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Sparseint$

Sparseint$-fix

Fixing function for sparseint$ structures.

Signature
(sparseint$-fix x) → new-x
Arguments: x — Guard (sparseint$-p x).
Returns: new-x — Type (sparseint$-p new-x).

Definitions and Theorems

Function: sparseint$-fix$inline

(defun sparseint$-fix$inline (x)
 (declare (xargs :guard (sparseint$-p x)))
 (let ((__function__ 'sparseint$-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (case (sparseint$-kind x)
      (:leaf (b* ((val (ifix x))) val))
      (:concat
           (b* ((width (pos-fix (logtail 2 (car x))))
                (lsbs-taller (acl2::bool-fix (logbitp 0 (car x))))
                (msbs-taller (acl2::bool-fix (logbitp 1 (car x))))
                (lsbs (sparseint$-fix (cadr x)))
                (msbs (sparseint$-fix (cddr x))))
             (cons (logcons (bool->bit lsbs-taller)
                            (logcons (bool->bit msbs-taller) width))
                   (cons lsbs msbs)))))
    :exec x)))

Theorem: sparseint$-p-of-sparseint$-fix

(defthm sparseint$-p-of-sparseint$-fix
  (b* ((new-x (sparseint$-fix$inline x)))
    (sparseint$-p new-x))
  :rule-classes :rewrite)

Theorem: sparseint$-fix-when-sparseint$-p

(defthm sparseint$-fix-when-sparseint$-p
  (implies (sparseint$-p x)
           (equal (sparseint$-fix x) x)))

Function: sparseint$-equiv$inline

(defun sparseint$-equiv$inline (x y)
  (declare (xargs :guard (and (sparseint$-p x)
                              (sparseint$-p y))))
  (equal (sparseint$-fix x)
         (sparseint$-fix y)))

Theorem: sparseint$-equiv-is-an-equivalence

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

Theorem: sparseint$-equiv-implies-equal-sparseint$-fix-1

(defthm sparseint$-equiv-implies-equal-sparseint$-fix-1
  (implies (sparseint$-equiv x x-equiv)
           (equal (sparseint$-fix x)
                  (sparseint$-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sparseint$-fix-under-sparseint$-equiv

(defthm sparseint$-fix-under-sparseint$-equiv
  (sparseint$-equiv (sparseint$-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sparseint$-fix-1-forward-to-sparseint$-equiv

(defthm equal-of-sparseint$-fix-1-forward-to-sparseint$-equiv
  (implies (equal (sparseint$-fix x) y)
           (sparseint$-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sparseint$-fix-2-forward-to-sparseint$-equiv

(defthm equal-of-sparseint$-fix-2-forward-to-sparseint$-equiv
  (implies (equal x (sparseint$-fix y))
           (sparseint$-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sparseint$-equiv-of-sparseint$-fix-1-forward

(defthm sparseint$-equiv-of-sparseint$-fix-1-forward
  (implies (sparseint$-equiv (sparseint$-fix x) y)
           (sparseint$-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sparseint$-equiv-of-sparseint$-fix-2-forward

(defthm sparseint$-equiv-of-sparseint$-fix-2-forward
  (implies (sparseint$-equiv x (sparseint$-fix y))
           (sparseint$-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sparseint$-kind$inline-of-sparseint$-fix-x

(defthm sparseint$-kind$inline-of-sparseint$-fix-x
  (equal (sparseint$-kind$inline (sparseint$-fix x))
         (sparseint$-kind$inline x)))

Theorem: sparseint$-kind$inline-sparseint$-equiv-congruence-on-x

(defthm sparseint$-kind$inline-sparseint$-equiv-congruence-on-x
  (implies (sparseint$-equiv x x-equiv)
           (equal (sparseint$-kind$inline x)
                  (sparseint$-kind$inline x-equiv)))
  :rule-classes :congruence)