

Fixing function for integer-range-listp.

Signature
(integer-range-list-fix lower upper x) → fixed-x
Arguments: lower — Guard (integerp lower).; upper — Guard (integerp upper).; x — Guard (integer-range-listp (ifix lower) (ifix upper) x).
Returns: fixed-x — Type (integer-range-listp lower upper fixed-x), given (and (integerp lower) (integerp upper) (< lower upper)).

This lifts integer-range-fix to lists. See that function for more information, in particular about the fixing of lower and upper to integers and the treatment of empty ranges.

Definitions and Theorems

Function: integer-range-list-fix

(defun integer-range-list-fix (lower upper x)
 (declare (xargs :guard (and (integerp lower)
                             (integerp upper)
                             (integer-range-listp (ifix lower)
                                                  (ifix upper)
                                                  x))))
 (let ((__function__ 'integer-range-list-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (cond ((atom x) nil)
            (t (cons (integer-range-fix lower upper (car x))
                     (integer-range-list-fix lower upper (cdr x)))))
      :exec x)))

Theorem: return-type-of-integer-range-list-fix

(defthm return-type-of-integer-range-list-fix
  (implies (and (integerp lower)
                (integerp upper)
                (< lower upper))
           (b* ((fixed-x (integer-range-list-fix lower upper x)))
             (integer-range-listp lower upper fixed-x)))
  :rule-classes :rewrite)

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

(defthm integer-listp-of-integer-range-list-fix
  (b* ((fixed-x (integer-range-list-fix lower upper x)))
    (integer-listp fixed-x))
  :rule-classes :rewrite)

Theorem: integer-range-list-fix-when-integer-range-p

(defthm integer-range-list-fix-when-integer-range-p
  (implies (and (integer-range-listp (ifix lower)
                                     (ifix upper)
                                     x)
                (< (ifix lower) (ifix upper)))
           (equal (integer-range-list-fix lower upper x)
                  x)))

Theorem: integer-range-list-fix-of-nil

(defthm integer-range-list-fix-of-nil
  (equal (integer-range-list-fix lower upper nil)
         nil))

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

(defthm integer-range-list-fix-of-cons
  (equal (integer-range-list-fix lower upper (cons x y))
         (cons (integer-range-fix lower upper x)
               (integer-range-list-fix lower upper y))))

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

(defthm integer-range-list-fix-of-append
  (equal (integer-range-list-fix lower upper (append x y))
         (append (integer-range-list-fix lower upper x)
                 (integer-range-list-fix lower upper y))))

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

(defthm len-of-integer-range-list-fix
  (equal (len (integer-range-list-fix lower upper x))
         (len x)))

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

(defthm consp-of-integer-range-list-fix
  (equal (consp (integer-range-list-fix lower upper x))
         (consp x)))

Theorem: car-of-integer-range-list-fix

(defthm car-of-integer-range-list-fix
  (implies (consp x)
           (equal (car (integer-range-list-fix lower upper x))
                  (integer-range-fix lower upper (car x)))))

Theorem: cdr-of-integer-range-list-fix

(defthm cdr-of-integer-range-list-fix
  (implies (consp x)
           (equal (cdr (integer-range-list-fix lower upper x))
                  (integer-range-list-fix lower upper (cdr x)))))

Theorem: rev-of-integer-range-list-fix

(defthm rev-of-integer-range-list-fix
  (equal (rev (integer-range-list-fix lower upper x))
         (integer-range-list-fix lower upper (rev x))))

Theorem: integer-range-list-fix-of-ifix-lower

(defthm integer-range-list-fix-of-ifix-lower
  (equal (integer-range-list-fix (ifix lower)
                                 upper x)
         (integer-range-list-fix lower upper x)))

Theorem: integer-range-list-fix-int-equiv-congruence-on-lower

(defthm integer-range-list-fix-int-equiv-congruence-on-lower
  (implies (int-equiv lower lower-equiv)
           (equal (integer-range-list-fix lower upper x)
                  (integer-range-list-fix lower-equiv upper x)))
  :rule-classes :congruence)

Theorem: integer-range-list-fix-of-ifix-upper

(defthm integer-range-list-fix-of-ifix-upper
  (equal (integer-range-list-fix lower (ifix upper)
                                 x)
         (integer-range-list-fix lower upper x)))

Theorem: integer-range-list-fix-int-equiv-congruence-on-upper

(defthm integer-range-list-fix-int-equiv-congruence-on-upper
  (implies (int-equiv upper upper-equiv)
           (equal (integer-range-list-fix lower upper x)
                  (integer-range-list-fix lower upper-equiv x)))
  :rule-classes :congruence)